chapter 5 resource masters©glencoe/mcgraw-hill iv glencoe algebra 2 teacher’s guide to using the...
TRANSCRIPT
Chapter 5Resource Masters
Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks.
Study Guide and Intervention Workbook 0-07-828029-XSkills Practice Workbook 0-07-828023-0Practice Workbook 0-07-828024-9
ANSWERS FOR WORKBOOKS The answers for Chapter 5 of these workbookscan be found in the back of this Chapter Resource Masters booklet.
Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teacher, and families without charge; and be used solely in conjunction with Glencoe’s Algebra 2. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.
Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027
ISBN: 0-07-828008-7 Algebra 2Chapter 5 Resource Masters
1 2 3 4 5 6 7 8 9 10 066 11 10 09 08 07 06 05 04 03 02
Glenc oe/ M cGr aw-H ill
© Glencoe/McGraw-Hill iii Glencoe Algebra 2
Contents
Vocabulary Builder . . . . . . . . . . . . . . . . vii
Lesson 5-1Study Guide and Intervention . . . . . . . . 239–240Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 241Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 242Reading to Learn Mathematics . . . . . . . . . . 243Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 244
Lesson 5-2Study Guide and Intervention . . . . . . . . 245–246Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 247Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 248Reading to Learn Mathematics . . . . . . . . . . 249Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 250
Lesson 5-3Study Guide and Intervention . . . . . . . . 251–252Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 253Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 254Reading to Learn Mathematics . . . . . . . . . . 255Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 256
Lesson 5-4Study Guide and Intervention . . . . . . . . 257–258Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 259Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 260Reading to Learn Mathematics . . . . . . . . . . 261Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 262
Lesson 5-5Study Guide and Intervention . . . . . . . . 263–264Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 265Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 266Reading to Learn Mathematics . . . . . . . . . . 267Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 268
Lesson 5-6Study Guide and Intervention . . . . . . . . 269–270Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 271Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 272Reading to Learn Mathematics . . . . . . . . . . 273Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 274
Lesson 5-7Study Guide and Intervention . . . . . . . . 275–276Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 277Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 278Reading to Learn Mathematics . . . . . . . . . . 279Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 280
Lesson 5-8Study Guide and Intervention . . . . . . . . 281–282Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 283Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 284Reading to Learn Mathematics . . . . . . . . . . 285Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 286
Lesson 5-9Study Guide and Intervention . . . . . . . . 287–288Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 289Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 290Reading to Learn Mathematics . . . . . . . . . . 291Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 292
Chapter 5 AssessmentChapter 5 Test, Form 1 . . . . . . . . . . . . 293–294Chapter 5 Test, Form 2A . . . . . . . . . . . 295–296Chapter 5 Test, Form 2B . . . . . . . . . . . 297–298Chapter 5 Test, Form 2C . . . . . . . . . . . 299–300Chapter 5 Test, Form 2D . . . . . . . . . . . 301–302Chapter 5 Test, Form 3 . . . . . . . . . . . . 303–304Chapter 5 Open-Ended Assessment . . . . . . 305Chapter 5 Vocabulary Test/Review . . . . . . . 306Chapter 5 Quizzes 1 & 2 . . . . . . . . . . . . . . . 307Chapter 5 Quizzes 3 & 4 . . . . . . . . . . . . . . . 308Chapter 5 Mid-Chapter Test . . . . . . . . . . . . . 309Chapter 5 Cumulative Review . . . . . . . . . . . 310Chapter 5 Standardized Test Practice . .311–312
Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1
ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A38
© Glencoe/McGraw-Hill iv Glencoe Algebra 2
Teacher’s Guide to Using theChapter 5 Resource Masters
The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 5 Resource Masters includes the core materials neededfor Chapter 5. These materials include worksheets, extensions, and assessment options.The answers for these pages appear at the back of this booklet.
All of the materials found in this booklet are included for viewing and printing in theAlgebra 2 TeacherWorks CD-ROM.
Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.
WHEN TO USE Give these pages tostudents before beginning Lesson 5-1.Encourage them to add these pages to theirAlgebra 2 Study Notebook. Remind them to add definitions and examples as theycomplete each lesson.
Study Guide and InterventionEach lesson in Algebra 2 addresses twoobjectives. There is one Study Guide andIntervention master for each objective.
WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.
Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.
WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.
Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.
WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.
Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.
WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.
Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.
WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.
© Glencoe/McGraw-Hill v Glencoe Algebra 2
Assessment OptionsThe assessment masters in the Chapter 5Resource Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.
Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions
and is intended for use with basic levelstudents.
• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.
• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.
• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.
All of the above tests include a free-response Bonus question.
• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.
• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.
Intermediate Assessment• Four free-response quizzes are included
to offer assessment at appropriateintervals in the chapter.
• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.
Continuing Assessment• The Cumulative Review provides
students an opportunity to reinforce andretain skills as they proceed throughtheir study of Algebra 2. It can also beused as a test. This master includes free-response questions.
• The Standardized Test Practice offerscontinuing review of algebra concepts invarious formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and quantitative-comparison questions. Bubble-in andgrid-in answer sections are provided onthe master.
Answers• Page A1 is an answer sheet for the
Standardized Test Practice questionsthat appear in the Student Edition onpages 282–283. This improves students’familiarity with the answer formats theymay encounter in test taking.
• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.
• Full-size answer keys are provided forthe assessment masters in this booklet.
Reading to Learn MathematicsVocabulary Builder
NAME ______________________________________________ DATE ____________ PERIOD _____
55
© Glencoe/McGraw-Hill vii Glencoe Algebra 2
Voca
bula
ry B
uild
erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 5.As you study the chapter, complete each term’s definition or description. Rememberto add the page number where you found the term. Add these pages to your AlgebraStudy Notebook to review vocabulary at the end of the chapter.
Vocabulary Term Found on Page Definition/Description/Example
binomial
coefficient
KOH·uh·FIH·shuhnt
complex conjugates
KAHN·jih·guht
complex number
degree
extraneous solution
ehk·STRAY·nee·uhs
FOIL method
imaginary unit
like radical expressions
like terms
(continued on the next page)
© Glencoe/McGraw-Hill viii Glencoe Algebra 2
Vocabulary Term Found on Page Definition/Description/Example
monomial
nth root
polynomial
power
principal root
pure imaginary number
radical equation
radical inequality
rationalizing the denominator
synthetic division
sihn·THEH·tihk
trinomial
Reading to Learn MathematicsVocabulary Builder (continued)
NAME ______________________________________________ DATE ____________ PERIOD _____
55
Study Guide and InterventionMonomials
NAME ______________________________________________ DATE ____________ PERIOD _____
5-15-1
© Glencoe/McGraw-Hill 239 Glencoe Algebra 2
Less
on
5-1
Monomials A monomial is a number, a variable, or the product of a number and one ormore variables. Constants are monomials that contain no variables.
Negative Exponent a!n " and " an for any real number a # 0 and any integer n.
When you simplify an expression, you rewrite it without parentheses or negativeexponents. The following properties are useful when simplifying expressions.
Product of Powers am $ an " am % n for any real number a and integers m and n.
Quotient of Powers " am ! n for any real number a # 0 and integers m and n.
For a, b real numbers and m, n integers:(am)n " amn
Properties of Powers(ab)m " ambm
! "n" , b # 0
! "!n" ! "n
or , a # 0, b # 0
Simplify. Assume that no variable equals 0.
bn&an
b&a
a&b
an&bn
a&b
am&an
1&a!n
1&an
ExampleExample
a. (3m4n!2)(!5mn)2
(3m4n!2)(!5mn)2 " 3m4n!2 $ 25m2n2
" 75m4m2n!2n2
" 75m4 % 2n!2 % 2
" 75m6
b.
"
" !m12 $ 4m4
" !4m16
!m12&&4m
14&
(!m4)3&(2m2)!2
(!m4)3""(2m2)!2
ExercisesExercises
Simplify. Assume that no variable equals 0.
1. c12 $ c!4 $ c6 c14 2. b6 3. (a4)5 a20
4. 5. ! "!16. ! "2
7. (!5a2b3)2(abc)2 5a6b8c2 8. m7 $ m8 m15 9.
10. 2 11. 4j(2j!2k2)(3j3k!7) 12. m23"2
2mn2(3m2n)2&&
12m3n424j2"
k523c4t2&22c4t2
2m2"n
8m3n2&4mn3
1&5
x2"y4
x2y&xy3
b"a5
a2b&a!3b2
y2"x6
x!2y&x4y!1
b8&b2
© Glencoe/McGraw-Hill 240 Glencoe Algebra 2
Scientific Notation
Scientific notation A number expressed in the form a ' 10n, where 1 ( a ) 10 and n is an integer
Express 46,000,000 in scientific notation.
46,000,000 " 4.6 ' 10,000,000 1 ( 4.6 ) 10" 4.6 ' 107 Write 10,000,000 as a power of ten.
Evaluate . Express the result in scientific notation.
" '
" 0.7 ' 106
" 7 ' 105
Express each number in scientific notation.
1. 24,300 2. 0.00099 3. 4,860,0002.43 # 104 9.9 # 10!4 4.86 # 106
4. 525,000,000 5. 0.0000038 6. 221,0005.25 # 108 3.8 # 10!6 2.21 # 105
7. 0.000000064 8. 16,750 9. 0.0003696.4 # 10!8 1.675 # 104 3.69 # 10!4
Evaluate. Express the result in scientific notation.
10. (3.6 ' 104)(5 ' 103) 11. (1.4 ' 10!8)(8 ' 1012) 12. (4.2 ' 10!3)(3 ' 10!2)1.8 # 108 1.12 # 105 1.26 # 10!4
13. 14. 15.
2.5 # 109 9 # 10!8 7.4 # 103
16. (3.2 ' 10!3)2 17. (4.5 ' 107)2 18. (6.8 ' 10!5)2
1.024 # 10!5 2.025 # 1015 4.624 # 10!9
19. ASTRONOMY Pluto is 3,674.5 million miles from the sun. Write this number inscientific notation. Source: New York Times Almanac 3.6745 # 109 miles
20. CHEMISTRY The boiling point of the metal tungsten is 10,220°F. Write thistemperature in scientific notation. Source: New York Times Almanac 1.022 # 104
21. BIOLOGY The human body contains 0.0004% iodine by weight. How many pounds ofiodine are there in a 120-pound teenager? Express your answer in scientific notation.Source: Universal Almanac 4.8 # 10!4 lb
4.81 ' 108&&6.5 ' 104
1.62 ' 10!2&&
1.8 ' 1059.5 ' 107&&3.8 ' 10!2
104&10!2
3.5&5
3.5 ' 104&&5 ' 10!2
3.5 ' 104""5 ' 10!2
Study Guide and Intervention (continued)
Monomials
NAME ______________________________________________ DATE ____________ PERIOD _____
5-15-1
Example 1Example 1
Example 2Example 2
ExercisesExercises
Skills PracticeMonomials
NAME ______________________________________________ DATE ____________ PERIOD _____
5-15-1
© Glencoe/McGraw-Hill 241 Glencoe Algebra 2
Less
on
5-1
Simplify. Assume that no variable equals 0.
1. b4 $ b3 b7 2. c5 $ c2 $ c2 c9
3. a!4 $ a!3 4. x5 $ x!4 $ x x2
5. (g4)2 g8 6. (3u)3 27u3
7. (!x)4 x4 8. !5(2z)3 !40z3
9. !(!3d)4 !81d4 10. (!2t2)3 !8t6
11. (!r7)3 !r 21 12. s3
13. 14. (!3f 3g)3 !27f 9g3
15. (2x)2(4y)2 64x2y2 16. !2gh( g3h5) !2g4h6
17. 10x2y3(10xy8) 100x3y11 18.
19. ! 20.
Express each number in scientific notation.
21. 53,000 5.3 # 104 22. 0.000248 2.48 # 10!4
23. 410,100,000 4.101 # 108 24. 0.00000805 8.05 # 10!6
Evaluate. Express the result in scientific notation.
25. (4 ' 103)(1.6 ' 10!6) 6.4 # 10!3 26. 6.4 # 10109.6 ' 107&&1.5 ' 10!3
2q2"p2
!10pq4r&&!5p3q2r
c7"6a3b
!6a4bc8&&36a7b2c
8z2"w2
24wz7&3w3z5
1"k
k9&k10
s15&s12
1"a7
© Glencoe/McGraw-Hill 242 Glencoe Algebra 2
Simplify. Assume that no variable equals 0.
1. n5 $ n2 n7 2. y7 $ y3 $ y2 y12
3. t9 $ t!8 t 4. x!4 $ x!4 $ x4
5. (2f 4)6 64f 24 6. (!2b!2c3)3 !
7. (4d2t5v!4)(!5dt!3v!1) ! 8. 8u(2z)3 64uz3
9. ! 10. !
11. 12. ! "2
13. !(4w!3z!5)(8w)2 ! 14. (m4n6)4(m3n2p5)6 m34n36p30
15. ! d2f 4"4!! d5f"3 !12d23f 19 16. ! "!2
17. 18. !
Express each number in scientific notation.
19. 896,000 20. 0.000056 21. 433.7 ' 108
8.96 # 105 5.6 # 10!5 4.337 # 1010
Evaluate. Express the result in scientific notation.
22. (4.8 ' 102)(6.9 ' 104) 23. (3.7 ' 109)(8.7 ' 102) 24.
3.312 # 107 3.219 # 1012 3 # 10!5
25. COMPUTING The term bit, short for binary digit, was first used in 1946 by John Tukey.A single bit holds a zero or a one. Some computers use 32-bit numbers, or strings of 32 consecutive bits, to identify each address in their memories. Each 32-bit numbercorresponds to a number in our base-ten system. The largest 32-bit number is nearly4,295,000,000. Write this number in scientific notation. 4.295 # 109
26. LIGHT When light passes through water, its velocity is reduced by 25%. If the speed oflight in a vacuum is 1.86 ' 105 miles per second, at what velocity does it travel throughwater? Write your answer in scientific notation. 1.395 # 105 mi/s
27. TREES Deciduous and coniferous trees are hard to distinguish in a black-and-whitephoto. But because deciduous trees reflect infrared energy better than coniferous trees,the two types of trees are more distinguishable in an infrared photo. If an infraredwavelength measures about 8 ' 10!7 meters and a blue wavelength measures about 4.5 ' 10!7 meters, about how many times longer is the infrared wavelength than theblue wavelength? about 1.8 times
2.7 ' 106&&9 ' 1010
4v2"m2
!20(m2v)(!v)3&&
5(!v)2(!m4)15x11"y3
(3x!2y3)(5xy!8)&&
(x!3)4y!2
y6"4x2
2x3y2&!x2y5
4&3
3&2
256"wz5
4"9r4s6z12
2&3r2s3z6
27x6"16
!27x3(!x7)&&
16x4
s4"3x4
!6s5x3&18sx7
4m7y2"3
12m8y6&!9my4
20d 3t2"v5
8c9"b6
1"x4
Practice (Average)
Monomials
NAME ______________________________________________ DATE ____________ PERIOD _____
5-15-1
Reading to Learn MathematicsMonomials
NAME ______________________________________________ DATE ____________ PERIOD _____
5-15-1
© Glencoe/McGraw-Hill 243 Glencoe Algebra 2
Less
on
5-1
Pre-Activity Why is scientific notation useful in economics?
Read the introduction to Lesson 5-1 at the top of page 222 in your textbook.
Your textbook gives the U.S. public debt as an example from economics thatinvolves large numbers that are difficult to work with when written instandard notation. Give an example from science that involves very largenumbers and one that involves very small numbers. Sample answer:distances between Earth and the stars, sizes of molecules and atoms
Reading the Lesson
1. Tell whether each expression is a monomial or not a monomial. If it is a monomial, tellwhether it is a constant or not a constant.
a. 3x2 monomial; not a constant b. y2 % 5y ! 6 not a monomial
c. !73 monomial; constant d. not a monomial
2. Complete the following definitions of a negative exponent and a zero exponent.
For any real number a # 0 and any integer n, a!n " .
For any real number a # 0, a0 " .
3. Name the property or properties of exponents that you would use to simplify eachexpression. (Do not actually simplify.)
a. quotient of powers
b. y6 $ y9 product of powers
c. (3r2s)4 power of a product and power of a power
Helping You Remember
4. When writing a number in scientific notation, some students have trouble rememberingwhen to use positive exponents and when to use negative ones. What is an easy way toremember this? Sample answer: Use a positive exponent if the number is10 or greater. Use a negative number if the number is less than 1.
m8&m3
1
"a1n"
1&z
© Glencoe/McGraw-Hill 244 Glencoe Algebra 2
Properties of ExponentsThe rules about powers and exponents are usually given with letters such as m, n,and k to represent exponents. For example, one rule states that am $ an " am % n.
In practice, such exponents are handled as algebraic expressions and the rules of algebra apply.
Simplify 2a2(an $ 1 $ a4n).
2a2(an % 1 % a4n) " 2a2 $ an % 1 % 2a2 $ a4n Use the Distributive Law.
" 2a2 % n % 1 % 2a2 % 4n Recall am $ an " am % n.
" 2an % 3 % 2a2 % 4n Simplify the exponent 2 % n % 1 as n % 3.
It is important always to collect like terms only.
Simplify (an $ bm)2.
(an % bm)2 " (an % bm)(an % bm)F O I L
" an $ an % an $ bm % an $ bm % bm $ bm The second and third terms are like terms.
" a2n % 2anbm % b2m
Simplify each expression by performing the indicated operations.
1. 232m 2. (a3)n 3. (4nb2)k
4. (x3aj )m 5. (!ayn)3 6. (!bkx)2
7. (c2)hk 8. (!2dn)5 9. (a2b)(anb2)
10. (xnym)(xmyn) 11. 12.
13. (ab2 ! a2b)(3an % 4bn)
14. ab2(2a2bn ! 1 % 4abn % 6bn % 1)
12x3&4xn
an&2
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
5-15-1
Example 1Example 1
Example 2Example 2
Study Guide and InterventionPolynomials
NAME ______________________________________________ DATE ____________ PERIOD _____
5-25-2
© Glencoe/McGraw-Hill 245 Glencoe Algebra 2
Less
on
5-2
Add and Subtract Polynomials
Polynomial a monomial or a sum of monomials
Like Terms terms that have the same variable(s) raised to the same power(s)
To add or subtract polynomials, perform the indicated operations and combine like terms.
Simplify !6rs $ 18r2 ! 5s2 ! 14r2 $ 8rs ! 6s2.
!6rs % 18r2 ! 5s2 !14r2 % 8rs! 6s2
" (18r2 ! 14r2) % (!6rs % 8rs) % (!5s2 ! 6s2) Group like terms." 4r2 % 2rs ! 11s2 Combine like terms.
Simplify 4xy2 $ 12xy ! 7x2y ! (20xy $ 5xy2 ! 8x2y).
4xy2 % 12xy ! 7x2y ! (20xy % 5xy2 ! 8x2y)" 4xy2 % 12xy ! 7x2y ! 20xy ! 5xy2 % 8x2y Distribute the minus sign." (!7x2y % 8x2y ) % (4xy2 ! 5xy2) % (12xy ! 20xy) Group like terms." x2y ! xy2 ! 8xy Combine like terms.
Simplify.
1. (6x2 ! 3x % 2) ! (4x2 % x ! 3) 2. (7y2 % 12xy ! 5x2) % (6xy ! 4y2 ! 3x2)2x2 ! 4x $ 5 3y2$ 18xy ! 8x2
3. (!4m2 ! 6m) ! (6m % 4m2) 4. 27x2 ! 5y2 % 12y2 ! 14x2
!8m2 ! 12m 13x2 $ 7y2
5. (18p2 % 11pq ! 6q2) ! (15p2 ! 3pq % 4q2) 6. 17j2 ! 12k2 % 3j2 ! 15j2 % 14k2
3p2 $ 14pq ! 10q2 5j2 $ 2k2
7. (8m2 ! 7n2) ! (n2 ! 12m2) 8. 14bc % 6b ! 4c % 8b ! 8c % 8bc20m2 ! 8n2 14b $ 22bc ! 12c
9. 6r2s % 11rs2 % 3r2s ! 7rs2 % 15r2s ! 9rs2 10. !9xy % 11x2 ! 14y2 ! (6y2 ! 5xy ! 3x2)24r2s ! 5rs2 14x2 ! 4xy ! 20y2
11. (12xy ! 8x % 3y) % (15x ! 7y ! 8xy) 12. 10.8b2 ! 5.7b % 7.2 ! (2.9b2 ! 4.6b ! 3.1)7x $ 4xy ! 4y 7.9b2 ! 1.1b $ 10.3
13. (3bc ! 9b2 ! 6c2) % (4c2 ! b2 % 5bc) 14. 11x2 % 4y2 % 6xy % 3y2 ! 5xy ! 10x2
!10b2 $ 8bc ! 2c2 x2 $ xy $ 7y2
15. x2 ! xy % y2 ! xy % y2 ! x2 16. 24p3 ! 15p2 % 3p ! 15p3 % 13p2 ! 7p
! x2 ! xy $ y2 9p3 ! 2p2 ! 4p3"4
7"8
1"8
3&8
1&4
1&2
1&2
3&8
1&4
Example 1Example 1
Example 2Example 2
ExercisesExercises
© Glencoe/McGraw-Hill 246 Glencoe Algebra 2
Multiply Polynomials You use the distributive property when you multiplypolynomials. When multiplying binomials, the FOIL pattern is helpful.
To multiply two binomials, add the products ofF the first terms,
FOIL Pattern O the outer terms,I the inner terms, andL the last terms.
Find 4y(6 ! 2y $ 5y2).
4y(6 ! 2y % 5y2) " 4y(6) % 4y(!2y) % 4y(5y2) Distributive Property" 24y ! 8y2 % 20y3 Multiply the monomials.
Find (6x ! 5)(2x $ 1).
(6x ! 5)(2x % 1) " 6x $ 2x % 6x $ 1 % (!5) $ 2x % (!5) $ 1First terms Outer terms Inner terms Last terms
" 12x2 % 6x ! 10x ! 5 Multiply monomials." 12x2 ! 4x ! 5 Add like terms.
Find each product.
1. 2x(3x2 ! 5) 2. 7a(6 ! 2a ! a2) 3. !5y2( y2 % 2y ! 3)6x3 ! 10x 42a ! 14a2 ! 7a3 !5y4 ! 10y3 $ 15y2
4. (x ! 2)(x % 7) 5. (5 ! 4x)(3 ! 2x) 6. (2x ! 1)(3x % 5)x2 $ 5x ! 14 15 ! 22x $ 8x2 6x2 $ 7x ! 5
7. (4x % 3)(x % 8) 8. (7x ! 2)(2x ! 7) 9. (3x ! 2)(x % 10)4x2 $ 35x $ 24 14x2 ! 53x $ 14 3x2 $ 28x ! 20
10. 3(2a % 5c) ! 2(4a ! 6c) 11. 2(a ! 6)(2a % 7) 12. 2x(x % 5) ! x2(3 ! x)!2a $ 27c 4a2 ! 10a ! 84 x3 ! x2 $ 10x
13. (3t2 ! 8)(t2 % 5) 14. (2r % 7)2 15. (c % 7)(c ! 3)3t4 $ 7t2 ! 40 4r2 $ 28r $ 49 c2 $ 4c ! 21
16. (5a % 7)(5a ! 7) 17. (3x2 ! 1)(2x2 % 5x)25a2 ! 49 6x4 $ 15x3 ! 2x2 ! 5x
18. (x2 ! 2)(x2 ! 5) 19. (x % 1)(2x2 ! 3x % 1)x4 ! 7x2 $ 10 2x3 ! x2 ! 2x $ 1
20. (2n2 ! 3)(n2 % 5n ! 1) 21. (x ! 1)(x2 ! 3x % 4)2n4 $ 10n3 ! 5n2 ! 15n $ 3 x3 ! 4x2 $ 7x ! 4
Study Guide and Intervention (continued)
Polynomials
NAME ______________________________________________ DATE ____________ PERIOD _____
5-25-2
Example 1Example 1
Example 2Example 2
ExercisesExercises
Skills PracticePolynomials
NAME ______________________________________________ DATE ____________ PERIOD _____
5-25-2
© Glencoe/McGraw-Hill 247 Glencoe Algebra 2
Less
on
5-2
Determine whether each expression is a polynomial. If it is a polynomial, state thedegree of the polynomial.
1. x2 % 2x % 2 yes; 2 2. no 3. 8xz % y yes; 2
Simplify.
4. (g % 5) % (2g % 7) 5. (5d % 5) ! (d % 1)3g $ 12 4d $ 4
6. (x2 ! 3x ! 3) % (2x2 % 7x ! 2) 7. (!2f 2 ! 3f ! 5) % (!2f 2 ! 3f % 8)3x2 $ 4x ! 5 !4f 2 ! 6f $ 3
8. (4r2 ! 6r % 2) ! (!r2 % 3r % 5) 9. (2x2 ! 3xy) ! (3x2 ! 6xy ! 4y2)5r2 ! 9r ! 3 !x2 $ 3xy $ 4y2
10. (5t ! 7) % (2t2 % 3t % 12) 11. (u ! 4) ! (6 % 3u2 ! 4u)2t2 $ 8t $ 5 !3u2 $ 5u ! 10
12. !5(2c2 ! d2) 13. x2(2x % 9)!10c2 $ 5d2 2x3 $ 9x2
14. 2q(3pq % 4q4) 15. 8w(hk2 % 10h3m4 ! 6k5w3)6pq2 $ 8q5 8hk2w $ 80h3m4w ! 48k5w4
16. m2n3(!4m2n2 ! 2mnp ! 7) 17. !3s2y(!2s4y2 % 3sy3 % 4)!4m4n5 ! 2m3n4p ! 7m2n3 6s6y3 ! 9s3y4 ! 12s2y
18. (c % 2)(c % 8) 19. (z ! 7)(z % 4)c2 $ 10c $ 16 z2 ! 3z ! 28
20. (a ! 5)2 21. (2x ! 3)(3x ! 5)a2 ! 10a $ 25 6x2 ! 19x $ 15
22. (r ! 2s)(r % 2s) 23. (3y % 4)(2y ! 3)r2 ! 4s2 6y2 ! y ! 12
24. (3 ! 2b)(3 % 2b) 25. (3w % 1)2
9 ! 4b2 9w2 $ 6w $ 1
1&2
b2c&d4
© Glencoe/McGraw-Hill 248 Glencoe Algebra 2
Determine whether each expression is a polynomial. If it is a polynomial, state thedegree of the polynomial.
1. 5x3 % 2xy4 % 6xy yes; 5 2. ! ac ! a5d3 yes; 8 3. no
4. 25x3z ! x#78$ yes; 4 5. 6c!2 % c ! 1 no 6. % no
Simplify.
7. (3n2 % 1) % (8n2 ! 8) 8. (6w ! 11w2) ! (4 % 7w2)11n2 ! 7 !18w2 $ 6w ! 4
9. (!6n ! 13n2) % (!3n % 9n2) 10. (8x2 ! 3x) ! (4x2 % 5x ! 3)!9n ! 4n2 4x2 ! 8x $ 3
11. (5m2 ! 2mp ! 6p2) ! (!3m2 % 5mp % p2) 12. (2x2 ! xy % y2) % (!3x2 % 4xy % 3y2)8m2 ! 7mp ! 7p2 !x2 $ 3xy $ 4y2
13. (5t ! 7) % (2t2 % 3t % 12) 14. (u ! 4) ! (6 % 3u2 ! 4u)2t2 $ 8t $ 5 !3u2 $ 5u ! 10
15. !9( y2 ! 7w) 16. !9r4y2(!3ry7 % 2r3y4 ! 8r10)!9y2 $ 63w 27r 5y9 ! 18r7y6 $ 72r14y2
17. !6a2w(a3w ! aw4) 18. 5a2w3(a2w6 ! 3a4w2 % 9aw6)!6a5w2 $ 6a3w5 5a4w9 ! 15a6w5 $ 45a3w9
19. 2x2(x2 % xy ! 2y2) 20. ! ab3d2(!5ab2d5 ! 5ab)
2x4 $ 2x3y ! 4x2y2 3a2b5d7 $ 3a2b4d 2
21. (v2 ! 6)(v2 % 4) 22. (7a % 9y)(2a ! y)v4 ! 2v2 ! 24 14a2 $ 11ay ! 9y2
23. ( y ! 8)2 24. (x2 % 5y)2
y2 ! 16y $ 64 x4 $ 10x2y $ 25y2
25. (5x % 4w)(5x ! 4w) 26. (2n4 ! 3)(2n4 % 3)25x2 ! 16w2 4n8 ! 9
27. (w % 2s)(w2 ! 2ws % 4s2) 28. (x % y)(x2 ! 3xy % 2y2)w3 $ 8s3 x3 ! 2x2y ! xy2 $ 2y3
29. BANKING Terry invests $1500 in two mutual funds. The first year, one fund grows 3.8%and the other grows 6%. Write a polynomial to represent the amount Terry’s $1500grows to in that year if x represents the amount he invested in the fund with the lessergrowth rate. !0.022x $ 1590
30. GEOMETRY The area of the base of a rectangular box measures 2x2 % 4x ! 3 squareunits. The height of the box measures x units. Find a polynomial expression for thevolume of the box. 2x3 $ 4x2 ! 3x units3
3&5
6&s
5&r
12m8n9&&(m ! n)2
4&3
Practice (Average)
Polynomials
NAME ______________________________________________ DATE ____________ PERIOD _____
5-25-2
Reading to Learn MathematicsPolynomials
NAME ______________________________________________ DATE ____________ PERIOD _____
5-25-2
© Glencoe/McGraw-Hill 249 Glencoe Algebra 2
Less
on
5-2
Pre-Activity How can polynomials be applied to financial situations?
Read the introduction to Lesson 5-2 at the top of page 229 in your textbook.
Suppose that Shenequa decides to enroll in a five-year engineering programrather than a four-year program. Using the model given in your textbook,how could she estimate the tuition for the fifth year of her program? (Donot actually calculate, but describe the calculation that would be necessary.)Multiply $15,604 by 1.04.
Reading the Lesson
1. State whether the terms in each of the following pairs are like terms or unlike terms.
a. 3x2, 3y2 unlike terms b. !m4, 5m4 like termsc. 8r3, 8s3 unlike terms d. !6, 6 like terms
2. State whether each of the following expressions is a monomial, binomial, trinomial, ornot a polynomial. If the expression is a polynomial, give its degree.
a. 4r4 ! 2r % 1 trinomial; degree 4 b. #3x$ not a polynomialc. 5x % 4y binomial; degree 1 d. 2ab % 4ab2 ! 6ab3 trinomial; degree 4
3. a. What is the FOIL method used for in algebra? to multiply binomialsb. The FOIL method is an application of what property of real numbers?
Distributive Propertyc. In the FOIL method, what do the letters F, O, I, and L mean?
first, outer, inner, lastd. Suppose you want to use the FOIL method to multiply (2x % 3)(4x % 1). Show the
terms you would multiply, but do not actually multiply them.
F (2x)(4x)O (2x)(1)I (3)(4x)L (3)(1)
Helping You Remember
4. You can remember the difference between monomials, binomials, and trinomials bythinking of common English words that begin with the same prefixes. Give two wordsunrelated to mathematics that start with mono-, two that begin with bi-, and two thatbegin with tri-. Sample answer: monotonous, monogram; bicycle, bifocal;tricycle, tripod
© Glencoe/McGraw-Hill 250 Glencoe Algebra 2
Polynomials with Fractional CoefficientsPolynomials may have fractional coefficients as long as there are no variables in the denominators. Computing with fractional coefficients is performed in the same way as computing with whole-number coefficients.
Simpliply. Write all coefficients as fractions.
1. !&35&m ! &
27&p ! &
13&n" ! !&
73&p ! &
52&m ! &
34&n"
2.!&32&x ! &
43&y ! &
54&z" % !!&
14&x % y % &
25&z" % !!&
78&x ! &
67&y % &
12&z" "
38"
3.!&12&a2 ! &
13&ab % &
14&b2" % !&
56&a2 % &
23&ab ! &
34&b2" "
43
4.!&12&a2 ! &
13&ab % &
14&b2" ! !&
13&a2 ! &
12&ab % &
56&b2" "
5.!&12&a2 ! &
13&ab % &
14&b2" $ !&
12&a ! &
23&b" "
14"
6.!&23&a2 ! &
15&a % &
27&" $ !&
23&a3 % &
15&a2 ! &
27&a" "
7.!&23&x2 ! &
34&x ! 2" $ !&
45&x ! &
16&x2 ! &
12&"
8.!&16& % &
13&x % &
16&x4 ! &
12&x2" $ !&
16&x3 ! &
13& ! &
13&x" "3
1
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
5-25-2
Study Guide and InterventionDividing Polynomials
NAME ______________________________________________ DATE ____________ PERIOD _____
5-35-3
© Glencoe/McGraw-Hill 251 Glencoe Algebra 2
Less
on
5-3
Use Long Division To divide a polynomial by a monomial, use the properties of powersfrom Lesson 5-1.
To divide a polynomial by a polynomial, use a long division pattern. Remember that onlylike terms can be added or subtracted.
Simplify .
" ! !
" p3!2t2!1r1!1 ! p2!2qt1!1r2!1 ! p3!2t1!1r1!1
" 4pt !7qr ! 3p
Use long division to find (x3 ! 8x2 $ 4x ! 9) % (x ! 4).
x2 ! 4x ! 12x ! 4%x$3$!$ 8$x$2$%$ 4$x$ !$ 9$
(!)x3 ! 4x2
!4x2 % 4x(!)!4x2 % 16x
!12x ! 9(!)!12x % 48
!57The quotient is x2 ! 4x ! 12, and the remainder is !57.
Therefore " x2 ! 4x ! 12 ! .
Simplify.
1. 2. 3.
6a2 $ 10a ! 10 5ab ! $ 7a4
4. (2x2 ! 5x ! 3) * (x ! 3) 5. (m2 ! 3m ! 7) * (m % 2)
2x $ 1 m ! 5 $
6. (p3 ! 6) * (p ! 1) 7. (t3 ! 6t2 % 1) * (t % 2)
p2 $ p $ 1 ! t2 ! 8t $ 16 !
8. (x5 ! 1) * (x ! 1) 9. (2x3 ! 5x2 % 4x ! 4) * (x ! 2)
x4 $ x3 $ x2 $ x $ 1 2x2 ! x $ 2
31"t $ 2
5"p ! 1
3"m $ 2
4b2"a
6n3"m
60a2b3 ! 48b4 % 84a5b2&&&
12ab224mn6 ! 40m2n3&&&
4m2n318a3 % 30a2&&3a
57&x ! 4
x3 ! 8x2 % 4x ! 9&&&x ! 4
9&3
21&3
12&3
9p3tr&3p2tr
21p2qtr2&&
3p2tr12p3t2r&
3p2tr12p3t2r ! 21p2qtr2 ! 9p3tr&&&&
3p2tr
12p3t2r ! 21p2qtr2 ! 9p3tr""""
3p2trExample 1Example 1
Example 2Example 2
ExercisesExercises
© Glencoe/McGraw-Hill 252 Glencoe Algebra 2
Use Synthetic Division
Synthetic division a procedure to divide a polynomial by a binomial using coefficients of the dividend and the value of r in the divisor x ! r
Use synthetic division to find (2x3 ! 5x2 % 5x ! 2) * (x ! 1).
Step 1 Write the terms of the dividend so that the degrees of the terms are in 2x3 ! 5x2 % 5x ! 2 descending order. Then write just the coefficients. 2 !5 5 !2
Step 2 Write the constant r of the divisor x ! r to the left, In this case, r " 1. 1 2 !5 5 !2Bring down the first coefficient, 2, as shown.
2
Step 3 Multiply the first coefficient by r, 1 $ 2 " 2. Write their product under the 1 2 !5 5 !2 second coefficient. Then add the product and the second coefficient: 2 !5 % 2 " ! 3. 2 !3
Step 4 Multiply the sum, !3, by r : !3 $ 1 " !3. Write the product under the next 1 2 !5 5 !2 coefficient and add: 5 % (!3) " 2. 2 !3
2 !3 2
Step 5 Multiply the sum, 2, by r : 2 $ 1 " 2. Write the product under the next 1 2 !5 5 !2 coefficient and add: !2 % 2 " 0. The remainder is 0. 2 !3 2
2 !3 2 0
Thus, (2x3 ! 5x2 % 5x ! 2) * (x ! 1) " 2x2 ! 3x % 2.
Simplify.
1. (3x3 ! 7x2 % 9x ! 14) * (x ! 2) 2. (5x3 % 7x2 ! x ! 3) * (x % 1)
3x2 ! x $ 7 5x2 $ 2x ! 3
3. (2x3 % 3x2 ! 10x ! 3) * (x % 3) 4. (x3 ! 8x2 % 19x ! 9) * (x ! 4)
2x2 ! 3x ! 1 x2 ! 4x $ 3 $
5. (2x3 % 10x2 % 9x % 38) * (x % 5) 6. (3x3 ! 8x2 % 16x ! 1) * (x ! 1)
2x2 $ 9 ! 3x2 ! 5x $ 11 $
7. (x3 ! 9x2 % 17x ! 1) * (x ! 2) 8. (4x3 ! 25x2 % 4x % 20) * (x ! 6)
x2 ! 7x $ 3 $ 4x2 ! x ! 2 $
9. (6x3 % 28x2 ! 7x % 9) * (x % 5) 10. (x4 ! 4x3 % x2 % 7x ! 2) * (x ! 2)
6x2 ! 2x $ 3 ! x3 ! 2x2 ! 3x $ 1
11. (12x4 % 20x3 ! 24x2 % 20x % 35) * (3x % 5) 4x3 ! 8x $ 20 $ !65"3x $ 5
6"x $ 5
8"x ! 6
5"x ! 2
10"x ! 1
7"x $ 5
3"x ! 4
Study Guide and Intervention (continued)
Dividing Polynomials
NAME ______________________________________________ DATE ____________ PERIOD _____
5-35-3
ExercisesExercises
Skills PracticeDividing Polynomials
NAME ______________________________________________ DATE ____________ PERIOD _____
5-35-3
© Glencoe/McGraw-Hill 253 Glencoe Algebra 2
Less
on
5-3
Simplify.
1. 5c $ 3 2. 3x $ 5
3. 5y2 $ 2y $ 1 4. 3x ! 1 !
5. (15q6 % 5q2)(5q4)!1 6. (4f 5 ! 6f4 % 12f 3 ! 8f 2)(4f 2)!1
3q2 $ f 3 ! $ 3f ! 2
7. (6j2k ! 9jk2) * 3jk 8. (4a2h2 ! 8a3h % 3a4) * (2a2)
2j ! 3k 2h2 ! 4ah $
9. (n2 % 7n % 10) * (n % 5) 10. (d2 % 4d % 3) * (d % 1)
n $ 2 d $ 3
11. (2s2 % 13s % 15) * (s % 5) 12. (6y2 % y ! 2)(2y ! 1)!1
2s $ 3 3y $ 2
13. (4g2 ! 9) * (2g % 3) 14. (2x2 ! 5x ! 4) * (x ! 3)
2g ! 3 2x $ 1 !
15. 16.
u $ 8 $ 2x $ 1 !
17. (3v2 ! 7v ! 10)(v ! 4)!1 18. (3t4 % 4t3 ! 32t2 ! 5t ! 20)(t % 4)!1
3v $ 5 $ 3t3 ! 8t2 ! 5
19. 20.
y2 ! 3y $ 6 ! 2x2 $ 5x ! 4 $
21. (4p3 ! 3p2 % 2p) * ( p ! 1) 22. (3c4 % 6c3 ! 2c % 4)(c % 2)!1
4p2 $ p $ 3 $ 3c3 ! 2 $
23. GEOMETRY The area of a rectangle is x3 % 8x2 % 13x ! 12 square units. The width ofthe rectangle is x % 4 units. What is the length of the rectangle? x2 $ 4x ! 3 units
8"c $ 2
3"p ! 1
3"x ! 3
18"y $ 2
2x3 ! x2 ! 19x % 15&&&x ! 3
y3 ! y2 ! 6&&y % 2
10"v ! 4
1"x ! 3
12"u ! 3
2x2 ! 5x ! 4&&x ! 3
u2 % 5u ! 12&&u ! 3
1"x ! 3
3a2"2
3f 2"2
1"q2
2"x
12x2 ! 4x ! 8&&4x
15y3 % 6y2 % 3y&&3y
12x % 20&&4
10c % 6&2
© Glencoe/McGraw-Hill 254 Glencoe Algebra 2
Simplify.
1. 3r6 ! r4 $ 2. ! 6k2 $
3. (!30x3y % 12x2y2 ! 18x2y) * (!6x2y) 4. (!6w3z4 ! 3w2z5 % 4w % 5z) * (2w2z)
5x ! 2y $ 3 !3wz3 ! $ $
5. (4a3 ! 8a2 % a2)(4a)!1 6. (28d3k2 % d2k2 ! 4dk2)(4dk2)!1
a2 ! 2a $ "a4" 7d2 $ "
d4" ! 1
7. f $ 5 8. 2x $ 7
9. (a3 ! 64) * (a ! 4) a2 $ 4a $ 16 10. (b3 % 27) * (b % 3) b2 ! 3b $ 9
11. 2x2 ! 8x $ 38 12. 2x2 ! 6x $ 22 !
13. (3w3 % 7w2 ! 4w % 3) * (w % 3) 14. (6y4 % 15y3 ! 28y ! 6) * (y % 2)
3w2 ! 2w $ 2 ! "w $3
3" 6y3 $ 3y2 ! 6y ! 16 $ "y2$6
2"
15. (x4 ! 3x3 ! 11x2 % 3x % 10) * (x ! 5) 16. (3m5 % m ! 1) * (m % 1)
x3 $ 2x2 ! x ! 2 3m4 ! 3m3 $ 3m2 ! 3m $ 4 !
17. (x4 ! 3x3 % 5x ! 6)(x % 2)!1 18. (6y2 ! 5y ! 15)(2y % 3)!1
x3 ! 5x2 $ 10x ! 15 $ "x2$4
2" 3y ! 7 $ "2y6$ 3"
19. 20.
2x $ 2 $ "2x1!2
3" 2x ! 1 ! "3x6$ 1"
21. (2r3 % 5r2 ! 2r ! 15) * (2r ! 3) 22. (6t3 % 5t2 ! 2t % 1) * (3t % 1)r2 $ 4r $ 5 2t2 $ t ! 1 $ "3t
2$ 1"
23. 24.
2p3 $ 3p2 ! 4p $ 1 2h2 ! h $ 325. GEOMETRY The area of a rectangle is 2x2 ! 11x % 15 square feet. The length of the
rectangle is 2x ! 5 feet. What is the width of the rectangle? x ! 3 ft
26. GEOMETRY The area of a triangle is 15x4 % 3x3 % 4x2 ! x ! 3 square meters. Thelength of the base of the triangle is 6x2 ! 2 meters. What is the height of the triangle?5x2 $ x $ 3 m
2h4 ! h3 % h2 % h ! 3&&&
h2 ! 14p4 ! 17p2 % 14p ! 3&&&2p ! 3
6x2 ! x ! 7&&3x % 1
4x2 ! 2x % 6&&2x ! 3
5"m $ 1
72"x $ 3
2x3 % 4x ! 6&&x % 3
2x3 % 6x % 152&&x % 4
2x2 % 3x ! 14&&x ! 2
f2 % 7f % 10&&f % 2
5"2w2
2"wz
3z4"2
9m"2k
3k"m
6k2m ! 12k3m2 % 9m3&&&
2km28"r2
15r10 ! 5r8 % 40r2&&&
5r4
Practice (Average)
Dividing Polynomials
NAME ______________________________________________ DATE ____________ PERIOD _____
5-35-3
Reading to Learn MathematicsDividing Polynomials
NAME ______________________________________________ DATE ____________ PERIOD _____
5-35-3
© Glencoe/McGraw-Hill 255 Glencoe Algebra 2
Less
on
5-3
Pre-Activity How can you use division of polynomials in manufacturing?
Read the introduction to Lesson 5-3 at the top of page 233 in your textbook.
Using the division symbol (*), write the division problem that you woulduse to answer the question asked in the introduction. (Do not actuallydivide.) (32x2 $ x) % (8x)
Reading the Lesson
1. a. Explain in words how to divide a polynomial by a monomial. Divide each term ofthe polynomial by the monomial.
b. If you divide a trinomial by a monomial and get a polynomial, what kind ofpolynomial will the quotient be? trinomial
2. Look at the following division example that uses the division algorithm for polynomials.
2x % 4x ! 4%$$2x2 ! 4x % 7
2x2 ! 8x4x % 74x ! 16
23Which of the following is the correct way to write the quotient? CA. 2x % 4 B. x ! 4 C. 2x % 4 % D.
3. If you use synthetic division to divide x3 % 3x2 ! 5x ! 8 by x ! 2, the division will looklike this:
2 1 3 !5 !82 10 10
1 5 5 2
Which of the following is the answer for this division problem? BA. x2 % 5x % 5 B. x2 % 5x % 5 %
C. x3 % 5x2 % 5x % D. x3 % 5x2 % 5x % 2
Helping You Remember
4. When you translate the numbers in the last row of a synthetic division into the quotientand remainder, what is an easy way to remember which exponents to use in writing theterms of the quotient? Sample answer: Start with the power that is one lessthan the degree of the dividend. Decrease the power by one for eachterm after the first. The final number will be the remainder. Drop any termthat is represented by a 0.
2&x ! 2
2&x ! 2
23&x ! 4
23&x ! 4
© Glencoe/McGraw-Hill 256 Glencoe Algebra 2
Oblique AsymptotesThe graph of y " ax % b, where a # 0, is called an oblique asymptote of y " f(x) if the graph of f comes closer and closer to the line as x → ∞ or x → !∞. ∞ is themathematical symbol for infinity, which means endless.
For f(x) " 3x % 4 % &2x&, y " 3x % 4 is an oblique asymptote because
f(x) ! 3x ! 4 " &2x&, and &
2x& → 0 as x → ∞ or !∞. In other words, as | x |
increases, the value of &2x& gets smaller and smaller approaching 0.
Find the oblique asymptote for f(x) & "x2 $
x8$x
2$ 15
".
!2 1 8 15 Use synthetic division.!2 !12
1 6 3
y " &x2 %
x8%x
2% 15& " x % 6 % &x %
32&
As | x | increases, the value of &x %3
2& gets smaller. In other words, since
&x %3
2& → 0 as x → ∞ or x → !∞, y " x % 6 is an oblique asymptote.
Use synthetic division to find the oblique asymptote for each function.
1. y " &8x2 !
x %4x
5% 11
&
2. y " &x2 %
x3!x
2! 15&
3. y " &x2 !
x2!x
3! 18&
4. y " &ax2
x%
!bx
d% c
&
5. y " &ax2
x%
%bx
d% c
&
Enrichment
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5-35-3
ExampleExample
Study Guide and InterventionFactoring Polynomials
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5-45-4
© Glencoe/McGraw-Hill 257 Glencoe Algebra 2
Less
on
5-4
Factor Polynomials
For any number of terms, check for:greatest common factor
For two terms, check for:Difference of two squares
a2 ! b2 " (a % b)(a ! b)Sum of two cubes
a3 % b3 " (a % b)(a2 ! ab % b2)Difference of two cubes
a3 ! b3 " (a ! b)(a2 % ab % b2)Techniques for Factoring Polynomials For three terms, check for:
Perfect square trinomialsa2 % 2ab % b2 " (a % b)2
a2 ! 2ab % b2 " (a ! b)2
General trinomialsacx2 % (ad % bc)x % bd " (ax % b)(cx % d )
For four terms, check for:Groupingax % bx % ay % by " x(a % b) % y(a % b)
" (a % b)(x % y)
Factor 24x2 ! 42x ! 45.
First factor out the GCF to get 24x2 ! 42x ! 45 " 3(8x2 ! 14x ! 15). To find the coefficientsof the x terms, you must find two numbers whose product is 8 $ (!15) " !120 and whosesum is !14. The two coefficients must be !20 and 6. Rewrite the expression using !20x and6x and factor by grouping.
8x2 ! 14x ! 15 " 8x2 ! 20x % 6x ! 15 Group to find a GCF." 4x(2x ! 5) % 3(2x ! 5) Factor the GCF of each binomial." (4x % 3)(2x ! 5) Distributive Property
Thus, 24x2 ! 42x ! 45 " 3(4x % 3)(2x ! 5).
Factor completely. If the polynomial is not factorable, write prime.
1. 14x2y2 % 42xy3 2. 6mn % 18m ! n ! 3 3. 2x2 % 18x % 16
14xy2(x $ 3y) (6m ! 1)(n $ 3) 2(x $ 8)(x $ 1)
4. x4 ! 1 5. 35x3y4 ! 60x4y 6. 2r3 % 250
(x2 $ 1)(x $ 1)(x ! 1) 5x3y(7y3 ! 12x) 2(r $ 5)(r2 ! 5r $ 25)
7. 100m8 ! 9 8. x2 % x % 1 9. c4 % c3 ! c2 ! c
(10m4 ! 3)(10m4 $ 3) prime c(c $ 1)2(c ! 1)
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 258 Glencoe Algebra 2
Simplify Quotients In the last lesson you learned how to simplify the quotient of twopolynomials by using long division or synthetic division. Some quotients can be simplified byusing factoring.
Simplify .
" Factor the numerator and denominator.
" Divide. Assume x # 8, ! .
Simplify. Assume that no denominator is equal to 0.
1. 2. 3.
4. 5. 6.
7. 8. 9.
10. 11. 12.
13. 14. 15.
16. 17. 18.
9x2 $ 6x $ 4""3x $ 2
y2 $ 4y $ 16""3y ! 5
2x ! 3""4x2 ! 2x $ 1
27x3 ! 8&&9x2 ! 4
y3 ! 64&&3y2 ! 17y % 20
4x2 ! 4x ! 3&&8x3 % 1
2x ! 3"x $ 8
7x ! 3"x ! 2
x $ 9"2x ! 5
4x2 ! 12x % 9&&2x2 % 13x ! 24
7x2 % 11x ! 6&&x2 ! 4
x2 ! 81&&2x2 ! 23x % 45
x ! 7"x $ 5
3x ! 5"2x ! 3
2x $ 5"x ! 1
x2 ! 14x % 49&&x2 ! 2x ! 35
3x2 % 4x ! 15&&2x2 % 3x ! 9
4x2 % 16x % 15&&2x2 % x ! 3
x ! 2"3x $ 7
6x $ 1"x $ 2
2x ! 1"x ! 2
x2 ! 7x % 10&&3x2 ! 8x ! 35
6x2 % 25x % 4&&x2 % 6x % 8
4x2 % 4x ! 3&&2x2 ! x ! 6
5x ! 1"x $ 3
2x ! 1"4x ! 1
x $ 3"x ! 5
5x2 % 9x ! 2&&x2 % 5x % 6
2x2 % 5x ! 3&&4x2 % 11x ! 3
x2 % x ! 6&&x2 ! 7x % 10
x ! 5"x $ 1
x $ 5"2x! 3
x ! 4"x $ 2
x2 ! 11x % 30&&x2 ! 5x ! 6
x2 % 6x % 5&&2x2 ! x ! 3
x2 ! 7x % 12&&x2 ! x ! 6
3&2
x % 4&x ! 8
(2x % 3)( x % 4)&&(x ! 8)(2x % 3)
8x2 % 11x % 12&&2x2 ! 13x ! 24
8x2 % 11x % 12""2x2 ! 13x ! 24
Study Guide and Intervention (continued)
Factoring Polynomials
NAME ______________________________________________ DATE ____________ PERIOD _____
5-45-4
ExampleExample
ExercisesExercises
Skills PracticeFactoring Polynomials
NAME ______________________________________________ DATE ____________ PERIOD _____
5-45-4
© Glencoe/McGraw-Hill 259 Glencoe Algebra 2
Less
on
5-4
Factor completely. If the polynomial is not factorable, write prime.
1. 7x2 ! 14x 2. 19x3 ! 38x2
7x(x ! 2) 19x2(x ! 2)
3. 21x3 ! 18x2y % 24xy2 4. 8j3k ! 4jk3 ! 73x(7x2 ! 6xy $ 8y2) prime
5. a2 % 7a ! 18 6. 2ak ! 6a % k ! 3(a $ 9)(a ! 2) (2a $ 1)(k ! 3)
7. b2 % 8b % 7 8. z2 ! 8z ! 10(b $ 7)(b $ 1) prime
9. m2 % 7m ! 18 10. 2x2 ! 3x ! 5(m ! 2)(m $ 9) (2x ! 5)(x $ 1)
11. 4z2 % 4z ! 15 12. 4p2 % 4p ! 24(2z $ 5)(2z ! 3) 4(p ! 2)(p $ 3)
13. 3y2 % 21y % 36 14. c2 ! 1003(y $ 4)(y $ 3) (c $ 10)(c ! 10)
15. 4f2 ! 64 16. d2 ! 12d % 364(f $ 4)(f ! 4) (d ! 6)2
17. 9x2 % 25 18. y2 % 18y % 81prime (y $ 9)2
19. n3 ! 125 20. m4 ! 1(n ! 5)(n2 $ 5n $ 25) (m2 $ 1)(m ! 1)(m $ 1)
Simplify. Assume that no denominator is equal to 0.
21. 22.
23. 24.x ! 1"x ! 7
x2 % 6x ! 7&&
x2 ! 49x ! 5"x
x2 ! 10x % 25&&
x2 ! 5x
x $ 1"x $ 3
x2 % 4x % 3&&x2 % 6x % 9
x ! 2"x ! 5
x2 % 7x ! 18&&x2 % 4x ! 45
© Glencoe/McGraw-Hill 260 Glencoe Algebra 2
Factor completely. If the polynomial is not factorable, write prime.
1. 15a2b ! 10ab2 2. 3st2 ! 9s3t % 6s2t2 3. 3x3y2 ! 2x2y % 5xy5ab(3a ! 2b) 3st(t ! 3s2 $ 2st) xy(3x2y ! 2x $ 5)
4. 2x3y ! x2y % 5xy2 % xy3 5. 21 ! 7t % 3r ! rt 6. x2 ! xy % 2x ! 2yxy(2x2 ! x $ 5y $ y2) (7 $ r )(3 ! t) (x $ 2)(x ! y)
7. y2 % 20y % 96 8. 4ab % 2a % 6b % 3 9. 6n2 ! 11n ! 2(y $ 8)(y $ 12) (2a $ 3)(2b $ 1) (6n $ 1)(n ! 2)
10. 6x2 % 7x ! 3 11. x2 ! 8x ! 8 12. 6p2 ! 17p ! 45(3x ! 1)(2x $ 3) prime (2p ! 9)(3p $ 5)
13. r3 % 3r2 ! 54r 14. 8a2 % 2a ! 6 15. c2 ! 49r (r $ 9)(r ! 6) 2(4a ! 3)(a $ 1) (c ! 7)(c $ 7)
16. x3 % 8 17. 16r2 ! 169 18. b4 ! 81(x $ 2)(x2 ! 2x $ 4) (4r $ 13)(4r ! 13) (b2 $ 9)(b $ 3)(b ! 3)
19. 8m3 ! 25 prime 20. 2t3 % 32t2 % 128t 2t(t $ 8)2
21. 5y5 % 135y2 5y2(y $ 3)(y2 ! 3y $ 9) 22. 81x4 ! 16 (9x2 $ 4)(3x $ 2)(3x ! 2)
Simplify. Assume that no denominator is equal to 0.
23. 24. 25.
26. DESIGN Bobbi Jo is using a software package to create a drawing of a cross section of a brace as shown at the right.Write a simplified, factored expression that represents the area of the cross section of the brace. x(20.2 ! x) cm2
27. COMBUSTION ENGINES In an internal combustion engine, the up and down motion of the pistons is converted into the rotary motion of the crankshaft, which drives the flywheel. Let r1 represent the radius of the flywheel at the right and let r2 represent the radius of thecrankshaft passing through it. If the formula for the area of a circle is A " +r2, write a simplified, factored expression for the area of the cross section of the flywheel outside the crankshaft. ' (r1 ! r2)(r1 $ r2)
r1r2
x cm
x cm
8.2 cm
12 c
m
3(x $ 3)""x2 $ 3x $ 9
3x2 ! 27&&x3 ! 27
x ! 8"x $ 9
x2 ! 16x % 64&&
x2 % x ! 72x $ 4"x $ 5
x2 ! 16&&x2 % x ! 20
Practice (Average)
Factoring Polynomials
NAME ______________________________________________ DATE ____________ PERIOD _____
5-45-4
Reading to Learn MathematicsFactoring Polynomials
NAME ______________________________________________ DATE ____________ PERIOD _____
5-45-4
© Glencoe/McGraw-Hill 261 Glencoe Algebra 2
Less
on
5-4
Pre-Activity How does factoring apply to geometry?
Read the introduction to Lesson 5-4 at the top of page 239 in your textbook.
If a trinomial that represents the area of a rectangle is factored into twobinomials, what might the two binomials represent? the length andwidth of the rectangle
Reading the Lesson
1. Name three types of binomials that it is always possible to factor. difference of twosquares, sum of two cubes, difference of two cubes
2. Name a type of trinomial that it is always possible to factor. perfect squaretrinomial
3. Complete: Since x2 % y2 cannot be factored, it is an example of a polynomial.
4. On an algebra quiz, Marlene needed to factor 2x2 ! 4x ! 70. She wrote the followinganswer: (x % 5)(2x ! 14). When she got her quiz back, Marlene found that she did notget full credit for her answer. She thought she should have gotten full credit because shechecked her work by multiplication and showed that (x % 5)(2x ! 14) " 2x2 ! 4x ! 70.
a. If you were Marlene’s teacher, how would you explain to her that her answer was notentirely correct? Sample answer: When you are asked to factor apolynomial, you must factor it completely. The factorization was notcomplete, because 2x ! 14 can be factored further as 2(x ! 7).
b. What advice could Marlene’s teacher give her to avoid making the same kind of errorin factoring in the future? Sample answer: Always look for a commonfactor first. If there is a common factor, factor it out first, and then seeif you can factor further.
Helping You Remember
5. Some students have trouble remembering the correct signs in the formulas for the sumand difference of two cubes. What is an easy way to remember the correct signs?Sample answer: In the binomial factor, the operation sign is the same asin the expression that is being factored. In the trinomial factor, theoperation sign before the middle term is the opposite of the sign in theexpression that is being factored. The sign before the last term is alwaysa plus.
prime
© Glencoe/McGraw-Hill 262 Glencoe Algebra 2
Using Patterns to FactorStudy the patterns below for factoring the sum and the difference of cubes.
a3 % b3 " (a % b)(a2 ! ab % b2)a3 ! b3 " (a ! b)(a2 % ab % b2)
This pattern can be extended to other odd powers. Study these examples.
Factor a5 $ b5.Extend the first pattern to obtain a5 % b5 " (a % b)(a4 ! a3b % a2b2 ! ab3 % b4).Check: (a % b)(a4 ! a3b % a2b2 ! ab3 % b4) " a5 ! a4b % a3b2 ! a2b3 % ab4
% a4b ! a3b2 % a2b3 ! ab4 % b5
" a5 % b5
Factor a5 ! b5.Extend the second pattern to obtain a5 ! b5 " (a ! b)(a4 % a3b % a2b2 % ab3 % b4).Check: (a ! b)(a4 % a3b % a2b2 % ab3 % b4) " a5 % a4b % a3b2 % a2b3 % ab4
! a4b ! a3b2 ! a2b3 ! ab4 ! b5
" a5 ! b5
In general, if n is an odd integer, when you factor an % bn or an ! bn, one factor will beeither (a % b) or (a ! b), depending on the sign of the original expression. The other factorwill have the following properties:• The first term will be an ! 1 and the last term will be bn ! 1.• The exponents of a will decrease by 1 as you go from left to right.• The exponents of b will increase by 1 as you go from left to right.• The degree of each term will be n ! 1.• If the original expression was an % bn, the terms will alternately have % and ! signs.• If the original expression was an ! bn, the terms will all have % signs.
Use the patterns above to factor each expression.
1. a7 % b7
2. c9 ! d9
3. e11 % f 11
To factor x10 ! y10, change it to (x5 $ y5)(x5 ! y5) and factor each binomial. Usethis approach to factor each expression.
4. x10 ! y10
5. a14 ! b14
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
5-45-4
Example 1Example 1
Example 2Example 2
Study Guide and InterventionRoots of Real Numbers
NAME ______________________________________________ DATE ____________ PERIOD _____
5-55-5
© Glencoe/McGraw-Hill 263 Glencoe Algebra 2
Less
on
5-5
Simplify Radicals
Square Root For any real numbers a and b, if a2 " b, then a is a square root of b.
nth Root For any real numbers a and b, and any positive integer n, if an " b, then a is an n throot of b.
1. If n is even and b , 0, then b has one positive root and one negative root.Real nth Roots of b, 2. If n is odd and b , 0, then b has one positive root.n!b", !
n!b" 3. If n is even and b ) 0, then b has no real roots.4. If n is odd and b ) 0, then b has one negative root.
Simplify !49z8".
#49z8$ " #(7z4)2$ " 7z4
z4 must be positive, so there is no need totake the absolute value.
Simplify !3!(2a !" 1)6"!
3#(2a !$ 1)6$ " !3#[(2a !$ 1)2]3$ " (2a ! 1)2
Example 1Example 1 Example 2Example 2
ExercisesExercises
Simplify.
1. #81$ 2. 3#!343$ 3. #144p6$9 !7 12|p3|
4. -#4a10$ 5. 5#243p1$0$ 6. !3#m6n9$
(2a5 3p2 !m2n3
7. 3#!b12$ 8. #16a10$b8$ 9. #121x6$!b4 4|a5|b4 11|x3|
10. #(4k)4$ 11. -#169r4$ 12. !3#!27p$6$
16k2 (13r 2 3p2
13. !#625y2$z4$ 14. #36q34$ 15. #100x2$y4z6$!25|y |z2 6|q17| 10|x |y2|z3|
16. 3#!0.02$7$ 17. !#!0.36$ 18. #0.64p$10$!0.3 not a real number 0.8|p5|
19. 4#(2x)8$ 20. #(11y2)$4$ 21. 3#(5a2b)$6$4x2 121y4 25a4b2
22. #(3x !$ 1)2$ 23. 3#(m !$5)6$ 24. #36x2 !$ 12x %$ 1$|3x ! 1| (m ! 5)2 |6x ! 1|
© Glencoe/McGraw-Hill 264 Glencoe Algebra 2
Approximate Radicals with a Calculator
Irrational Number a number that cannot be expressed as a terminating or a repeating decimal
Radicals such as #2$ and #3$ are examples of irrational numbers. Decimal approximationsfor irrational numbers are often used in applications. These approximations can be easilyfound with a calculator.
Approximate 5!18.2" with a calculator.
5#18.2$ & 1.787
Use a calculator to approximate each value to three decimal places.
1. #62$ 2. #1050$ 3. 3#0.054$7.874 32.404 0.378
4. !4#5.45$ 5. #5280$ 6. #18,60$0$
!1.528 72.664 136.382
7. #0.095$ 8. 3#!15$ 9. 5#100$0.308 !2.466 2.512
10. 6#856$ 11. #3200$ 12. #0.05$3.081 56.569 0.224
13. #12,50$0$ 14. #0.60$ 15. !4#500$
111.803 0.775 !4.729
16. 3#0.15$ 17. 6#4200$ 18. #75$0.531 4.017 8.660
19. LAW ENFORCEMENT The formula r " 2#5L$ is used by police to estimate the speed rin miles per hour of a car if the length L of the car’s skid mark is measures in feet.Estimate to the nearest tenth of a mile per hour the speed of a car that leaves a skidmark 300 feet long. 77.5 mi/h
20. SPACE TRAVEL The distance to the horizon d miles from a satellite orbiting h milesabove Earth can be approximated by d " #8000h$ % h2$. What is the distance to thehorizon if a satellite is orbiting 150 miles above Earth? about 1100 ft
Study Guide and Intervention (continued)
Roots of Real Numbers
NAME ______________________________________________ DATE ____________ PERIOD _____
5-55-5
ExampleExample
ExercisesExercises
Skills PracticeRoots of Real Numbers
NAME ______________________________________________ DATE ____________ PERIOD _____
5-55-5
© Glencoe/McGraw-Hill 265 Glencoe Algebra 2
Less
on
5-5
Use a calculator to approximate each value to three decimal places.
1. #230$ 15.166 2. #38$ 6.164
3. !#152$ !12.329 4. #5.6$ 2.366
5. 3#88$ 4.448 6. 3#!222$ !6.055
7. !4#0.34$ !0.764 8. 5#500$ 3.466
Simplify.
9. -#81$ (9 10. #144$ 12
11. #(!5)2$ 5 12. #!52$ not a real number
13. #0.36$ 0.6 14. !'( !
15. 3#!8$ !2 16. !3#27$ !3
17. 3#0.064$ 0.4 18. 5#32$ 2
19. 4#81$ 3 20. #y2$ |y |
21. 3#125s3$ 5s 22. #64x6$ 8|x3|
23. 3#!27a$6$ !3a2 24. #m8n4$ m4n2
25. !#100p4$q2$ !10p2|q | 26. 4#16w4v$8$ 2|w |v2
27. #(!3c)4$ 9c2 28. #(a % b$)2$ |a $ b |
2"3
4&9
© Glencoe/McGraw-Hill 266 Glencoe Algebra 2
Use a calculator to approximate each value to three decimal places.
1. #7.8$ 2. !#89$ 3. 3#25$ 4. 3#!4$2.793 !9.434 2.924 !1.587
5. 4#1.1$ 6. 5#!0.1$ 7. 6#5555$ 8. 4#(0.94)$2$1.024 !0.631 4.208 0.970
Simplify.
9. #0.81$ 10. !#324$ 11. !4#256$ 12. 6#64$
0.9 !18 !4 213. 3#!64$ 14. 3#0.512$ 15. 5#!243$ 16. !
4#1296$!4 0.8 !3 !6
17. 5'( 18. 5#243x1$0$ 19. #(14a)2$ 20. #!(14a$)2$ not a
!"43" 3x2 14|a| real number
21. #49m2t$8$ 22. '( 23. 3#!64r6$w15$ 24. #(2x)8$
7|m |t4 !4r2w5 16x4
25. !4#625s8$ 26. 3#216p3$q9$ 27. #676x4$y6$ 28. 3#!27x9$y12$
!5s2 6pq3 26x2|y3| !3x3y4
29. !#144m$8n6$ 30. 5#!32x5$y10$ 31. 6#(m %$4)6$ 32. 3#(2x %$ 1)3$!12m4|n3| !2xy2 |m $ 4| 2x $ 1
33. !#49a10$b16$ 34. 4#(x ! 5$)8$ 35. 3#343d6$ 36. #x2 % 1$0x % 2$5$!7|a5|b8 (x ! 5)2 7d2 |x $ 5|
37. RADIANT TEMPERATURE Thermal sensors measure an object’s radiant temperature,which is the amount of energy radiated by the object. The internal temperature of an object is called its kinetic temperature. The formula Tr " Tk
4#e$ relates an object’s radianttemperature Tr to its kinetic temperature Tk. The variable e in the formula is a measureof how well the object radiates energy. If an object’s kinetic temperature is 30°C and e " 0.94, what is the object’s radiant temperature to the nearest tenth of a degree?29.5)C
38. HERO’S FORMULA Salvatore is buying fertilizer for his triangular garden. He knowsthe lengths of all three sides, so he is using Hero’s formula to find the area. Hero’sformula states that the area of a triangle is #s(s !$a)(s !$ b)(s !$ c)$, where a, b, and c arethe lengths of the sides of the triangle and s is half the perimeter of the triangle. If thelengths of the sides of Salvatore’s garden are 15 feet, 17 feet, and 20 feet, what is thearea of the garden? Round your answer to the nearest whole number. 124 ft2
4|m|"5
16m2&25
!1024&243
Practice (Average)
Roots of Real Numbers
NAME ______________________________________________ DATE ____________ PERIOD _____
5-55-5
Reading to Learn MathematicsRoots of Real Numbers
NAME ______________________________________________ DATE ____________ PERIOD _____
5-55-5
© Glencoe/McGraw-Hill 267 Glencoe Algebra 2
Less
on
5-5
Pre-Activity How do square roots apply to oceanography?
Read the introduction to Lesson 5-5 at the top of page 245 in your textbook.
Suppose the length of a wave is 5 feet. Explain how you would estimate thespeed of the wave to the nearest tenth of a knot using a calculator. (Do notactually calculate the speed.) Sample answer: Using a calculator,find the positive square root of 5. Multiply this number by 1.34.Then round the answer to the nearest tenth.
Reading the Lesson
1. For each radical below, identify the radicand and the index.
a. 3#23$ radicand: index:
b. #15x2$ radicand: index:
c. 5#!343$ radicand: index:
2. Complete the following table. (Do not actually find any of the indicated roots.)
Number Number of Positive Number of Negative Number of Positive Number of NegativeSquare Roots Square Roots Cube Roots Cube Roots
27 1 1 1 0
!16 0 0 0 1
3. State whether each of the following is true or false.
a. A negative number has no real fourth roots. true
b. -#121$ represents both square roots of 121. true
c. When you take the fifth root of x5, you must take the absolute value of x to identifythe principal fifth root. false
Helping You Remember
4. What is an easy way to remember that a negative number has no real square roots buthas one real cube root? Sample answer: The square of a positive or negativenumber is positive, so there is no real number whose square is negative.However, the cube of a negative number is negative, so a negativenumber has one real cube root, which is a negative number.
5!343
215x2
323
© Glencoe/McGraw-Hill 268 Glencoe Algebra 2
Approximating Square RootsConsider the following expansion.
(a % &2ba&)2
" a2 % &22aab
& % &4ba
2
2&
" a2 % b % &4ba
2
2&
Think what happens if a is very great in comparison to b. The term &4ba
2
2& is very small and can be disregarded in an approximation.
(a % &2ba&)2
) a2 % b
a % &2ba& ) #a2 % b$
Suppose a number can be expressed as a2 % b, a , b. Then an approximate value
of the square root is a % &2ba&. You should also see that a ! &2
ba& ) #a2 ! b$.
Use the formula !a2 ("b" # a ( "2ba" to approximate !101" and !622".
a. #101$ " #100 %$ 1$ " #102 %$ 1$ b. #622$ " #625 !$ 3$ " #252 !$ 3$
Let a " 10 and b " 1. Let a " 25 and b " 3.
#101$ ) 10 % &2(110)& #622$ ) 25 ! &2(
325)&
) 10.05 ) 24.94
Use the formula to find an approximation for each square root to the nearest hundredth. Check your work with a calculator.
1. #626$ 2. #99$ 3. #402$
4. #1604$ 5. #223$ 6. #80$
7. #4890$ 8. #2505$ 9. #3575$
10. #1,441$,100$ 11. #290$ 12. #260$
13. Show that a ! &2ba& ) #a2 ! b$ for a , b. $
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
5-55-5
ExampleExample
Study Guide and InterventionRadical Expressions
NAME ______________________________________________ DATE ____________ PERIOD _____
5-65-6
© Glencoe/McGraw-Hill 269 Glencoe Algebra 2
Less
on
5-6
Simplify Radical Expressions
For any real numbers a and b, and any integer n , 1:Product Property of Radicals 1. if n is even and a and b are both nonnegative, then n#ab$ "
n#a$ $n#b$.
2. if n is odd, then n#ab$ "n#a$ $
n#b$.
To simplify a square root, follow these steps:1. Factor the radicand into as many squares as possible.2. Use the Product Property to isolate the perfect squares.3. Simplify each radical.
Quotient Property of RadicalsFor any real numbers a and b # 0, and any integer n , 1, n'( " , if all roots are defined.
To eliminate radicals from a denominator or fractions from a radicand, multiply thenumerator and denominator by a quantity so that the radicand has an exact root.
n#a$&n#b$
a&b
Simplify 3!!16a"5b7".
3#!16a$5b7$ "3#(!2)3$$ 2 $ a$3 $ a2$$ (b2)3$ $ b$
" !2ab2 3#2a2b$
Simplify %&.
'( " Quotient Property
" Factor into squares.
" Product Property
" Simplify.
" $ Rationalize thedenominator.
" Simplify.2| x|#10xy$&&
15y3
#5y$y$
2| x|#2x$&3y2#5y$
2| x|#2x$&3y2#5y$
#(2x)2$ $ #2x$&&#(3y2)2$ $ #5y$
#(2x)2 $$ 2x$&&#(3y2)2$ $ 5y$
#8x3$-y5$
8x3&45y5
8x3"45y5Example 1Example 1 Example 2Example 2
ExercisesExercises
Simplify.
1. 5#54$ 15!6" 2. 4#32a9b$20$ 2a2|b5| 4!2a" 3. #75x4y$7$ 5x2y3!5y"
4. '( 5. '( 6. 3'( pq 3!5p2"""10
p5q3&40
|a3|b!2b"""14a6b3&98
6!5""2536&125
© Glencoe/McGraw-Hill 270 Glencoe Algebra 2
Operations with Radicals When you add expressions containing radicals, you canadd only like terms or like radical expressions. Two radical expressions are called likeradical expressions if both the indices and the radicands are alike.
To multiply radicals, use the Product and Quotient Properties. For products of the form (a#b$ % c#d$) $ (e#f$ % g#h$), use the FOIL method. To rationalize denominators, use conjugates. Numbers of the form a#b$ % c#d$ and a#b$ ! c#d$, where a, b, c, and d arerational numbers, are called conjugates. The product of conjugates is always a rationalnumber.
Simplify 2!50" $ 4!500" ! 6!125".
2#50$ % 4#500$ ! 6#125$ " 2#52 $ 2$ % 4#102 $ 5$ ! 6#52 $ 5$ Factor using squares.
" 2 $ 5 $ #2$ % 4 $ 10 $ #5$ ! 6 $ 5 $ #5$ Simplify square roots.
" 10#2$ % 40#5$ ! 30#5$ Multiply.
" 10#2$ % 10#5$ Combine like radicals.
Study Guide and Intervention (continued)
Radical Expressions
NAME ______________________________________________ DATE ____________ PERIOD _____
5-65-6
Example 1Example 1
Simplify (2!3" ! 4!2")(!3" $ 2!2").(2#3$ ! 4#2$)(#3$ % 2#2$)
" 2#3$ $ #3$ % 2#3$ $ 2#2$ ! 4#2$ $ #3$ ! 4#2$ $ 2#2$" 6 % 4#6$ ! 4#6$ ! 16" !10
Simplify .
" $
"
"
"11 ! 5#5$&&4
6 ! 5#5$ % 5&&9 ! 5
6 ! 2#5$ ! 3#5$ % (#5$ )2&&&
32 ! (#5$)2
3 ! #5$&3 ! #5$
2 ! #5$&3 % #5$
2 ! #5$&3 % #5$
2 ! !5""3 $ !5"
Example 2Example 2 Example 3Example 3
ExercisesExercises
Simplify.
1. 3#2$ % #50$ ! 4#8$ 2. #20$ % #125$ ! #45$ 3. #300$ ! #27$ ! #75$
0 4!5" 2!3"
4. 3#81$ $3#24$ 5. 3#2$( 3#4$ %
3#12$) 6. 2#3$(#15$ % #60$ )
6 3!9" 2 $ 2 3!3" 18!5"
7. (2 % 3#7$)(4 % #7$) 8. (6#3$ ! 4#2$)(3#3$ % #2$) 9. (4#2$ ! 3#5$)(2#20 %$5$)
29 $ 14!7" 46 ! 6!6" 40!2" ! 30!5"
10. 5 11. 5 $ 3!2" 12. 13!3" ! 23""115 ! 3#3$&&1 % 2#3$
4 % #2$&2 ! #2$
5#48$ % #75$&&
5#3$
Skills PracticeRadical Expressions
NAME ______________________________________________ DATE ____________ PERIOD _____
5-65-6
© Glencoe/McGraw-Hill 271 Glencoe Algebra 2
Less
on
5-6
Simplify.
1. #24$ 2!6" 2. #75$ 5!3"
3. 3#16$ 2 3!2" 4. !4#48$ !2 4!3"
5. 4#50x5$ 20x2!2x" 6. 4#64a4b$4$ 2|ab | 4!4"
7. 3'(! d2f5 ! f 3!d 2f 2" 8. '(s2t |s |!t"
9. !'( ! 10. 3'(
11. '(&25gz3& 12. (3#3$)(5#3$) 45
13. (4#12$ )(3#20$ ) 48!15" 14. #2$ % #8$ % #50$ 8!2"
15. #12$ ! 2#3$ % #108$ 6!3" 16. 8#5$ ! #45$ ! #80$ !5"
17. 2#48$ ! #75$ ! #12$ !3" 18. (2 % #3$)(6 ! #2$) 12 ! 2!2" $ 6!3" ! !6"
19. (1 ! #5$)(1 % #5$) !4 20. (3 ! #7$)(5 % #2$) 15 $ 3!2" ! 5!7" ! !14"
21. (#2$ ! #6$)2 8 ! 4!3" 22.
23. 24. 40 $ 5!6"""585
&8 ! #6$
12 ! 4!2"""74
&3 % #2$
21 $ 3!2"""473
&7 ! #2$
g!10gz"""5z
3!6""32&9
!21""73&7
5"6
25&36
1"2
1&8
© Glencoe/McGraw-Hill 272 Glencoe Algebra 2
Simplify.
1. #540$ 6!15" 2. 3#!432$ !6 3!2" 3. 3#128$ 4 3!2"
4. !4#405$ !3 4!5" 5. 3#!500$0$ !10 3!5" 6. 5#!121$5$ !3 5!5"
7. 3#125t6w$2$ 5t 2 3!w2" 8. 4#48v8z$13$ 2v2z3 4!3z" 9. 3#8g3k8$ 2gk2 3!k2"
10. #45x3y$8$ 3xy4!5x" 11. '( 12. 3'( 3!9"
13. '(c4d7 c2d 3!2d" 14. '( 15. 4'(16. (3#15$ )(!4#45$ ) 17. (2#24$ )(7#18$ ) 18. #810$ % #240$ ! #250$
!180!3" 168!3" 4!10" $ 4!15"
19. 6#20$ % 8#5$ ! 5#45$ 20. 8#48$ ! 6#75$ % 7#80$ 21. (3#2$ % 2#3$)2
5!5" 2!3" $ 28!5" 30 $ 12!6"
22. (3 ! #7$)2 23. (#5$ ! #6$)(#5$ % #2$) 24. (#2$ % #10$ )(#2$ ! #10$ )16 ! 6!7" 5 $ !10" ! !30" ! 2!3" !8
25. (1 % #6$)(5 ! #7$) 26. (#3$ % 4#7$)2 27. (#108$ ! 6#3$)2
5 ! !7" $ 5!6" ! !42" 115 $ 8!21" 0
28. !15" $ 2!3" 29. 6!2" $ 6 30.
31. 32. 27 $ 11!6" 33.
34. BRAKING The formula s " 2#5!$ estimates the speed s in miles per hour of a car whenit leaves skid marks ! feet long. Use the formula to write a simplified expression for s if ! " 85. Then evaluate s to the nearest mile per hour. 10!17"; 41 mi/h
35. PYTHAGOREAN THEOREM The measures of the legs of a right triangle can berepresented by the expressions 6x2y and 9x2y. Use the Pythagorean Theorem to find asimplified expression for the measure of the hypotenuse. 3x2 |y |!13"
6 $ 5!x" $ x""4 ! x3 % #x$&2 ! #x$
3 % #6$&&5 ! #24$
8 $ 5!2"""2
3 % #2$&2 ! #2$
17 ! !3"""135 % #3$&4 % #3$
6$ ! 1
#3$$ ! 2
4!72a""3a8
&9a33a2!a""8b2
9a5&64b4
1"16
1&128
216&24
!11""311&9
Practice (Average)
Radical Expressions
NAME ______________________________________________ DATE ____________ PERIOD _____
5-65-6
Reading to Learn MathematicsRadical Expressions
NAME ______________________________________________ DATE ____________ PERIOD _____
5-65-6
© Glencoe/McGraw-Hill 273 Glencoe Algebra 2
Less
on
5-6
Pre-Activity How do radical expressions apply to falling objects?
Read the introduction to Lesson 5-6 at the top of page 250 in your textbook.
Describe how you could use the formula given in your textbook and acalculator to find the time, to the nearest tenth of a second, that it wouldtake for the water balloons to drop 22 feet. (Do not actually calculate thetime.) Sample answer: Multiply 22 by 2 (giving 44) and divideby 32. Use the calculator to find the square root of the result.Round this square root to the nearest tenth.
Reading the Lesson
1. Complete the conditions that must be met for a radical expression to be in simplified form.
• The n is as as possible.
• The contains no (other than 1) that are nth
powers of a(n) or polynomial.
• The radicand contains no .
• No appear in the .
2. a. What are conjugates of radical expressions used for? to rationalize binomialdenominators
b. How would you use a conjugate to simplify the radical expression ?
Multiply numerator and denominator by 3 $ !2".c. In order to simplify the radical expression in part b, two multiplications are
necessary. The multiplication in the numerator would be done by themethod, and the multiplication in the denominator would be done by finding the
of .
Helping You Remember
3. One way to remember something is to explain it to another person. When rationalizing the
denominator in the expression , many students think they should multiply numerator
and denominator by . How would you explain to a classmate why this is incorrect
and what he should do instead. Sample answer: Because you are working withcube roots, not square roots, you need to make the radicand in thedenominator a perfect cube, not a perfect square. Multiply numerator and denominator by to make the denominator 3!8", which equals 2.
3!4""3!4"
3#2$&3#2$
1&3#2$
squarestwodifference
FOIL
1 % #2$&3 ! #2$
denominatorradicalsfractions
integerfactorsradicand
smallindex
© Glencoe/McGraw-Hill 274 Glencoe Algebra 2
Special Products with RadicalsNotice that (#3$)(#3$) " 3, or (#3$)2 " 3.
In general, (#x$)2 " x when x . 0.
Also, notice that (#9$)(#4$) " #36$.
In general, (#x$)(#y$) " #xy$ when x and y are not negative.
You can use these ideas to find the special products below.
(#a$ % #b$)(#a$ ! #b$) " (#a$)2 ! (#b$)2 " a ! b
(#a$ % #b$)2 " (#a$)2 % 2#ab$ % (#b$)2 " a % 2#ab$ % b
(#a$ ! #b$)2 " (#a$)2 ! 2#ab$ % (#b$)2 " a ! 2#ab$ % b
Find the product: (!2" $ !5")(!2" ! !5").(#2$ % #5$)(#2$ ! #5$) " (#2$)2 ! (#5$)2 " 2 ! 5 " !3
Evaluate (!2" $ !8")2.
(#2$ % #8$)2 " (#2$)2 % 2#2$#8$ % (#8$)2
" 2 % 2#16$ % 8 " 2 % 2(4) % 8 " 2 % 8 % 8 " 18
Multiply.
1. (#3$ ! #7$)(#3$ % #7$) 2. (#10$ % #2$)(#10$ ! #2$)
3. (#2x$ ! #6$)(#2x$ ! #6$) 4. (#3$ ! 27)2
5. (#1000$ % #10$ )2 6. (#y$ % #5$)(#y$ ! #5$)
7. (#50$ ! #x$)2 8. (#x$ % 20)2
You can extend these ideas to patterns for sums and differences of cubes.Study the pattern below.
( 3#8$ ! 3#x$)( 3#82$ %
3#8x$ %3#x2$) "
3#83$ ! 3#x3$ " 8 ! x
Multiply.
9. ( 3#2$ !3#5$)( 3#22$ %
3#10$ %3#52$)
10. ( 3#y$ %3#w$)( 3#y2$ !
3#yw$ %3#w2$ )
11. ( 3#7$ %3#20$ )( 3#72$ !
3#140$ %3#202$ )
12. ( 3#11$ !3#8$)( 3#112$ %
3#88$ %3#82$)
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
5-65-6
Example 1Example 1
Example 2Example 2
Study Guide and InterventionRational Exponents
NAME ______________________________________________ DATE ____________ PERIOD _____
5-75-7
© Glencoe/McGraw-Hill 275 Glencoe Algebra 2
Less
on
5-7
Rational Exponents and Radicals
Definition of b"n1" For any real number b and any positive integer n,
b&1n&
"n#b$, except when b ) 0 and n is even.
Definition of b"mn" For any nonzero real number b, and any integers m and n, with n , 1,
b&mn&
"n#bm$ " ( n#b$)m, except when b ) 0 and n is even.
Write 28"12" in radical form.
Notice that 28 , 0.
28&12
&" #28$" #22 $ 7$" #22$ $ #7$" 2#7$
Evaluate $ '"13"
.
Notice that !8 ) 0, !125 ) 0, and 3 is odd.
! "&13
&"
"
"2&5
!2&!5
3#!8$&3#!125$
!8&!125
!8"!125
Example 1Example 1 Example 2Example 2
ExercisesExercises
Write each expression in radical form.
1. 11&17
&2. 15
&13
&3. 300
&32
&
7!11" 3!15" !3003"
Write each radical using rational exponents.
4. #47$ 5. 3#3a5b2$ 6. 4#162p5$
47"12" 3"
13"a"
53"b"
23" 3 * 2"
14"
* p"54"
Evaluate each expression.
7. !27&23
&8. 9. (0.0004)
&12
&
9 0.02
10. 8&23
&$ 4
&32
&11. 12.
32 1"2
1"4
16!&12
&
&(0.25)
&12
&
144!&12
&
&
27!&13
&
1"10
5!&12
&
&2#5$
© Glencoe/McGraw-Hill 276 Glencoe Algebra 2
Simplify Expressions All the properties of powers from Lesson 5-1 apply to rationalexponents. When you simplify expressions with rational exponents, leave the exponent inrational form, and write the expression with all positive exponents. Any exponents in thedenominator must be positive integers
When you simplify radical expressions, you may use rational exponents to simplify, but youranswer should be in radical form. Use the smallest index possible.
Study Guide and Intervention (continued)
Rational Exponents
NAME ______________________________________________ DATE ____________ PERIOD _____
5-75-7
Simplify y"23"
* y"38".
y&23
&$ y
&38
&" y
&23
& % &38
&" y
&2254&
Simplify 4!144x6".
4#144x6$ " (144x6)&14
&
" (24 $ 32 $ x6)&14
&
" (24)&14
&$ (32)&
14
&$ (x6)&
14
&
" 2 $ 3&12
&$ x
&32
&" 2x $ (3x)&
12
&" 2x#3x$
Example 1Example 1 Example 2Example 2
ExercisesExercises
Simplify each expression.
1. x&45
&$ x
&65
& 2. !y&23
&" &34
& 3. p&45
&$ p
&170&
x2 y "12" p"
32"
4. !m!&65
&" &25
& 5. x!&38
&$ x
&43
& 6. !s!&16
&"!&43
&
x"2234" s"
29"
7. 8. !a&23
&" &65
&$ !a&
25
&"3 9.
p"23" a2
10. 6#128$ 11. 4#49$ 12. 5#288$
2!2" !7" 2 5!9"
13. #32$ $ 3#16$ 14. 3#25$ $ #125$ 15. 6#16$
48!2" 25 6!5" 3!4"
16. 17. #$3#48$ 18.
6!48" !a" 6!b5""bx!3" !
6!35"""6
a3#b4$
&#ab3$
x !3#3$
$
x"56
"
"x
x!&12
&
&x!&
13
&
p&p
&13
&
1"m3
Skills PracticeRational Exponents
NAME ______________________________________________ DATE ____________ PERIOD _____
5-75-7
© Glencoe/McGraw-Hill 277 Glencoe Algebra 2
Less
on
5-7
Write each expression in radical form.
1. 3&16
& 6!3" 2. 8&15
& 5!8"
3. 12&23
& 3!122" or ( 3!12")24. (s3)&
35
& s 5!s4"
Write each radical using rational exponents.
5. #51$ 51"12"
6. 3#37$ 37"13"
7. 4#153$ 15"34"
8. 3#6xy2$ 6"13"x"
13"y"
23"
Evaluate each expression.
9. 32&15
& 2 10. 81&14
& 3
11. 27!&13
& 12. 4!&12
&
13. 16&32
& 64 14. (!243)&45
& 81
15. 27&13
&$ 27&
53
& 729 16. ! "&32
&
Simplify each expression.
17. c&152&
$ c&35
& c3 18. m&29
&$ m
&196& m2
19. !q&12
&"3 q"32"
20. p!&15
&
21. x!&161& 22. x"1
52"
23. 24.
25.12#64$ !2" 26. 8#49a8b$2$ |a | 4!7b"
n"23
"
"nn
&13
&
&
n&16
&$ n
&12
&
y"14
"
"yy!&
12
&
&
y&14
&
x&23
&
&
x&14
&
x"151"
"x
p"45
"
"p
8"27
4&9
1"2
1"3
© Glencoe/McGraw-Hill 278 Glencoe Algebra 2
Write each expression in radical form.
1. 5&13
& 2. 6&25
& 3. m&47
& 4. (n3)&25
&
3!5" 5!62" or ( 5!6")2 7!m4" or ( 7!m")4 n 5!n"
Write each radical using rational exponents.
5. #79$ 6. 4#153$ 7. 3#27m6n$4$ 8. 5#2a10b$
79"12" 153"
14" 3m2n"
43" 5 * 2"
12" |a5 |b"
12"
Evaluate each expression.
9. 81&14
& 3 10. 1024!&15
& 11. 8!&53
&
12. !256!&34
& ! 13. (!64)!&23
& 14. 27&13
&$ 27&
43
& 243
15. ! "&23
& 16. 17. !25&12
&"!!64!&13
&" !
Simplify each expression.
18. g&47
&$ g
&37
& g 19. s&34
&$ s
&143& s4 20. !u!&
13
&"!&45
& u"145" 21. y!&
12
&
22. b!&35
& 23. q "15"
24. 25.
26.10#85$ 2!2" 27. #12$ $
5#123$ 28. 4#6$ $ 3 4#6$ 29.
1210!12" 3!6"
30. ELECTRICITY The amount of current in amperes I that an appliance uses can be
calculated using the formula I " ! "&12
&, where P is the power in watts and R is the
resistance in ohms. How much current does an appliance use if P " 500 watts and R " 10 ohms? Round your answer to the nearest tenth. 7.1 amps
31. BUSINESS A company that produces DVDs uses the formula C " 88n&13
&% 330 to
calculate the cost C in dollars of producing n DVDs per day. What is the company’s costto produce 150 DVDs per day? Round your answer to the nearest dollar. $798
P&R
a!3b""3ba
b$
2z $ 2z"12
"
""z ! 12z
&12
&
&
z&12
&! 1
t"11
12"
"5t
&23
&
&
5t&12
&$ t!&
34
&
q&35
&
&q
&25
&
b"25
"
"b
y"12
"
"y
5"4
16"49
64&23
&
&
343&23
&
25"36
125&216
1"16
1"64
1"32
1"4
Practice (Average)
Rational Exponents
NAME ______________________________________________ DATE ____________ PERIOD _____
5-75-7
Reading to Learn MathematicsRational Exponents
NAME ______________________________________________ DATE ____________ PERIOD _____
5-75-7
© Glencoe/McGraw-Hill 279 Glencoe Algebra 2
Less
on
5-7
Pre-Activity How do rational exponents apply to astronomy?
Read the introduction to Lesson 5-7 at the top of page 257 in your textbook.
The formula in the introduction contains the exponent . What do you think it might mean to raise a number to the power?
Sample answer: Take the fifth root of the number and square it.
Reading the Lesson
1. Complete the following definitions of rational exponents.
• For any real number b and for any positive integer n, b&n1
&" except
when b and n is .
• For any nonzero real number b, and any integers m and n, with n ,
b&mn
&" " , except when b and
n is .
2. Complete the conditions that must be met in order for an expression with rationalexponents to be simplified.
• It has no exponents.
• It has no exponents in the .
• It is not a fraction.
• The of any remaining is the number possible.
3. Margarita and Pierre were working together on their algebra homework. One exercise
asked them to evaluate the expression 27&43
&. Margarita thought that they should raise 27 to the fourth power first and then take the cube root of the result. Pierre thought thatthey should take the cube root of 27 first and then raise the result to the fourth power.Whose method is correct? Both methods are correct.
Helping You Remember
4. Some students have trouble remembering which part of the fraction in a rationalexponent gives the power and which part gives the root. How can your knowledge ofinteger exponents help you to keep this straight? Sample answer: An integer exponent can be written as a rational exponent. For example, 23 & 2"
31".
You know that this means that 2 is raised to the third power, so thenumerator must give the power, and, therefore, the denominator mustgive the root.
leastradicalindexcomplex
denominatorfractionalnegative
even+ 0( n!b")mn
!bm", 1
even+ 0
n!b"
2&5
2&5
© Glencoe/McGraw-Hill 280 Glencoe Algebra 2
Lesser-Known Geometric FormulasMany geometric formulas involve radical expressions.
Make a drawing to illustrate each of the formulas given on this page.Then evaluate the formula for the given value of the variable. Round answers to the nearest hundredth.
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
5-75-7
1. The area of an isosceles triangle. Twosides have length a; the other side haslength c. Find A when a " 6 and c " 7.
A " &4c
a2 !$ c2$A ( 17.06
3. The area of a regular pentagon with aside of length a. Find A when a " 4.
A " &a42 %$10#5$$
A ( 27.53
5. The volume of a regular tetrahedronwith an edge of length a. Find V when a " 2.
V " &1a23$
V ( 0.94
7. Heron’s Formula for the area of a triangle uses the semi-perimeter s,where s " &
a %2b % c&. The sides of the
triangle have lengths a, b, and c. Find Awhen a " 3, b " 4, and c " 5.
A " #s(s !$a)(s !$ b)(s !$ c)$A & 6
2. The area of an equilateral triangle witha side of length a. Find A when a " 8.
A " &a42$
A ( 27.71
4. The area of a regular hexagon with aside of length a. Find A when a " 9.
A " &32a2$
A ( 210.44
6. The area of the curved surface of a rightcone with an altitude of h and radius ofbase r. Find S when r " 3 and h " 6.
S " +r#r2 % h$2$S ( 63.22
8. The radius of a circle inscribed in a giventriangle also uses the semi-perimeter.Find r when a " 6, b " 7, and c " 9.
r "
r ( 1.91
#s(s !$a)(s !$ b)(s !$ c)$&&&s
Study Guide and InterventionRadical Equations and Inequalities
NAME ______________________________________________ DATE ____________ PERIOD _____
5-85-8
© Glencoe/McGraw-Hill 281 Glencoe Algebra 2
Less
on
5-8
Solve Radical Equations The following steps are used in solving equations that havevariables in the radicand. Some algebraic procedures may be needed before you use thesesteps.
Step 1 Isolate the radical on one side of the equation.Step 2 To eliminate the radical, raise each side of the equation to a power equal to the index of the radical.Step 3 Solve the resulting equation.Step 4 Check your solution in the original equation to make sure that you have not obtained any extraneous roots.
Solve 2!4x $"8" ! 4 & 8.
2#4x % 8$ ! 4 " 8 Original equation
2#4x % 8$ " 12 Add 4 to each side.
#4x % 8$ " 6 Isolate the radical.
4x % 8 " 36 Square each side.
4x " 28 Subtract 8 from each side.
x " 7 Divide each side by 4.
Check2#4(7) %$ 8$ ! 4 ! 8
2#36$ ! 4 ! 82(6) ! 4 ! 8
8 " 8The solution x " 7 checks.
Solve !3x $"1" & !5x" ! 1.
#3x % 1$ " #5x$ ! 1 Original equation
3x % 1 " 5x ! 2#5x$ % 1 Square each side.
2#5x$ " 2x Simplify.
#5x$ " x Isolate the radical.
5x " x2 Square each side.
x2 ! 5x " 0 Subtract 5x from each side.
x(x ! 5) " 0 Factor.
x " 0 or x " 5Check#3(0) %$ 1$ " 1, but #5(0)$ ! 1 " !1, so 0 isnot a solution.#3(5) %$ 1$ " 4, and #5(5)$ ! 1 " 4, so thesolution is x " 5.
Example 1Example 1 Example 2Example 2
ExercisesExercises
Solve each equation.
1. 3 % 2x#3$ " 5 2. 2#3x % 4$ % 1 " 15 3. 8 % #x % 1$ " 2
15 no solution
4. #5 ! x$ ! 4 " 6 5. 12 % #2x ! 1$ " 4 6. #12 !$x$ " 0
!95 no solution 12
7. #21$ ! #5x ! 4$ " 0 8. 10 ! #2x$ " 5 9. #x2 % 7$x$ " #7x ! 9$
5 12.5 no solution
10. 4 3#2x % 1$1$ ! 2 " 10 11. 2#x % 11$ " #x % 2$ % #3x ! 6$ 12. #9x ! 1$1$ " x % 1
8 14 3, 4
!3""3
© Glencoe/McGraw-Hill 282 Glencoe Algebra 2
Solve Radical Inequalities A radical inequality is an inequality that has a variablein a radicand. Use the following steps to solve radical inequalities.
Step 1 If the index of the root is even, identify the values of the variable for which the radicand is nonnegative.Step 2 Solve the inequality algebraically.Step 3 Test values to check your solution.
Solve 5 ! !20x $" 4" - !3.
Study Guide and Intervention (continued)
Radical Equations and Inequalities
NAME ______________________________________________ DATE ____________ PERIOD _____
5-85-8
ExampleExampleSince the radicand of a square rootmust be greater than or equal tozero, first solve20x % 4 . 0.20x % 4 . 0
20x . !4
x . !1&5
Now solve 5 ! #20x %$ 4$ . ! 3.
5 ! #20x %$ 4$ . !3 Original inequality
#20x %$ 4$ ( 8 Isolate the radical.
20x % 4 ( 64 Eliminate the radical by squaring each side.
20x ( 60 Subtract 4 from each side.
x ( 3 Divide each side by 20.
It appears that ! ( x ( 3 is the solution. Test some values.
x & !1 x & 0 x & 4
#20(!1$) % 4$ is not a real 5 ! #20(0)$% 4$ " 3, so the 5 ! #20(4)$% 4$ & !4.2, sonumber, so the inequality is inequality is satisfied. the inequality is notnot satisfied. satisfied
Therefore the solution ! ( x ( 3 checks.
Solve each inequality.
1. #c ! 2$ % 4 . 7 2. 3#2x ! 1$ % 6 ) 15 3. #10x %$ 9$ ! 2 , 5
c - 11 . x + 5 x , 4
4. 5 3#x % 2$ ! 8 ) 2 5. 8 ! #3x % 4$ . 3 6. #2x % 8$ ! 4 , 2
x + 6 ! . x . 7 x , 14
7. 9 ! #6x % 3$ . 6 8. ( 4 9. 2#5x ! 6$ ! 1 ) 5
! . x . 1 x - 8 . x + 3
10. #2x % 1$2$ % 4 . 12 11. #2d %$1$ % #d$ ( 5 12. 4#b % 3$ ! #b ! 2$ . 10
x - 26 0 . d . 4 b - 6
6"5
1"2
20&x % 1$
4"3
1"2
1&5
1&5
ExercisesExercises
Skills PracticeRadical Equations and Inequalities
NAME ______________________________________________ DATE ____________ PERIOD _____
5-85-8
© Glencoe/McGraw-Hill 283 Glencoe Algebra 2
Less
on
5-8
Solve each equation or inequality.
1. #x$ " 5 25 2. #x$ % 3 " 7 16
3. 5#j$ " 1 4. v&12
&% 1 " 0 no solution
5. 18 ! 3y&12
&" 25 no solution 6. 3#2w$ " 4 32
7. #b ! 5$ " 4 21 8. #3n %$1$ " 5 8
9. 3#3r ! 6$ " 3 11 10. 2 % #3p %$7$ " 6 3
11. #k ! 4$ ! 1 " 5 40 12. (2d % 3)&13
&" 2
13. (t ! 3)&13
&" 2 11 14. 4 ! (1 ! 7u)&
13
&" 0 !9
15. #3z ! 2$ " #z ! 4$ no solution 16. #g % 1$ " #2g !$7$ 8
17. #x ! 1$ " 4#x % 1$ no solution 18. 5 % #s ! 3$ ( 6 3 . s . 4
19. !2 % #3x % 3$ ) 7 !1 + x + 26 20. !#2a %$4$ . !6 !2 . a . 16
21. 2#4r ! 3$ , 10 r , 7 22. 4 ! #3x % 1$ , 3 ! + x + 0
23. #y % 4$ ! 3 . 3 y - 32 24. !3#11r %$ 3$ . !15 ! . r . 23"11
1"3
5"2
1"25
© Glencoe/McGraw-Hill 284 Glencoe Algebra 2
Solve each equation or inequality.
1. #x$ " 8 64 2. 4 ! #x$ " 3 1
3. #2p$ % 3 " 10 4. 4#3h$ ! 2 " 0
5. c&12
&% 6 " 9 9 6. 18 % 7h
&12
&" 12 no solution
7. 3#d % 2$ " 7 341 8. 5#w ! 7$ " 1 8
9. 6 % 3#q ! 4$ " 9 31 10. 4#y ! 9$ % 4 " 0 no solution
11. #2m !$ 6$ ! 16 " 0 131 12. 3#4m %$ 1$ ! 2 " 2
13. #8n !$5$ ! 1 " 2 14. #1 ! 4$t$ ! 8 " !6 !
15. #2t ! 5$ ! 3 " 3 16. (7v ! 2)&14
&% 12 " 7 no solution
17. (3g % 1)&12
&! 6 " 4 33 18. (6u ! 5)&
13
&% 2 " !3 !20
19. #2d !$5$ " #d ! 1$ 4 20. #4r ! 6$ " #r$ 2
21. #6x ! 4$ " #2x % 1$0$ 22. #2x % 5$ " #2x % 1$ no solution
23. 3#a$ . 12 a - 16 24. #z % 5$ % 4 ( 13 !5 . z . 76
25. 8 % #2q$ ( 5 no solution 26. #2a !$3$ ) 5 + a + 14
27. 9 ! #c % 4$ ( 6 c - 5 28. 3#x ! 1$ ) !2 x + !7
29. STATISTICS Statisticians use the formula ! " #v$ to calculate a standard deviation !,where v is the variance of a data set. Find the variance when the standard deviation is 15. 225
30. GRAVITATION Helena drops a ball from 25 feet above a lake. The formula
t " #25 !$h$ describes the time t in seconds that the ball is h feet above the water.
How many feet above the water will the ball be after 1 second? 9 ft
1&4
3"2
7"2
41"2
3"4
7"4
63"4
1"12
49"2
Practice (Average)
Radical Equations and Inequalities
NAME ______________________________________________ DATE ____________ PERIOD _____
5-85-8
Reading to Learn MathematicsRadical Equations and Inequalities
NAME ______________________________________________ DATE ____________ PERIOD _____
5-85-8
© Glencoe/McGraw-Hill 285 Glencoe Algebra 2
Less
on
5-8
Pre-Activity How do radical equations apply to manufacturing?
Read the introduction to Lesson 5-8 at the top of page 263 in your textbook.
Explain how you would use the formula in your textbook to find the cost ofproducing 125,000 computer chips. (Describe the steps of the calculation in theorder in which you would perform them, but do not actually do the calculation.)
Sample answer: Raise 125,000 to the power by taking the cube root of 125,000 and squaring the result (or raise 125,000 to the power by entering 125,000 ^ (2/3) on a calculator).Multiply the number you get by 10 and then add 1500.
Reading the Lesson
1. a. What is an extraneous solution of a radical equation? Sample answer: a numberthat satisfies an equation obtained by raising both sides of the originalequation to a higher power but does not satisfy the original equation
b. Describe two ways you can check the proposed solutions of a radical equation in orderto determine whether any of them are extraneous solutions. Sample answer: Oneway is to check each proposed solution by substituting it into theoriginal equation. Another way is to use a graphing calculator to graphboth sides of the original equation. See where the graphs intersect.This can help you identify solutions that may be extraneous.
2. Complete the steps that should be followed in order to solve a radical inequality.
Step 1 If the of the root is , identify the values of
the variable for which the is .
Step 2 Solve the algebraically.
Step 3 Test to check your solution.
Helping You Remember
3. One way to remember something is to explain it to another person. Suppose that yourfriend Leora thinks that she does not need to check her solutions to radical equations bysubstitution because she knows she is very careful and seldom makes mistakes in herwork. How can you explain to her that she should nevertheless check every proposedsolution in the original equation? Sample answer: Squaring both sides of anequation can produce an equation that is not equivalent to the originalone. For example, the only solution of x & 5 is 5, but the squaredequation x2 & 25 has two solutions, 5 and !5.
valuesinequality
nonnegativeradicandevenindex
2"3
2"3
© Glencoe/McGraw-Hill 286 Glencoe Algebra 2
Truth TablesIn mathematics, the basic operations are addition, subtraction, multiplication,division, finding a root, and raising to a power. In logic, the basic operations are the following: not (*), and (+), or (,), and implies (→).
If P and Q are statements, then *P means not P; *Q means not Q; P + Qmeans P and Q; P , Q means P or Q; and P → Q means P implies Q. The operations are defined by truth tables. On the left below is the truth table for the statement *P. Notice that there are two possible conditions for P, true (T) or false (F). If P is true, *P is false; if P is false, *P is true. Also shown are the truth tables for P + Q, P , Q, and P → Q.
P *P P Q P + Q P Q P , Q P Q P → QT F T T T T T T T T TF T T F F T F T T F F
F T F F T T F T TF F F F F F F F T
You can use this information to find out under what conditions a complex statement is true.
Under what conditions is )P * Q true?
Create the truth table for the statement. Use the information from the truth table above for P * Q to complete the last column.
P Q *P *P , QT T F TT F F FF T T TF F T T
The truth table indicates that *P , Q is true in all cases except where P is true and Q is false.
Use truth tables to determine the conditions under which each statement is true.
1. *P , *Q 2. *P → (P → Q)
3. (P , Q) , (*P + *Q) 4. (P → Q) , (Q → P)
5. (P → Q) + (Q → P) 6. (*P + *Q) → *(P , Q)
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
5-85-8
ExampleExample
To solve a quadratic equation that does not have real solutions, you can use the fact thati2 " !1 to find complex solutions.
Solve 2x2 $ 24 & 0.
2x2 % 24 " 0 Original equation
2x2 " !24 Subtract 24 from each side.
x2 " !12 Divide each side by 2.
x " -#!12$ Take the square root of each side.
x " -2i#3$ #!12$ " #4$ $ #!1$ $ #3$
Simplify.
1. (!4 % 2i) % (6 ! 3i) 2. (5 ! i) ! (3 ! 2i) 3. (6 ! 3i) % (4 ! 2i)2 ! i 2 $ i 10 ! 5i
4. (!11 % 4i) ! (1 ! 5i) 5. (8 % 4i) % (8 ! 4i) 6. (5 % 2i) ! (!6 ! 3i)!12 $ 9i 16 11 $ 5i
7. (12 ! 5i) ! (4 % 3i) 8. (9 % 2i) % (!2 % 5i) 9. (15 ! 12i) % (11 ! 13i)8 ! 8i 7 $ 7i 26 ! 25i
10. i4 11. i6 12. i15
1 !1 !i
Solve each equation.
13. 5x2 % 45 " 0 14. 4x2 % 24 " 0 15. !9x2 " 9(3i (i!6" (i
Study Guide and InterventionComplex Numbers
NAME ______________________________________________ DATE ____________ PERIOD _____
5-95-9
© Glencoe/McGraw-Hill 287 Glencoe Algebra 2
Less
on
5-9
Add and Subtract Complex Numbers
A complex number is any number that can be written in the form a % bi, Complex Number where a and b are real numbers and i is the imaginary unit (i 2 " !1).
a is called the real part, and b is called the imaginary part.
Addition and Combine like terms.Subtraction of (a % bi) % (c % di) " (a % c) % (b % d )iComplex Numbers (a % bi) ! (c % di) " (a ! c) % (b ! d )i
Simplify (6 $ i) $ (4 ! 5i).
(6 % i) % (4 ! 5i)" (6 % 4) % (1 ! 5)i" 10 ! 4i
Simplify (8 $ 3i) ! (6 ! 2i).
(8 % 3i) ! (6 ! 2i)" (8 ! 6) % [3 ! (!2)]i" 2 % 5i
Example 1Example 1 Example 2Example 2
Example 3Example 3
ExercisesExercises
© Glencoe/McGraw-Hill 288 Glencoe Algebra 2
Multiply and Divide Complex Numbers
Multiplication of Complex Numbers Use the definition of i2 and the FOIL method:(a % bi)(c % di) " (ac ! bd ) % (ad % bc)i
To divide by a complex number, first multiply the dividend and divisor by the complexconjugate of the divisor.
Complex Conjugate a % bi and a ! bi are complex conjugates. The product of complex conjugates is always a real number.
Simplify (2 ! 5i) * (!4 $ 2i).
(2 ! 5i) $ (!4 % 2i)" 2(!4) % 2(2i) % (!5i)(!4) % (!5i)(2i) FOIL
" !8 % 4i % 20i ! 10i2 Multiply.
" !8 % 24i ! 10(!1) Simplify.
" 2 % 24i Standard form
Simplify .
" $ Use the complex conjugate of the divisor.
" Multiply.
" i2 " !1
" ! i Standard form
Simplify.
1. (2 % i)(3 ! i) 7 $ i 2. (5 ! 2i)(4 ! i) 18 ! 13i 3. (4 ! 2i)(1 ! 2i) !10i
4. (4 ! 6i)(2 % 3i) 26 5. (2 % i)(5 ! i) 11 $ 3i 6. (5 ! 3i)(!1 ! i) !8 ! 2i
7. (1 ! i)(2 % 2i)(3 ! 3i) 8. (4 ! i)(3 ! 2i)(2 % i) 9. (5 ! 2i)(1 ! i)(3 % i)12 ! 12i 31 ! 12i 16 ! 18i
10. ! i 11. ! ! i 12. ! ! 2i
13. 1 ! i 14. ! ! 2i 15. ! $ i
16. $ 17. !1 ! 2i!2" 18. $ 2i!6""3!3""3
#6$ % i#3$&&
#2$ ! i4 ! i#2$&&
i#2$3i!5""7
2"7
3 % i#5$&&3 ! i#5$
31"41
8"41
3 % 4i&4 ! 5i
1"2
!5 ! 3i&2 ! 2i
4 ! 2i&3 % i
5"3
6 ! 5i&3i
7"2
13"2
7 ! 13i&2i
1"2
3"2
5&3 % i
11&13
3&13
3 ! 11i&13
6 ! 9i ! 2i % 3i2&&&
4 ! 9i2
2 ! 3i&2 ! 3i
3 ! i&2 % 3i
3 ! i&2 % 3i
3 ! i"2 % 3i
Study Guide and Intervention (continued)
Complex Numbers
NAME ______________________________________________ DATE ____________ PERIOD _____
5-95-9
Example 1Example 1
Example 2Example 2
ExercisesExercises
Skills PracticeComplex Numbers
NAME ______________________________________________ DATE ____________ PERIOD _____
5-95-9
© Glencoe/McGraw-Hill 289 Glencoe Algebra 2
Less
on
5-9
Simplify.
1. #!36$ 6i 2. #!196$ 14i
3. #!81x6$ 9|x3|i 4. #!23$ $ #!46$ !23!2"
5. (3i)(!2i)(5i) 30i 6. i11 !i
7. i65 i 8. (7 ! 8i) % (!12 ! 4i) !5 ! 12i
9. (!3 % 5i) % (18 ! 7i) 15 ! 2i 10. (10 ! 4i) ! (7 % 3i) 3 ! 7i
11. (2 % i)(2 % 3i) 1 $ 8i 12. (2 % i)(3 ! 5i) 11 ! 7i
13. (7 ! 6i)(2 ! 3i) !4 ! 33i 14. (3 % 4i)(3 ! 4i) 25
15. 16.
Solve each equation.
17. 3x2 % 3 " 0 (i 18. 5x2 % 125 " 0 (5i
19. 4x2 % 20 " 0 (i!5" 20. !x2 ! 16 " 0 (4i
21. x2 % 18 " 0 (3i!2" 22. 8x2 % 96 " 0 (2i!3"
Find the values of m and n that make each equation true.
23. 20 ! 12i " 5m % 4ni 4, !3 24. m ! 16i " 3 ! 2ni 3, 8
25. (4 % m) % 2ni " 9 % 14i 5, 7 26. (3 ! n) % (7m ! 14)i " 1 % 7i 3, 2
3 $ 6i"10
3i&4 % 2i
!6 ! 8i"3
8 ! 6i&3i
© Glencoe/McGraw-Hill 290 Glencoe Algebra 2
Simplify.
1. #!49$ 7i 2. 6#!12$ 12i!3" 3. #!121$s8$ 11s4i
4. #!36a$3b4$ 5. #!8$ $ #!32$ 6. #!15$ $ #!25$6|a|b2i!a" !16 !5!15"
7. (!3i)(4i)(!5i) 8. (7i)2(6i) 9. i42
!60i !294i !1
10. i55 11. i89 12. (5 ! 2i) % (!13 ! 8i)!i i !8 ! 10i
13. (7 ! 6i) % (9 % 11i) 14. (!12 % 48i) % (15 % 21i) 15. (10 % 15i) ! (48 ! 30i)16 $ 5i 3 $ 69i !38 $ 45i
16. (28 ! 4i) ! (10 ! 30i) 17. (6 ! 4i)(6 % 4i) 18. (8 ! 11i)(8 ! 11i)18 $ 26i 52 !57 ! 176i
19. (4 % 3i)(2 ! 5i) 20. (7 % 2i)(9 ! 6i) 21.
23 ! 14i 75 ! 24i
22. 23. 24. !1 ! i
Solve each equation.
25. 5n2 % 35 " 0 (i!7" 26. 2m2 % 10 " 0 (i!5"
27. 4m2 % 76 " 0 (i!19" 28. !2m2 ! 6 " 0 (i!3"
29. !5m2 ! 65 " 0 (i!13" 30. x2 % 12 " 0 (4i
Find the values of m and n that make each equation true.
31. 15 ! 28i " 3m % 4ni 5, !7 32. (6 ! m) % 3ni " !12 % 27i 18, 9
33. (3m % 4) % (3 ! n)i " 16 ! 3i 4, 6 34. (7 % n) % (4m ! 10)i " 3 ! 6i 1, !4
35. ELECTRICITY The impedance in one part of a series circuit is 1 % 3j ohms and theimpedance in another part of the circuit is 7 ! 5j ohms. Add these complex numbers tofind the total impedance in the circuit. 8 ! 2 j ohms
36. ELECTRICITY Using the formula E " IZ, find the voltage E in a circuit when thecurrent I is 3 ! j amps and the impedance Z is 3 % 2j ohms. 11 $ 3j volts
3&4
2 ! 4i&1 % 3i
7 $ i"5
3 ! i&2 ! i
14 $ 16i""113
2&7 ! 8i
!5 $ 6i"2
6 % 5i&
!2i
Practice (Average)
Complex Numbers
NAME ______________________________________________ DATE ____________ PERIOD _____
5-95-9
Reading to Learn MathematicsComplex Numbers
NAME ______________________________________________ DATE ____________ PERIOD _____
5-95-9
© Glencoe/McGraw-Hill 291 Glencoe Algebra 2
Less
on
5-9
Pre-Activity How do complex numbers apply to polynomial equations?
Read the introduction to Lesson 5-9 at the top of page 270 in your textbook.
Suppose the number i is defined such that i2 " !1. Complete each equation.
2i2 " (2i)2 " i4 "
Reading the Lesson
1. Complete each statement.
a. The form a % bi is called the of a complex number.
b. In the complex number 4 % 5i, the real part is and the imaginary part is .
This is an example of a complex number that is also a(n) number.
c. In the complex number 3, the real part is and the imaginary part is .
This is example of complex number that is also a(n) number.
d. In the complex number 7i, the real part is and the imaginary part is .
This is an example of a complex number that is also a(n) number.
2. Give the complex conjugate of each number.
a. 3 % 7i
b. 2 ! i
3. Why are complex conjugates used in dividing complex numbers? The product ofcomplex conjugates is always a real number.
4. Explain how you would use complex conjugates to find (3 % 7i) * (2 ! i). Write thedivision in fraction form. Then multiply numerator and denominator by 2 $ i .
Helping You Remember
5. How can you use what you know about simplifying an expression such as to
help you remember how to simplify fractions with imaginary numbers in thedenominator? Sample answer: In both cases, you can multiply thenumerator and denominator by the conjugate of the denominator.
1 % #3$&2 ! #5$
2 $ i
3 ! 7i
pure imaginary 70
real03
imaginary54
standard form
1!4!2
© Glencoe/McGraw-Hill 292 Glencoe Algebra 2
Conjugates and Absolute ValueWhen studying complex numbers, it is often convenient to represent a complex number by a single variable. For example, we might let z " x % yi. We denote the conjugate of z by z$. Thus, z$ " x ! yi.
We can define the absolute value of a complex number as follows.
- z- " - x % yi- " #x2 % y$2$
There are many important relationships involving conjugates and absolute values of complex numbers.
Show +z +2 & zz" for any complex number z.
Let z " x % yi. Then,z " (x % yi)(x ! yi)
" x2 % y2
" #(x2 % y2$)2$" - z-2
Show is the multiplicative inverse for any nonzero
complex number z.
We know - z-2 " zz$. If z # 0, then we have z! " " 1.
Thus, is the multiplicative inverse of z.
For each of the following complex numbers, find the absolute value andmultiplicative inverse.
1. 2i 2. !4 ! 3i 3. 12 ! 5i
4. 5 ! 12i 5. 1 % i 6. #3$ ! i
7. % i 8. ! i 9. &12& ! i#3$
&2#2$&2
#2$&2
#3$&3
#3$&3
z$&- z-2
z$&- z-2
z""+z +2
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
5-95-9
Example 1Example 1
Example 2Example 2
© Glencoe/McGraw-Hill A2 Glencoe Algebra 2
Answers (Lesson 5-1)
Stu
dy
Gu
ide
and I
nte
rven
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Mon
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ls
NAM
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____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
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____
_
5-1
5-1
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and
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.
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itho
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0.
bn& an
b & aa & b
an& bn
a & bam& an
1& a!
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Exam
ple
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a.(3
m4 n
!2 )
(!5m
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(3m
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m2 n
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!3 )
217
.(4.
5 '
107 )
218
.(6.
8 '
10!
5 )2
1.02
4 #
10!
52.
025
#10
154.
624
#10
!9
19.A
STR
ON
OM
YP
luto
is 3
,674
.5 m
illio
n m
iles
from
the
sun
.Wri
te t
his
num
ber
insc
ient
ific
not
atio
n.So
urce
:New
York
Tim
es A
lman
ac3.
6745
#10
9m
iles
20.C
HEM
ISTR
YT
he b
oilin
g po
int
of t
he m
etal
tun
gste
n is
10,
220°
F.W
rite
thi
ste
mpe
ratu
re in
sci
enti
fic
nota
tion
.So
urce
:New
York
Tim
es A
lman
ac1.
022
#10
4
21.B
IOLO
GY
The
hum
an b
ody
cont
ains
0.0
004%
iodi
ne b
y w
eigh
t.H
ow m
any
poun
ds o
fio
dine
are
the
re in
a 1
20-p
ound
tee
nage
r? E
xpre
ss y
our
answ
er in
sci
enti
fic
nota
tion
.So
urce
:Uni
versa
l Alm
anac
4.8
#10
!4
lb
4.81
'10
8&
&6.
5 '
104
1.62
'10
!2
&&
1.8
'10
59.
5 '
107
&&
3.8
'10
!2
104
& 10!
23.
5&5
3.5
'10
4&
&5
'10
!2
3.5
'10
4"
"5
'10
!2
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Mon
omia
ls
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-1
5-1
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A3 Glencoe Algebra 2
An
swer
s
Answers (Lesson 5-1)
Skil
ls P
ract
ice
Mon
omia
ls
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-1
5-1
©G
lenc
oe/M
cGra
w-Hi
ll24
1G
lenc
oe A
lgeb
ra 2
Lesson 5-1
Sim
pli
fy.A
ssu
me
that
no
vari
able
equ
als
0.
1.b4
$b3
b72.
c5$
c2$
c2c9
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s3
13.
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2x)2
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264
x2y2
16.!
2gh(
g3h5
)!
2g4 h
6
17.1
0x2 y
3 (10
xy8 )
100x
3 y11
18.
19.
!20
.
Exp
ress
eac
h n
um
ber
in s
cien
tifi
c n
otat
ion
.
21.5
3,00
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3 #
104
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23.4
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luat
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lt i
n s
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c n
otat
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.
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4 '
103 )
(1.6
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6.4
#10
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26.
6.4
#10
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6 '
107
&&
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!10
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©G
lenc
oe/M
cGra
w-Hi
ll24
2G
lenc
oe A
lgeb
ra 2
Sim
pli
fy.A
ssu
me
that
no
vari
able
equ
als
0.
1.n5
$n2
n72.
y7$
y3$
y2y1
2
3.t9
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5 f"3
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d23 f
1916
. !"!
2
17.
18.
!
Exp
ress
eac
h n
um
ber
in s
cien
tifi
c n
otat
ion
.
19.8
96,0
0020
.0.0
0005
621
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108
8.96
#10
55.
6 #
10!
54.
337
#10
10
Eva
luat
e.E
xpre
ss t
he
resu
lt i
n s
cien
tifi
c n
otat
ion
.
22.(
4.8
'10
2 )(6
.9 '
104 )
23.(
3.7
'10
9 )(8
.7 '
102 )
24.
3.31
2 #
107
3.21
9 #
1012
3 #
10!
5
25.C
OM
PUTI
NG
The
ter
m b
it,s
hort
for
bin
ary
digi
t,w
as f
irst
use
d in
194
6 by
Joh
n T
ukey
.A
sin
gle
bit
hold
s a
zero
or
a on
e.So
me
com
pute
rs u
se 3
2-bi
t nu
mbe
rs,o
r st
ring
s of
32
con
secu
tive
bit
s,to
iden
tify
eac
h ad
dres
s in
the
ir m
emor
ies.
Eac
h 32
-bit
num
ber
corr
espo
nds
to a
num
ber
in o
ur b
ase-
ten
syst
em.T
he la
rges
t 32
-bit
num
ber
is n
earl
y4,
295,
000,
000.
Wri
te t
his
num
ber
in s
cien
tifi
c no
tati
on.
4.29
5 #
109
26.L
IGH
TW
hen
light
pas
ses
thro
ugh
wat
er,i
ts v
eloc
ity
is r
educ
ed b
y 25
%.I
f th
e sp
eed
oflig
ht in
a v
acuu
m is
1.8
6 '
105
mile
s pe
r se
cond
,at
wha
t ve
loci
ty d
oes
it t
rave
l thr
ough
wat
er?
Wri
te y
our
answ
er in
sci
enti
fic
nota
tion
. 1.3
95 #
105
mi/s
27
.TR
EES
Dec
iduo
us a
nd c
onif
erou
s tr
ees
are
hard
to
dist
ingu
ish
in a
bla
ck-a
nd-w
hite
phot
o.B
ut b
ecau
se d
ecid
uous
tre
es r
efle
ct in
frar
ed e
nerg
y be
tter
tha
n co
nife
rous
tre
es,
the
two
type
s of
tre
es a
re m
ore
dist
ingu
isha
ble
in a
n in
frar
ed p
hoto
.If
an in
frar
edw
avel
engt
h m
easu
res
abou
t 8
'10
!7
met
ers
and
a bl
ue w
avel
engt
h m
easu
res
abou
t 4.
5 '
10!
7m
eter
s,ab
out
how
man
y ti
mes
long
er is
the
infr
ared
wav
elen
gth
than
the
blue
wav
elen
gth?
abo
ut 1
.8 ti
mes
2.7
'10
6&
&9
'10
10
4v2
" m2
!20
(m2 v
)(!
v)3
&&
5(!
v)2 (
!m
4 )15
x11"
y3(3
x!2 y
3 )(5
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4 y!
2
y6" 4x
22x
3 y2
& !x2 y
54 & 3
3 & 2
256
" wz5
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4 s6 z
122
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3 z6
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9my4
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8c9
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Pra
ctic
e (A
vera
ge)
Mon
omia
ls
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-1
5-1
© Glencoe/McGraw-Hill A4 Glencoe Algebra 2
Answers (Lesson 5-1)
Rea
din
g t
o L
earn
Math
emati
csM
onom
ials
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-1
5-1
©G
lenc
oe/M
cGra
w-Hi
ll24
3G
lenc
oe A
lgeb
ra 2
Lesson 5-1
Pre-
Act
ivit
yW
hy
is s
cien
tifi
c n
otat
ion
use
ful
in e
con
omic
s?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 5-
1 at
the
top
of
page
222
in y
our
text
book
.
Your
tex
tboo
k gi
ves
the
U.S
.pub
lic d
ebt
as a
n ex
ampl
e fr
om e
cono
mic
s th
atin
volv
es la
rge
num
bers
tha
t ar
e di
ffic
ult
to w
ork
wit
h w
hen
wri
tten
inst
anda
rd n
otat
ion.
Giv
e an
exa
mpl
e fr
om s
cien
ce t
hat
invo
lves
ver
y la
rge
num
bers
and
one
tha
t in
volv
es v
ery
smal
l num
bers
.Sa
mpl
e an
swer
:di
stan
ces
betw
een
Eart
h an
d th
e st
ars,
size
s of
mol
ecul
es
and
atom
s
Rea
din
g t
he
Less
on
1.Te
ll w
heth
er e
ach
expr
essi
on is
a m
onom
ial o
r no
t a
mon
omia
l.If
it is
a m
onom
ial,
tell
whe
ther
it is
a c
onst
ant
or n
ot a
con
stan
t.
a.3x
2m
onom
ial;
not a
con
stan
tb.
y2%
5y!
6no
t a m
onom
ial
c.!
73m
onom
ial;
cons
tant
d.no
t a m
onom
ial
2.C
ompl
ete
the
follo
win
g de
fini
tion
s of
a n
egat
ive
expo
nent
and
a z
ero
expo
nent
.
For
any
real
num
ber
a#
0 an
d an
y in
tege
r n,
a!n
".
For
any
real
num
ber
a#
0,a0
".
3.N
ame
the
prop
erty
or
prop
erti
es o
f ex
pone
nts
that
you
wou
ld u
se t
o si
mpl
ify
each
expr
essi
on.(
Do
not
actu
ally
sim
plif
y.)
a.qu
otie
nt o
f pow
ers
b.y6
$y9
prod
uct o
f pow
ers
c.(3
r2s)
4po
wer
of a
pro
duct
and
pow
er o
f a p
ower
Hel
pin
g Y
ou
Rem
emb
er
4.W
hen
wri
ting
a n
umbe
r in
sci
enti
fic
nota
tion
,som
e st
uden
ts h
ave
trou
ble
rem
embe
ring
whe
n to
use
pos
itiv
e ex
pone
nts
and
whe
n to
use
neg
ativ
e on
es.W
hat
is a
n ea
sy w
ay t
ore
mem
ber
this
?Sa
mpl
e an
swer
:Use
a p
ositi
ve e
xpon
ent i
f the
num
ber i
s10
or g
reat
er.U
se a
neg
ativ
e nu
mbe
r if t
he n
umbe
r is
less
than
1.
m8
& m3
1
" a1 n"
1 & z
©G
lenc
oe/M
cGra
w-Hi
ll24
4G
lenc
oe A
lgeb
ra 2
Prop
ertie
s of
Exp
onen
tsT
he r
ules
abo
ut p
ower
s an
d ex
pone
nts
are
usua
lly g
iven
wit
h le
tter
s su
ch a
s m
,n,
and
kto
rep
rese
nt e
xpon
ents
.For
exa
mpl
e,on
e ru
le s
tate
s th
at a
m$
an"
am%
n .
In p
ract
ice,
such
exp
onen
ts a
re h
andl
ed a
s al
gebr
aic
expr
essi
ons
and
the
rule
s of
al
gebr
a ap
ply.
Sim
pli
fy 2
a2 (
an
$1
$a
4n).
2a2 (
an%
1%
a4n )
"2a
2$
an%
1%
2a2
$a4
nUs
e th
e Di
strib
utive
Law
.
"2a
2 %
n%
1%
2a2
%4n
Reca
ll am
$an
"am
%n .
"2a
n%
3%
2a2
%4n
Sim
plify
the
expo
nent
2 %
n%
1 as
n%
3.
It is
impo
rtan
t al
way
s to
col
lect
like
term
s on
ly.
Sim
pli
fy (
an
$bm
)2.
(an
%bm
)2"
(an
%bm
)(an
%bm
)F
OI
L"
an$
an%
an$
bm%
an$
bm%
bm$
bmTh
e se
cond
and
third
term
s ar
e lik
e te
rms.
" a
2n%
2an b
m%
b2m
Sim
pli
fy e
ach
exp
ress
ion
by
per
form
ing
the
ind
icat
ed o
per
atio
ns.
1.23
2m23
$m
2.(a
3 )n
a3n
3.(4
n b2 )
k4k
n b2k
4.(x
3 aj )
mx3
maj
m5.
(!ay
n )3
!a3
y3n
6.(!
bkx)
2!
b2k x
2
7.(c
2 )hk
c2hk
8.(!
2dn )
5!
32d5
n9.
(a2 b
)(an
b2)
a2 $
n b3
10.(
xnym
)(xm
yn)
11.
12.
xn$
myn
$m
an!
23x
3 !
n
13.(
ab2
!a2
b)(3
an%
4bn )
3an
$1 b
2$
4abn
$2
!3a
n$
2 b!
4a2 b
n$
1
14.a
b2(2
a2bn
!1
%4a
bn%
6bn
%1 )
2a3 b
n $
1$
4a2 b
n $
2$
6abn
$3
12x3
& 4xn
an& a2
En
rich
men
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-1
5-1
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
© Glencoe/McGraw-Hill A5 Glencoe Algebra 2
An
swer
s
Answers (Lesson 5-2)
Stu
dy
Gu
ide
and I
nte
rven
tion
Poly
nom
ials
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-2
5-2
©G
lenc
oe/M
cGra
w-Hi
ll24
5G
lenc
oe A
lgeb
ra 2
Lesson 5-2
Ad
d a
nd
Su
btr
act
Poly
no
mia
ls
Poly
nom
ial
a m
onom
ial o
r a s
um o
f mon
omia
ls
Like
Ter
ms
term
s th
at h
ave
the
sam
e va
riabl
e(s)
raise
d to
the
sam
e po
wer(s
)
To a
dd o
r su
btra
ct p
olyn
omia
ls,p
erfo
rm t
he in
dica
ted
oper
atio
ns a
nd c
ombi
ne li
ke t
erm
s.
Sim
pli
fy !
6rs
$18
r2!
5s2
! 1
4r2
$8r
s !
6s2 .
!6r
s%
18r2
!5s
2!
14r2
%8r
s!6s
2
"(1
8r2
!14
r2) %
(!6r
s%
8rs)
%(!
5s2
!6s
2 )G
roup
like
term
s."
4r2
%2r
s!
11s2
Com
bine
like
term
s.
Sim
pli
fy 4
xy2
$12
xy!
7x2 y
!(2
0xy
$5x
y2!
8x2 y
).
4xy2
%12
xy!
7x2 y
!(2
0xy
%5x
y2!
8x2 y
)"
4xy2
%12
xy!
7x2 y
!20
xy!
5xy2
%8x
2 yDi
strib
ute
the
min
us s
ign.
" (!
7x2 y
%8x
2 y) %
(4xy
2!
5xy2
) %(1
2xy
!20
xy)
Gro
up li
ke te
rms.
"x2
y!
xy2
!8x
yCo
mbi
ne li
ke te
rms.
Sim
pli
fy.
1.(6
x2!
3x%
2) !
(4x2
%x
!3)
2.(7
y2%
12xy
!5x
2 ) %
(6xy
!4y
2!
3x2 )
2x2
!4x
$5
3y2 $
18xy
!8x
2
3.(!
4m2
!6m
) !(6
m%
4m2 )
4.27
x2!
5y2
%12
y2!
14x2
!8m
2!
12m
13x2
$7y
2
5.(1
8p2
%11
pq!
6q2 )
!(1
5p2
!3p
q%
4q2 )
6.17
j2!
12k2
%3j
2!
15j2
%14
k2
3p2
$14
pq!
10q2
5j2
$2k
2
7.(8
m2
!7n
2 ) !
(n2
!12
m2 )
8.14
bc%
6b!
4c%
8b!
8c%
8bc
20m
2!
8n2
14b
$22
bc!
12c
9.6r
2 s%
11rs
2%
3r2 s
!7r
s2%
15r2
s!
9rs2
10.!
9xy
%11
x2!
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Exam
ple2
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Exer
cises
Exer
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©G
lenc
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6G
lenc
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lgeb
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Mu
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.Whe
n m
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ng b
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o bi
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add
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e fir
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)
Poly
nom
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NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-2
5-2
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A6 Glencoe Algebra 2
Answers (Lesson 5-2)
Skil
ls P
ract
ice
Poly
nom
ials
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-2
5-2
©G
lenc
oe/M
cGra
w-Hi
ll24
7G
lenc
oe A
lgeb
ra 2
Lesson 5-2
Det
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wh
eth
er e
ach
exp
ress
ion
is
a p
olyn
omia
l.If
it
is a
pol
ynom
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stat
e th
ed
egre
e of
th
e p
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no3.
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8G
lenc
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lgeb
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Det
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wh
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er e
ach
exp
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ion
is
a p
olyn
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l.If
it
is a
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ynom
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e th
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AN
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EOM
ETRY
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poly
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ial e
xpre
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n fo
r th
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Pra
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Poly
nom
ials
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-2
5-2
© Glencoe/McGraw-Hill A7 Glencoe Algebra 2
An
swer
s
Answers (Lesson 5-2)
Rea
din
g t
o L
earn
Math
emati
csPo
lyno
mia
ls
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-2
5-2
©G
lenc
oe/M
cGra
w-Hi
ll24
9G
lenc
oe A
lgeb
ra 2
Lesson 5-2
Pre-
Act
ivit
yH
ow c
an p
olyn
omia
ls b
e ap
pli
ed t
o fi
nan
cial
sit
uat
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s?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 5-
2 at
the
top
of
page
229
in y
our
text
book
.
Supp
ose
that
She
nequ
a de
cide
s to
enr
oll i
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five
-yea
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four
-yea
r pr
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sing
the
mod
el g
iven
in y
our
text
book
,ho
w c
ould
she
est
imat
e th
e tu
itio
n fo
r th
e fi
fth
year
of
her
prog
ram
? (D
ono
t ac
tual
ly c
alcu
late
,but
des
crib
e th
e ca
lcul
atio
n th
at w
ould
be
nece
ssar
y.)
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tiply
$15
,604
by
1.04
.
Rea
din
g t
he
Less
on
1.St
ate
whe
ther
the
ter
ms
in e
ach
of t
he f
ollo
win
g pa
irs
are
like
ter
ms
or u
nlik
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2 ,3y
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term
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ate
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eac
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low
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expr
essi
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nom
ial,
trin
omia
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not
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l.If
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give
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3.a.
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or in
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mul
tiply
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etho
d is
an
appl
icat
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hat
prop
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of
real
num
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?Di
strib
utiv
e Pr
oper
tyc.
In t
he F
OIL
met
hod,
wha
t do
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lett
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F,O
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.
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Hel
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ou
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omia
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tri
nom
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ng o
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h w
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t be
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d tw
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ple
answ
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onot
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onog
ram
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ycle
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tricy
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tripo
d
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lenc
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cGra
w-Hi
ll25
0G
lenc
oe A
lgeb
ra 2
Poly
nom
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with
Fra
ctio
nal C
oeffi
cien
tsPo
lyno
mia
ls m
ay h
ave
frac
tion
al c
oeff
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as lo
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s th
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men
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-2
5-2
© Glencoe/McGraw-Hill A8 Glencoe Algebra 2
Answers (Lesson 5-3)
Stu
dy
Gu
ide
and I
nte
rven
tion
Divi
ding
Pol
ynom
ials
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-3
5-3
©G
lenc
oe/M
cGra
w-Hi
ll25
1G
lenc
oe A
lgeb
ra 2
Lesson 5-3
Use
Lo
ng
Div
isio
nTo
div
ide
a po
lyno
mia
l by
a m
onom
ial,
use
the
prop
erti
es o
f po
wer
sfr
om L
esso
n 5-
1.
To d
ivid
e a
poly
nom
ial b
y a
poly
nom
ial,
use
a lo
ng d
ivis
ion
patt
ern.
Rem
embe
r th
at o
nly
like
term
s ca
n be
add
ed o
r su
btra
cted
.
Sim
pli
fy
.
"!
!
"p3
!2 t
2!
1 r1
!1
!p2
!2 q
t1!
1 r2
!1
!p3
!2 t
1!
1 r1
!1
"4p
t!
7qr
!3p
Use
lon
g d
ivis
ion
to
fin
d (
x3!
8x2
$4x
!9)
%(x
!4)
.
x2!
4x!
12x
!4 %
x$3 $!$
8$x$2 $
%$4$
x$!$
9$(!
)x3
!4x
2
!4x
2%
4x(!
)!4x
2%
16x
!12
x!
9(!
)!12
x%
48!
57T
he q
uoti
ent
is x
2!
4x!
12,a
nd t
he r
emai
nder
is !
57.
The
refo
re
"x2
!4x
!12
!.
Sim
pli
fy.
1.2.
3.
6a2
$10
a!
105a
b!
$7a
4
4.(2
x2!
5x!
3) *
(x!
3)5.
(m2
!3m
!7)
*(m
%2)
2x$
1m
!5
$
6.(p
3!
6) *
(p!
1)7.
(t3
!6t
2%
1) *
(t%
2)
p2$
p$
1 !
t2!
8t$
16 !
8.(x
5!
1) *
(x!
1)9.
(2x3
!5x
2%
4x!
4) *
(x !
2)
x4$
x3$
x2$
x$
12x
2!
x$
2
31" t$
25
" p!
1
3" m
$2
4b2
"a
6n3
" m
60a2 b
3!
48b4
%84
a5 b2
&&
&12
ab2
24m
n6!
40m
2 n3
&&
&4m
2 n3
18a3
%30
a2&
& 3a
57& x
!4
x3!
8x2
%4x
!9
&&
&x
!4
9 & 321 & 3
12 & 3
9p3 t
r& 3p
2 tr
21p2 q
tr2
&&
3p2 t
r12
p3 t2 r
&3p
2 tr
12p3 t
2 r!
21p2 q
tr2
!9p
3 tr
&&
&&
3p2 t
r
12p3
t2r
!21
p2qt
r2!
9p3 t
r"
""
"3p
2 tr
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-Hi
ll25
2G
lenc
oe A
lgeb
ra 2
Use
Syn
thet
ic D
ivis
ion
Synt
hetic
div
isio
na
proc
edur
e to
divi
de a
pol
ynom
ial b
y a
bino
mia
l usin
g co
effic
ient
s of
the
divid
end
and
the
valu
e of
rin
the
divis
or x
!r
Use
syn
thet
ic d
ivis
ion
to f
ind
(2x3
!5x
2%
5x!
2) *
(x!
1).
Step
1W
rite
the
term
s of
the
divid
end
so th
at th
e de
gree
s of
the
term
s ar
e in
2x3
!5x
2%
5x!
2 de
scen
ding
ord
er. T
hen
write
just
the
coef
ficie
nts.
2 !
5 5
!2
Step
2W
rite
the
cons
tant
rof
the
divis
or x
!rt
o th
e le
ft, In
this
case
, r"
1.
12
!5
5!
2Br
ing
down
the
first
coe
fficie
nt, 2
, as
show
n.2
Step
3M
ultip
ly th
e fir
st c
oeffi
cient
by
r, 1
$2
"2.
Writ
e th
eir p
rodu
ct u
nder
the
12
!5
5!
2 se
cond
coe
fficie
nt. T
hen
add
the
prod
uct a
nd th
e se
cond
coe
fficie
nt:
2 !
5 %
2 "
!3.
2!
3
Step
4M
ultip
ly th
e su
m, !
3, b
y r:
!3
$1
"!
3. W
rite
the
prod
uct u
nder
the
next
12
!5
5!
2 co
effic
ient
and
add
: 5 %
(!3)
"2.
2!
32
!3
2
Step
5M
ultip
ly th
e su
m, 2
, by
r: 2
$1
"2.
Writ
e th
e pr
oduc
t und
er th
e ne
xt1
2!
55
!2
coef
ficie
nt a
nd a
dd: !
2 %
2 "
0. T
he re
mai
nder
is 0
.2
!3
22
!3
20
Thu
s,(2
x3!
5x2
%5x
!2)
*(x
!1)
"2x
2!
3x%
2.
Sim
pli
fy.
1.(3
x3!
7x2
%9x
!14
) *
(x!
2)2.
(5x3
%7x
2!
x!
3) *
(x%
1)
3x2
!x
$7
5x2
$2x
!3
3.(2
x3%
3x2
!10
x !
3) *
(x%
3)4.
(x3
!8x
2%
19x
!9)
*(x
!4)
2x2
!3x
!1
x2!
4x$
3 $
5.(2
x3%
10x2
%9x
%38
) *
(x%
5)6.
(3x3
!8x
2%
16x
!1)
*(x
!1)
2x2
$9
!3x
2!
5x$
11 $
7.(x
3!
9x2
%17
x!
1) *
(x!
2)8.
(4x3
!25
x2%
4x%
20)
*(x
!6)
x2!
7x$
3 $
4x2
!x
!2
$
9.(6
x3%
28x2
!7x
%9)
*(x
%5)
10.(
x4!
4x3
%x2
%7x
!2)
*(x
!2)
6x2
!2x
$3
!x3
!2x
2!
3x$
1
11.(
12x4
%20
x3!
24x2
%20
x%
35)
*(3
x%
5)4x
3!
8x$
20 $
!65
" 3x$
5
6" x
$5
8" x
!6
5" x
!2
10" x
!1
7" x
$5
3" x
!4
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Divi
ding
Pol
ynom
ials
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-3
5-3
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A9 Glencoe Algebra 2
An
swer
s
Answers (Lesson 5-3)
Skil
ls P
ract
ice
Divi
ding
Pol
ynom
ials
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-3
5-3
©G
lenc
oe/M
cGra
w-Hi
ll25
3G
lenc
oe A
lgeb
ra 2
Lesson 5-3
Sim
pli
fy.
1.5c
$3
2.3x
$5
3.5y
2$
2y$
14.
3x!
1 !
5.(1
5q6
%5q
2 )(5
q4)!
16.
(4f5
!6f
4%
12f3
!8f
2 )(4
f2)!
1
3q2
$f3
!$
3f!
2
7.(6
j2k
!9j
k2) *
3jk
8.(4
a2h2
!8a
3 h%
3a4 )
*(2
a2)
2j!
3k2h
2!
4ah
$
9.(n
2%
7n%
10)
*(n
%5)
10.(
d2
%4d
%3)
*(d
%1)
n$
2d
$3
11.(
2s2
%13
s%
15)
*(s
%5)
12.(
6y2
%y
!2)
(2y
!1)
!1
2s$
33y
$2
13.(
4g2
!9)
*(2
g%
3)14
.(2x
2!
5x!
4) *
(x!
3)
2g!
32x
$1
!
15.
16.
u$
8 $
2x$
1 !
17.(
3v2
!7v
!10
)(v
!4)
!1
18.(
3t4
%4t
3!
32t2
!5t
!20
)(t
%4)
!1
3v$
5 $
3t3
!8t
2!
5
19.
20.
y2!
3y$
6 !
2x2
$5x
!4
$
21.(
4p3
!3p
2%
2p) *
(p!
1)22
.(3c
4%
6c3
!2c
%4)
(c%
2)!
1
4p2
$p
$3
$3c
3!
2 $
23.G
EOM
ETRY
The
are
a of
a r
ecta
ngle
is x
3%
8x2
%13
x!
12 s
quar
e un
its.
The
wid
th o
fth
e re
ctan
gle
is x
%4
unit
s.W
hat
is t
he le
ngth
of
the
rect
angl
e?x2
$4x
!3
units
8" c
$2
3" p
!1
3" x
!3
18" y
$2
2x3
!x2
!19
x%
15&
&&
x!
3y3
!y2
!6
&&
y%
2
10" v
!4
1" x
!3
12" u
!3
2x2
!5x
!4
&&
x!
3u2
%5u
!12
&&
u!
3
1" x
!33a
2"
2
3f2
"2
1 " q2
2 " x12
x2!
4x!
8&
& 4x15
y3%
6y2
%3y
&& 3y
12x
%20
&& 4
10c
%6
&2
©G
lenc
oe/M
cGra
w-Hi
ll25
4G
lenc
oe A
lgeb
ra 2
Sim
pli
fy.
1.3r
6!
r4$
2.!
6k2
$
3.(!
30x3
y%
12x2
y2!
18x2
y) *
(!6x
2 y)
4.(!
6w3 z
4!
3w2 z
5%
4w%
5z) *
(2w
2 z)
5x!
2y$
3!
3wz3
!$
$
5.(4
a3!
8a2
%a2
)(4a
)!1
6.(2
8d3 k
2%
d2 k
2!
4dk2
)(4d
k2)!
1
a2!
2a$
"a 4"7d
2$
"d 4"!
1
7.f$
58.
2x$
7
9.(a
3!
64)
*(a
!4)
a2$
4a$
1610
.(b3
%27
) *
(b%
3)b2
!3b
$9
11.
2x2
!8x
$38
12.
2x2
!6x
$22
!
13.(
3w3
%7w
2!
4w%
3) *
(w%
3)14
.(6y
4%
15y3
!28
y!
6) *
(y%
2)
3w2
!2w
$2
!" w
$33
"6y
3$
3y2
!6y
!16
$" y2 $6 2
"
15.(
x4!
3x3
!11
x2%
3x%
10)
*(x
!5)
16.(
3m5
%m
!1)
*(m
%1)
x3$
2x2
!x
!2
3m4
!3m
3$
3m2
!3m
$4
!
17.(
x4!
3x3
%5x
!6)
(x%
2)!
118
.(6y
2!
5y!
15)(
2y%
3)!
1
x3!
5x2
$10
x!
15 $
" x2 $4 2"
3y!
7 $
" 2y6 $
3"
19.
20.
2x$
2 $
" 2x1 !2
3"
2x!
1 !
" 3x6 $
1"
21.(
2r3
%5r
2!
2r!
15)
*(2
r!
3)22
.(6t
3%
5t2
!2t
%1)
*(3
t%
1)r2
$4r
$5
2t2
$t!
1 $
" 3t2 $
1"
23.
24.
2p3
$3p
2!
4p$
12h
2!
h$
325
.GEO
MET
RYT
he a
rea
of a
rec
tang
le is
2x2
!11
x%
15 s
quar
e fe
et.T
he le
ngth
of
the
rect
angl
e is
2x
!5
feet
.Wha
t is
the
wid
th o
f th
e re
ctan
gle?
x!
3 ft
26.G
EOM
ETRY
The
are
a of
a t
rian
gle
is 1
5x4
%3x
3%
4x2
!x
!3
squa
re m
eter
s.T
hele
ngth
of
the
base
of
the
tria
ngle
is 6
x2!
2 m
eter
s.W
hat
is t
he h
eigh
t of
the
tri
angl
e?5x
2$
x$
3 m
2h4
!h3
%h2
%h
!3
&&
&h2
!1
4p4
!17
p2%
14p
!3
&&
&2p
!3
6x2
!x
! 7
&&
3x%
14x
2!
2x%
6&
&2x
!3
5" m
$1
72" x
$3
2x3
%4x
!6
&&
x%
32x
3%
6x%
152
&&
x%
4
2x2
%3x
!14
&&
x!
2f2
%7f
%10
&&
f%
2
5" 2w
22 " wz
3z4
"2
9m " 2k3k " m
6k2 m
!12
k3 m2
%9m
3&
&&
2km
28 " r2
15r10
!5r
8%
40r2
&&
&5r
4Pra
ctic
e (A
vera
ge)
Divi
ding
Pol
ynom
ials
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-3
5-3
© Glencoe/McGraw-Hill A10 Glencoe Algebra 2
Answers (Lesson 5-3)
Rea
din
g t
o L
earn
Math
emati
csDi
vidi
ng P
olyn
omia
ls
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-3
5-3
©G
lenc
oe/M
cGra
w-Hi
ll25
5G
lenc
oe A
lgeb
ra 2
Lesson 5-3
Pre-
Act
ivit
yH
ow c
an y
ou u
se d
ivis
ion
of
pol
ynom
ials
in
man
ufa
ctu
rin
g?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 5-
3 at
the
top
of
page
233
in y
our
text
book
.
Usi
ng t
he d
ivis
ion
sym
bol (
*),
wri
te t
he d
ivis
ion
prob
lem
tha
t yo
u w
ould
use
to a
nsw
er t
he q
uest
ion
aske
d in
the
intr
oduc
tion
.(D
o no
t ac
tual
lydi
vide
.)(3
2x2
$x)
%(8
x)
Rea
din
g t
he
Less
on
1.a.
Exp
lain
in w
ords
how
to
divi
de a
pol
ynom
ial b
y a
mon
omia
l.Di
vide
eac
h te
rm o
fth
e po
lyno
mia
l by
the
mon
omia
l.b.
If y
ou d
ivid
e a
trin
omia
l by
a m
onom
ial a
nd g
et a
pol
ynom
ial,
wha
t ki
nd o
fpo
lyno
mia
l will
the
quo
tien
t be
?tri
nom
ial
2.L
ook
at t
he f
ollo
win
g di
visi
on e
xam
ple
that
use
s th
e di
visi
on a
lgor
ithm
for
pol
ynom
ials
.
2x%
4x
!4%$
$2x
2!
4x%
72x
2!
8x 4x%
74x
!16 23
Whi
ch o
f th
e fo
llow
ing
is t
he c
orre
ct w
ay t
o w
rite
the
quo
tien
t?C
A.
2x%
4B
.x
!4
C.
2x%
4 %
D.
3.If
you
use
syn
thet
ic d
ivis
ion
to d
ivid
e x3
%3x
2!
5x!
8 by
x!
2,th
e di
visi
on w
ill lo
oklik
e th
is:
21
3!
5!
82
1010
15
52
Whi
ch o
f th
e fo
llow
ing
is t
he a
nsw
er f
or t
his
divi
sion
pro
blem
?B
A.
x2%
5x%
5B
.x2
%5x
%5
%
C.
x3%
5x2
%5x
%D
.x3
%5x
2%
5x%
2
Hel
pin
g Y
ou
Rem
emb
er
4.W
hen
you
tran
slat
e th
e nu
mbe
rs in
the
last
row
of
a sy
nthe
tic
divi
sion
into
the
quo
tien
tan
d re
mai
nder
,wha
t is
an
easy
way
to
rem
embe
r w
hich
exp
onen
ts t
o us
e in
wri
ting
the
term
s of
the
quo
tien
t?Sa
mpl
e an
swer
:Sta
rt w
ith th
e po
wer
that
is o
ne le
ssth
an th
e de
gree
of t
he d
ivid
end.
Decr
ease
the
pow
er b
y on
e fo
r eac
hte
rm a
fter t
he fi
rst.
The
final
num
ber w
ill b
e th
e re
mai
nder
.Dro
p an
y te
rmth
at is
repr
esen
ted
by a
0.
2& x
!2
2& x
!2
23& x
!4
23& x
!4
©G
lenc
oe/M
cGra
w-Hi
ll25
6G
lenc
oe A
lgeb
ra 2
Obl
ique
Asy
mpt
otes
The
gra
ph o
f y"
ax%
b,w
here
a #
0,i
s ca
lled
an o
bliq
ue a
sym
ptot
e of
y"
f(x)
if
the
gra
ph o
f fco
mes
clo
ser
and
clos
er t
o th
e lin
e as
x →
∞or
x →
!∞
.∞is
the
mat
hem
atic
al s
ymbo
l for
in
fin
ity,
whi
ch m
eans
end
less
.
For
f(x)
"3x
%4
%&2 x& ,
y"
3x%
4 is
an
obliq
ue a
sym
ptot
e be
caus
e
f(x)
!3x
!4
"&2 x& ,
and
&2 x&→
0 a
s x →
∞or
!∞
.In
othe
r w
ords
,as
|x|
incr
ease
s,th
e va
lue
of &2 x&
gets
sm
alle
r an
d sm
alle
r ap
proa
chin
g 0.
Fin
d t
he
obli
que
asym
pto
te f
or f
(x)
&"x2
$ x8 $x
2$15
".
!2
18
15Us
e sy
nthe
tic d
ivisio
n.!
2!
121
63
y"&x2
% x8 %x2%
15&
"x
%6
%& x
%32
&
As
|x|i
ncre
ases
,the
val
ue o
f &x
%32
&ge
ts s
mal
ler.
In o
ther
wor
ds,s
ince
& x%3
2&
→ 0
as
x →
∞or
x →
!∞
,y"
x%
6 is
an
obliq
ue a
sym
ptot
e.
Use
syn
thet
ic d
ivis
ion
to
fin
d t
he
obli
que
asym
pto
te f
or e
ach
fu
nct
ion
.
1.y
"&8x
2! x
%4x5%
11&
y&
8x!
44
2.y
"&x2
% x3 !x2!
15&
y&
x$
5
3.y
"&x2
! x2 !x3!
18&
y&
x$
1
4.y
"&ax
2
x%!bx
d%
c&
y&
ax$
b$
ad
5.y
"&ax
2
x%%bx
d%
c&
y&
ax$
b!
ad
En
rich
men
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-3
5-3
Exam
ple
Exam
ple
© Glencoe/McGraw-Hill A11 Glencoe Algebra 2
An
swer
s
Answers (Lesson 5-4)
Stu
dy
Gu
ide
and I
nte
rven
tion
Fact
orin
g Po
lyno
mia
ls
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-4
5-4
©G
lenc
oe/M
cGra
w-Hi
ll25
7G
lenc
oe A
lgeb
ra 2
Lesson 5-4
Fact
or
Poly
no
mia
ls
For a
ny n
umbe
r of t
erm
s, c
heck
for:
grea
test
com
mon
fact
or
For t
wo te
rms,
che
ck fo
r:Di
ffere
nce
of tw
o sq
uare
sa2
!b2
"(a
%b)
(a!
b)Su
m o
f two
cub
esa3
%b3
"(a
%b)
(a2
!ab
%b2
)Di
ffere
nce
of tw
o cu
bes
a3!
b3"
(a!
b)(a
2%
ab%
b2)
Tech
niqu
es fo
r Fac
torin
g Po
lyno
mia
lsFo
r thr
ee te
rms,
che
ck fo
r:Pe
rfect
squ
are
trino
mia
lsa2
%2a
b%
b2"
(a%
b)2
a2!
2ab
%b2
"(a
!b)
2
Gen
eral
trin
omia
lsac
x2%
(ad
%bc
)x%
bd"
(ax
%b)
(cx
%d)
For f
our t
erm
s, c
heck
for:
Gro
upin
gax
%bx
%ay
%by
"x(
a%
b) %
y(a
%b)
"(a
%b)
(x%
y)
Fact
or 2
4x2
!42
x!
45.
Fir
st f
acto
r ou
t th
e G
CF
to
get
24x2
!42
x!
45 "
3(8x
2!
14x
!15
).To
fin
d th
e co
effi
cien
tsof
the
xte
rms,
you
mus
t fi
nd t
wo
num
bers
who
se p
rodu
ct is
8 $
(!15
) "
!12
0 an
d w
hose
sum
is !
14.T
he t
wo
coef
fici
ents
mus
t be
!20
and
6.R
ewri
te t
he e
xpre
ssio
n us
ing
!20
xan
d6x
and
fact
or b
y gr
oupi
ng.
8x2
!14
x!
15 "
8x2
!20
x%
6x!
15G
roup
to fi
nd a
GCF
."
4x(2
x!
5) %
3(2x
!5)
Fact
or th
e G
CF o
f eac
h bi
nom
ial.
"(4
x%
3)(2
x!
5)Di
strib
utive
Pro
perty
Thu
s,24
x2!
42x
!45
"3(
4x%
3)(2
x!
5).
Fact
or c
omp
lete
ly.I
f th
e p
olyn
omia
l is
not
fac
tora
ble,
wri
te p
rim
e.
1.14
x2y2
%42
xy3
2.6m
n%
18m
!n
!3
3.2x
2%
18x
%16
14xy
2 (x
$3y
)(6
m!
1)(n
$3)
2(
x$
8)(x
$1)
4.x4
!1
5.35
x3y4
!60
x4y
6.2r
3%
250
(x2
$1)
(x$
1)(x
!1)
5x
3 y(7
y3!
12x)
2(r$
5)(r
2!
5r$
25)
7.10
0m8
!9
8.x2
%x
%1
9.c4
%c3
!c2
!c
(10m
4!
3)(1
0m4
$3)
prim
ec(
c$
1)2 (
c!
1)
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-Hi
ll25
8G
lenc
oe A
lgeb
ra 2
Sim
plif
y Q
uo
tien
tsIn
the
last
less
on y
ou le
arne
d ho
w t
o si
mpl
ify
the
quot
ient
of
two
poly
nom
ials
by
usin
g lo
ng d
ivis
ion
or s
ynth
etic
div
isio
n.So
me
quot
ient
s ca
n be
sim
plif
ied
byus
ing
fact
orin
g.
Sim
pli
fy
.
"Fa
ctor
the
num
erat
or a
nd d
enom
inat
or.
"Di
vide.
Ass
ume
x#
8, !
.
Sim
pli
fy.A
ssu
me
that
no
den
omin
ator
is
equ
al t
o 0.
1.2.
3.
4.5.
6.
7.8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
9x2
$6x
$4
""
3x$
2y2
$4y
$16
""
3y!
52x
!3
""
4x2
!2x
$1
27x3
!8
&&
9x2
!4
y3!
64&
&3y
2!
17y
%20
4x2
!4x
!3
&&
8x3
%1
2x!
3" x
$8
7x!
3" x
!2
x$
9" 2x
!5
4x2
!12
x%
9&
&2x
2%
13x
!24
7x2
%11
x!
6&
&x2
!4
x2!
81&
&2x
2!
23x
%45
x!
7" x
$5
3x!
5" 2x
!3
2x$
5" x
!1
x2!
14x
%49
&&
x2!
2x!
353x
2%
4x!
15&
&2x
2%
3x!
94x
2%
16x
%15
&&
2x2
%x
!3
x!
2" 3x
$7
6x$
1" x
$2
2x!
1" x
!2
x2!
7x%
10&
&3x
2!
8x!
356x
2%
25x
%4
&&
x2%
6x%
84x
2%
4x!
3&
&2x
2!
x!
6
5x!
1" x
$3
2x!
1" 4x
!1
x$
3" x
!5
5x2
%9x
!2
&&
x2%
5x%
62x
2%
5x!
3&
&4x
2%
11x
!3
x2%
x!
6&
&x2
!7x
%10
x!
5" x
$1
x$
5" 2x
!3
x!
4" x
$2
x2!
11x
%30
&&
x2!
5x!
6x2
%6x
%5
&&
2x2
!x
!3
x2!
7x%
12&
&x2
!x
!6
3 & 2x
%4
& x!
8
(2x
%3)
( x%
4)&
&(x
!8)
(2x
%3)
8x2
%11
x%
12&
&2x
2!
13x
!24
8x2
%11
x%
12"
"2x
2!
13x
!24
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Fact
orin
g Po
lyno
mia
ls
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-4
5-4
Exam
ple
Exam
ple
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A12 Glencoe Algebra 2
Answers (Lesson 5-4)
Skil
ls P
ract
ice
Fact
orin
g Po
lyno
mia
ls
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-4
5-4
©G
lenc
oe/M
cGra
w-Hi
ll25
9G
lenc
oe A
lgeb
ra 2
Lesson 5-4
Fact
or c
omp
lete
ly.I
f th
e p
olyn
omia
l is
not
fac
tora
ble,
wri
te p
rim
e.
1.7x
2!
14x
2.19
x3!
38x2
7x(x
!2)
19x2
(x!
2)
3.21
x3!
18x2
y%
24xy
24.
8j3 k
!4j
k3!
73x
(7x2
!6x
y$
8y2 )
prim
e
5.a2
%7a
!18
6.2a
k!
6a%
k!
3(a
$9)
(a!
2)(2
a$
1)(k
!3)
7.b2
%8b
%7
8.z2
!8z
!10
(b$
7)(b
$1)
prim
e
9.m
2%
7m!
1810
.2x2
!3x
!5
(m!
2)(m
$9)
(2x
!5)
(x$
1)
11.4
z2%
4z!
1512
.4p2
%4p
!24
(2z
$5)
(2z
!3)
4(p
!2)
(p$
3)
13.3
y2%
21y
%36
14.c
2!
100
3(y
$4)
(y$
3)(c
$10
)(c!
10)
15.4
f2!
6416
.d2
!12
d%
364(
f$4)
(f!
4)(d
!6)
2
17.9
x2%
2518
.y2
%18
y%
81pr
ime
(y$
9)2
19.n
3!
125
20.m
4!
1(n
!5)
(n2
$5n
$25
)(m
2$
1)(m
!1)
(m$
1)
Sim
pli
fy.A
ssu
me
that
no
den
omin
ator
is
equ
al t
o 0.
21.
22.
23.
24.
x!
1" x
!7
x2%
6x!
7&
&x2
!49
x!
5"
xx2
!10
x%
25&
&x2
!5x
x$
1" x
$3
x2%
4x%
3&
&x2
%6x
%9
x!
2" x
!5
x2%
7x!
18&
&x2
%4x
!45
©G
lenc
oe/M
cGra
w-Hi
ll26
0G
lenc
oe A
lgeb
ra 2
Fact
or c
omp
lete
ly.I
f th
e p
olyn
omia
l is
not
fac
tora
ble,
wri
te p
rim
e.
1.15
a2b
!10
ab2
2.3s
t2!
9s3 t
%6s
2 t2
3.3x
3 y2
!2x
2 y%
5xy
5ab(
3a!
2b)
3st(
t!3s
2$
2st)
xy(3
x2y
!2x
$5)
4.2x
3 y!
x2y
%5x
y2%
xy3
5.21
!7t
%3r
!rt
6.x2
!xy
%2x
!2y
xy(2
x2!
x$
5y $
y2)
(7 $
r)(3
!t)
(x$
2)(x
!y)
7.y2
%20
y%
968.
4ab
%2a
%6b
%3
9.6n
2!
11n
!2
(y$
8)(y
$12
)(2
a$
3)(2
b$
1)(6
n$
1)(n
!2)
10.6
x2%
7x!
311
.x2
!8x
!8
12.6
p2!
17p
!45
(3x
!1)
(2x
$3)
prim
e(2
p!
9)(3
p$
5)13
.r3
%3r
2!
54r
14.8
a2%
2a!
615
.c2
!49
r(r$
9)(r
!6)
2(4a
!3)
(a$
1)(c
!7)
(c$
7)16
.x3
%8
17.1
6r2
!16
918
.b4
!81
(x$
2)(x
2!
2x$
4)(4
r$13
)(4r!
13)
(b2
$9)
(b$
3)(b
!3)
19.8
m3
!25
prim
e20
.2t3
%32
t2%
128t
2t(t
$8)
2
21.5
y5%
135y
25y
2 (y
$3)
(y2
!3y
$9)
22.8
1x4
!16
(9x2
$4)
(3x
$2)
(3x
!2)
Sim
pli
fy.A
ssu
me
that
no
den
omin
ator
is
equ
al t
o 0.
23.
24.
25.
26.D
ESIG
NB
obbi
Jo
is u
sing
a s
oftw
are
pack
age
to c
reat
e a
draw
ing
of a
cro
ss s
ecti
on o
f a
brac
e as
sho
wn
at t
he r
ight
.W
rite
a s
impl
ifie
d,fa
ctor
ed e
xpre
ssio
n th
at r
epre
sent
s th
e ar
ea o
f th
e cr
oss
sect
ion
of t
he b
race
.x(
20.2
!x)
cm
2
27.C
OM
BU
STIO
N E
NG
INES
In a
n in
tern
al c
ombu
stio
n en
gine
,the
up
and
dow
n m
otio
n of
the
pis
tons
is c
onve
rted
into
the
rot
ary
mot
ion
of
the
cran
ksha
ft,w
hich
dri
ves
the
flyw
heel
.Let
r1
repr
esen
t th
e ra
dius
of
the
fly
whe
el a
t th
e ri
ght
and
let
r 2re
pres
ent
the
radi
us o
f th
ecr
anks
haft
pas
sing
thr
ough
it.I
f th
e fo
rmul
a fo
r th
e ar
ea o
f a
circ
le
is A
"+
r2,w
rite
a s
impl
ifie
d,fa
ctor
ed e
xpre
ssio
n fo
r th
e ar
ea o
f th
e cr
oss
sect
ion
of t
he f
lyw
heel
out
side
the
cra
nksh
aft.
'(r 1
!r 2
)(r1
$r 2
)
r 1r 2
x cm
x cm
8.2
cm
12 cm
3(x
$3)
""
x2$
3x$
93x
2!
27&
&x3
!27
x!
8" x
$9
x2!
16x
%64
&&
x2%
x!
72x
$4
" x$
5x2
!16
&&
x2%
x!
20Pra
ctic
e (A
vera
ge)
Fact
orin
g Po
lyno
mia
ls
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-4
5-4
© Glencoe/McGraw-Hill A13 Glencoe Algebra 2
An
swer
s
Answers (Lesson 5-4)
Rea
din
g t
o L
earn
Math
emati
csFa
ctor
ing
Poly
nom
ials
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-4
5-4
©G
lenc
oe/M
cGra
w-Hi
ll26
1G
lenc
oe A
lgeb
ra 2
Lesson 5-4
Pre-
Act
ivit
yH
ow d
oes
fact
orin
g ap
ply
to
geom
etry
?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 5-
4 at
the
top
of
page
239
in y
our
text
book
.
If a
tri
nom
ial t
hat
repr
esen
ts t
he a
rea
of a
rec
tang
le is
fac
tore
d in
to t
wo
bino
mia
ls,w
hat
mig
ht t
he t
wo
bino
mia
ls r
epre
sent
?th
e le
ngth
and
wid
th o
f the
rect
angl
e
Rea
din
g t
he
Less
on
1.N
ame
thre
e ty
pes
of b
inom
ials
tha
t it
is a
lway
s po
ssib
le t
o fa
ctor
.di
ffere
nce
of tw
osq
uare
s,su
m o
f tw
o cu
bes,
diffe
renc
e of
two
cube
s
2.N
ame
a ty
pe o
f tr
inom
ial t
hat
it is
alw
ays
poss
ible
to
fact
or.
perfe
ct s
quar
etri
nom
ial
3.C
ompl
ete:
Sinc
e x2
%y2
cann
ot b
e fa
ctor
ed,i
t is
an
exam
ple
of a
po
lyno
mia
l.
4.O
n an
alg
ebra
qui
z,M
arle
ne n
eede
d to
fac
tor
2x2
!4x
!70
.She
wro
te t
he f
ollo
win
gan
swer
:(x
%5)
(2x
!14
).W
hen
she
got
her
quiz
bac
k,M
arle
ne f
ound
tha
t sh
e di
d no
tge
t fu
ll cr
edit
for
her
ans
wer
.She
tho
ught
she
sho
uld
have
got
ten
full
cred
it b
ecau
se s
hech
ecke
d he
r w
ork
by m
ulti
plic
atio
n an
d sh
owed
tha
t (x
%5)
(2x
!14
) "
2x2
!4x
!70
.
a.If
you
wer
e M
arle
ne’s
tea
cher
,how
wou
ld y
ou e
xpla
in t
o he
r th
at h
er a
nsw
er w
as n
oten
tire
ly c
orre
ct?
Sam
ple
answ
er:W
hen
you
are
aske
d to
fact
or a
poly
nom
ial,
you
mus
t fac
tor i
t com
plet
ely.
The
fact
oriz
atio
n w
as n
otco
mpl
ete,
beca
use
2x!
14 c
an b
e fa
ctor
ed fu
rthe
r as
2(x
!7)
.
b.W
hat
advi
ce c
ould
Mar
lene
’s t
each
er g
ive
her
to a
void
mak
ing
the
sam
e ki
nd o
f er
ror
in f
acto
ring
in t
he f
utur
e?Sa
mpl
e an
swer
:Alw
ays
look
for a
com
mon
fact
or fi
rst.
If th
ere
is a
com
mon
fact
or,f
acto
r it o
ut fi
rst,
and
then
see
if yo
u ca
n fa
ctor
furt
her.
Hel
pin
g Y
ou
Rem
emb
er
5.So
me
stud
ents
hav
e tr
oubl
e re
mem
beri
ng t
he c
orre
ct s
igns
in t
he f
orm
ulas
for
the
sum
and
diff
eren
ce o
f tw
o cu
bes.
Wha
t is
an
easy
way
to
rem
embe
r th
e co
rrec
t si
gns?
Sam
ple
answ
er:I
n th
e bi
nom
ial f
acto
r,th
e op
erat
ion
sign
is th
e sa
me
asin
the
expr
essi
on th
at is
bei
ng fa
ctor
ed.I
n th
e tri
nom
ial f
acto
r,th
eop
erat
ion
sign
bef
ore
the
mid
dle
term
is th
e op
posi
te o
f the
sig
n in
the
expr
essi
on th
at is
bei
ng fa
ctor
ed.T
he s
ign
befo
re th
e la
st te
rm is
alw
ays
a pl
us.
prim
e
©G
lenc
oe/M
cGra
w-Hi
ll26
2G
lenc
oe A
lgeb
ra 2
Usin
g Pa
ttern
s to
Fac
tor
Stud
y th
e pa
tter
ns b
elow
for
fac
tori
ng t
he s
um a
nd t
he d
iffe
renc
e of
cub
es.
a3%
b3"
(a%
b)(a
2!
ab%
b2)
a3!
b3"
(a!
b)(a
2%
ab%
b2)
Thi
s pa
tter
n ca
n be
ext
ende
d to
oth
er o
dd p
ower
s.St
udy
thes
e ex
ampl
es.
Fact
or a
5$
b5.
Ext
end
the
firs
t pa
tter
n to
obt
ain
a5%
b5"
(a%
b)(a
4!
a3b
%a2
b2!
ab3
%b4
).C
hec
k:(
a%
b)(a
4!
a3b
%a2
b2!
ab3
%b4
) "a5
!a4
b%
a3b2
!a2
b3%
ab4
%a4
b!
a3b2
%a2
b3!
ab4
%b5
"a5
%b5
Fact
or a
5!
b5.
Ext
end
the
seco
nd p
atte
rn t
o ob
tain
a5
!b5
"(a
!b)
(a4
%a3
b%
a2b2
%ab
3%
b4).
Ch
eck
:(a
!b)
(a4
%a3
b%
a2b2
%ab
3%
b4) "
a5%
a4b
%a3
b2%
a2b3
%ab
4
!a4
b!
a3b2
!a2
b3!
ab4
!b5
"a5
!b5
In g
ener
al,i
f nis
an
odd
inte
ger,
whe
n yo
u fa
ctor
an
%bn
or a
n!
bn,o
ne f
acto
r w
ill b
eei
ther
(a%
b) o
r (a
!b)
,dep
endi
ng o
n th
e si
gn o
f th
e or
igin
al e
xpre
ssio
n.T
he o
ther
fac
tor
will
hav
e th
e fo
llow
ing
prop
erti
es:
•T
he f
irst
ter
m w
ill b
e an
!1
and
the
last
ter
m w
ill b
e bn
!1 .
•T
he e
xpon
ents
of a
will
dec
reas
e by
1 a
s yo
u go
fro
m le
ft t
o ri
ght.
•T
he e
xpon
ents
of b
will
incr
ease
by
1 as
you
go
from
left
to
righ
t.•
The
deg
ree
of e
ach
term
will
be
n!
1.•
If t
he o
rigi
nal e
xpre
ssio
n w
as a
n%
bn,t
he t
erm
s w
ill a
lter
nate
ly h
ave
%an
d !
sign
s.•
If t
he o
rigi
nal e
xpre
ssio
n w
as a
n!
bn,t
he t
erm
s w
ill a
ll ha
ve %
sign
s.
Use
th
e p
atte
rns
abov
e to
fac
tor
each
exp
ress
ion
.
1.a7
%b7
(a$
b)(a
6!
a5b
$a4
b2!
a3b3
$a2
b4!
ab5
$b6
)2.
c9!
d9
(c!
d)(c
8$
c7d
$c6
d2
$c5
d3$
c4d4
$c3
d5$
c2d
6$
cd7
$d8
)3.
e11
%f1
1
(e$
f)(e
10!
e9f$
e8f2
!e7
f3$
e6f4
!e5
f5$
e4f6
!e3
f7$
e2f8
!ef
9$
f10 )
To f
acto
r x1
0!
y10 ,
chan
ge i
t to
(x5
$y5
)(x5
!y5
) an
d f
acto
r ea
ch b
inom
ial.
Use
this
ap
pro
ach
to
fact
or e
ach
exp
ress
ion
.
4.x1
0!
y10
(x$
y)(x
4!
x3y
$x2
y2!
xy3
$y4
)(x!
y)(x
4$
x3y
$x2
y2$
xy3
$y4
)5.
a14
!b1
4(a
$b)
(a6
!a5
b$
a4b2
!a3
b3$
a2b4
!ab
5$
b6)(a
!b)
(a6
$a5
b$
a4b2
$a3
b3$
a2b4
$ab
5$
b6)
En
rich
men
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-4
5-4
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
© Glencoe/McGraw-Hill A14 Glencoe Algebra 2
Answers (Lesson 5-5)
Stu
dy
Gu
ide
and I
nte
rven
tion
Root
s of
Rea
l Num
bers
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-5
5-5
©G
lenc
oe/M
cGra
w-Hi
ll26
3G
lenc
oe A
lgeb
ra 2
Lesson 5-5
Sim
plif
y R
adic
als
Squa
re R
oot
For a
ny re
al n
umbe
rs a
and
b, if
a2
"b,
then
ais
a sq
uare
root
of b
.
nth
Root
For a
ny re
al n
umbe
rs a
and
b, a
nd a
ny p
ositiv
e in
tege
r n, i
f an
"b,
then
ais
an n
thro
ot o
f b.
1.If
nis
even
and
b,
0, th
en b
has
one
posit
ive ro
ot a
nd o
ne n
egat
ive ro
ot.
Real
nth
Roo
ts o
f b,
2.If
nis
odd
and
b,
0, th
en b
has
one
posit
ive ro
ot.
n !b",
!n !
b"3.
If n
is ev
en a
nd b
)0,
then
bha
s no
real
root
s.4.
If n
is od
d an
d b
)0,
then
bha
s on
e ne
gativ
e ro
ot.
Sim
pli
fy !
49z8
".
#49
z8$
"#
(7z4
)2$
"7z
4
z4m
ust
be p
osit
ive,
so t
here
is n
o ne
ed t
ota
ke t
he a
bsol
ute
valu
e.
Sim
pli
fy !
3 !(2
a!
"1)
6"
!3 #
(2a
!$
1)6
$"
!3 #
[(2a
!$
1)2 ]
3$
"(2
a!
1)2
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Sim
pli
fy.
1.#
81$2.
3 #!
343
$3.
#14
4p6
$9
!7
12| p
3 |4.
-#
4a10
$5.
5 #24
3p1
$0 $
6.!
3 #m
6 n9
$(
2a5
3p2
!m
2 n3
7.3 #
!b1
2$
8.#
16a1
0$
b8 $9.
#12
1x6
$!
b44| a
5 | b4
11| x
3 |10
.#(4
k)4
$11
.-#
169r
4$
12.!
3 #!
27p
$6 $
16k2
(13
r23p
2
13.!
#62
5y2
$z4 $
14.#
36q3
4$
15.#
100x
2$
y4z6
$!
25| y
| z2
6|q1
7 |10
| x| y
2 | z3 |
16.
3 #!
0.02
$7$
17.!
#!
0.36
$18
.#0.
64p
$10 $
!0.
3no
t a re
al n
umbe
r0.
8|p5
|19
.4 #
(2x)
8$
20.#
(11y
2 )$
4 $21
.3 #
(5a2
b)$
6 $4x
212
1y4
25a4
b2
22.#
(3x
!$
1)2
$23
.3 #
(m!
$5)
6$
24.#
36x2
!$
12x
%$
1$| 3
x!
1|(m
!5)
2| 6x
!1|
©G
lenc
oe/M
cGra
w-Hi
ll26
4G
lenc
oe A
lgeb
ra 2
Ap
pro
xim
ate
Rad
ical
s w
ith
a C
alcu
lato
r
Irrat
iona
l Num
ber
a nu
mbe
r tha
t can
not b
e ex
pres
sed
as a
term
inat
ing
or a
repe
atin
g de
cimal
Rad
ical
s su
ch a
s #
2$an
d #
3$ar
e ex
ampl
es o
f ir
rati
onal
num
bers
.Dec
imal
app
roxi
mat
ions
for
irra
tion
al n
umbe
rs a
re o
ften
use
d in
app
licat
ions
.The
se a
ppro
xim
atio
ns c
an b
e ea
sily
foun
d w
ith
a ca
lcul
ator
.
Ap
pro
xim
ate
5 !18
.2"
wit
h a
cal
cula
tor.
5 #18
.2$
&1.
787
Use
a c
alcu
lato
r to
ap
pro
xim
ate
each
val
ue
to t
hre
e d
ecim
al p
lace
s.
1.#
62$2.
#10
50$
3.3 #
0.05
4$
7.87
432
.404
0.37
8
4.!
4 #5.
45$
5.#
5280
$6.
#18
,60
$0$
!1.
528
72.6
6413
6.38
2
7.#
0.09
5$
8.3 #
!15
$9.
5 #10
0$
0.30
8!
2.46
62.
512
10.
6 #85
6$
11.#
3200
$12
.#0.
05$
3.08
156
.569
0.22
4
13.#
12,5
0$
0$14
.#0.
60$
15.!
4 #50
0$
111.
803
0.77
5!
4.72
9
16.
3 #0.
15$
17.
6 #42
00$
18.#
75$0.
531
4.01
78.
660
19.L
AW
EN
FOR
CEM
ENT
The
for
mul
a r
"2#
5L$is
use
d by
pol
ice
to e
stim
ate
the
spee
d r
in m
iles
per
hour
of
a ca
r if
the
leng
th L
of t
he c
ar’s
ski
d m
ark
is m
easu
res
in f
eet.
Est
imat
e to
the
nea
rest
ten
th o
f a
mile
per
hou
r th
e sp
eed
of a
car
tha
t le
aves
a s
kid
mar
k 30
0 fe
et lo
ng.
77.5
mi/h
20.S
PAC
E TR
AV
ELT
he d
ista
nce
to t
he h
oriz
on d
mile
s fr
om a
sat
ellit
e or
biti
ng h
mile
sab
ove
Ear
th c
an b
e ap
prox
imat
ed b
y d
"#
8000
h$
%h2
$.W
hat
is t
he d
ista
nce
to t
heho
rizo
n if
a s
atel
lite
is o
rbit
ing
150
mile
s ab
ove
Ear
th?
abou
t 110
0 ft
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Root
s of
Rea
l Num
bers
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-5
5-5
Exam
ple
Exam
ple
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A15 Glencoe Algebra 2
An
swer
s
Answers (Lesson 5-5)
Skil
ls P
ract
ice
Root
s of
Rea
l Num
bers
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-5
5-5
©G
lenc
oe/M
cGra
w-Hi
ll26
5G
lenc
oe A
lgeb
ra 2
Lesson 5-5
Use
a c
alcu
lato
r to
ap
pro
xim
ate
each
val
ue
to t
hre
e d
ecim
al p
lace
s.
1.#
230
$15
.166
2.#
38$6.
164
3.!
#15
2$
!12
.329
4.#
5.6
$2.
366
5.3 #
88$4.
448
6.3 #
!22
2$
!6.
055
7.!
4 #0.
34$
!0.
764
8.5 #
500
$3.
466
Sim
pli
fy.
9.-
#81$
(9
10.#
144
$12
11.#
(!5)
2$
512
.#!
52$
not a
real
num
ber
13.#
0.36
$0.
614
.!'(
!
15.
3 #!
8$
!2
16.!
3 #27$
!3
17.
3 #0.
064
$0.
418
.5 #
32$2
19.
4 #81$
320
.#y2 $
| y|
21.
3 #12
5s3
$5s
22.#
64x6
$8| x
3 |
23.
3 #!
27a
$6 $
!3a
224
.#m
8 n4
$m
4 n2
25.!
#10
0p4
$q2 $
!10
p2| q
|26
.4 #
16w
4 v$
8 $2| w
| v2
27.#
(!3c
)4$
9c2
28.#
(a%
b$
)2 $| a
$b|
2 " 34 & 9
©G
lenc
oe/M
cGra
w-Hi
ll26
6G
lenc
oe A
lgeb
ra 2
Use
a c
alcu
lato
r to
ap
pro
xim
ate
each
val
ue
to t
hre
e d
ecim
al p
lace
s.
1.#
7.8
$2.
!#
89$3.
3 #25$
4.3 #
!4
$2.
793
!9.
434
2.92
4!
1.58
75.
4 #1.
1$
6.5 #
!0.
1$
7.6 #
5555
$8.
4 #(0
.94)
$2 $
1.02
4!
0.63
14.
208
0.97
0
Sim
pli
fy.
9.#
0.81
$10
.!#
324
$11
.!4 #
256
$12
.6 #
64$0.
9!
18!
42
13.
3 #!
64$
14.
3 #0.
512
$15
.5 #
!24
3$
16.!
4 #12
96$
!4
0.8
!3
!6
17.
5 '(18
.5 #
243x
1$
0 $19
.#(1
4a)2
$20
.#!
(14a
$)2 $
not a
!"4 3"
3x2
14| a|
real
num
ber
21.#
49m
2 t$
8 $22
. '(23
.3 #
!64
r6$
w15 $
24.#
(2x)
8$
7| m| t4
!4r
2 w5
16x4
25.!
4 #62
5s8
$26
.3 #
216p
3$
q9 $27
.#67
6x4
$y6 $
28.
3 #!
27x9
$y1
2$
!5s
26p
q326
x2| y
3 |!
3x3 y
4
29.!
#14
4m$
8 n6
$30
.5 #
!32
x5$
y10
$31
.6 #
(m%
$4)
6$
32.
3 #(2
x%
$1)
3$
!12
m4 | n
3 |!
2xy2
| m$
4|2x
$1
33.!
#49
a10
$b1
6$
34.
4 #(x
!5
$)8 $
35.
3 #34
3d6
$36
.#x2
%1
$0x
%2
$5$
!7| a
5 | b8
(x!
5)2
7d2
| x$
5|37
.RA
DIA
NT
TEM
PER
ATU
RE
The
rmal
sen
sors
mea
sure
an
obje
ct’s
rad
iant
tem
pera
ture
,w
hich
is t
he a
mou
nt o
f en
ergy
rad
iate
d by
the
obj
ect.
The
inte
rnal
tem
pera
ture
of
an
obje
ct is
cal
led
its
kine
tic
tem
pera
ture
.The
form
ula
T r"
T k4 #
e$re
late
s an
obj
ect’s
rad
iant
tem
pera
ture
Tr
to it
s ki
neti
c te
mpe
ratu
re T
k.T
he v
aria
ble
ein
the
for
mul
a is
a m
easu
reof
how
wel
l the
obj
ect
radi
ates
ene
rgy.
If a
n ob
ject
’s k
inet
ic t
empe
ratu
re is
30°
C a
nd
e"
0.94
,wha
t is
the
obj
ect’s
rad
iant
tem
pera
ture
to
the
near
est
tent
h of
a d
egre
e?29
.5)C
38.H
ERO
’S F
OR
MU
LASa
lvat
ore
is b
uyin
g fe
rtili
zer
for
his
tria
ngul
ar g
arde
n.H
e kn
ows
the
leng
ths
of a
ll th
ree
side
s,so
he
is u
sing
Her
o’s
form
ula
to f
ind
the
area
.Her
o’s
form
ula
stat
es t
hat
the
area
of
a tr
iang
le is
#s(
s!
$a)
(s!
$b)
(s!
$c)$
,whe
re a
,b,a
nd c
are
the
leng
ths
of t
he s
ides
of
the
tria
ngle
and
sis
hal
f th
e pe
rim
eter
of
the
tria
ngle
.If
the
leng
ths
of t
he s
ides
of
Salv
ator
e’s
gard
en a
re 1
5 fe
et,1
7 fe
et,a
nd 2
0 fe
et,w
hat
is t
hear
ea o
f th
e ga
rden
? R
ound
you
r an
swer
to
the
near
est
who
le n
umbe
r.12
4 ft2
4|m
|"
516m
2&
25
!10
24&
243
Pra
ctic
e (A
vera
ge)
Root
s of
Rea
l Num
bers
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-5
5-5
© Glencoe/McGraw-Hill A16 Glencoe Algebra 2
Answers (Lesson 5-5)
Rea
din
g t
o L
earn
Math
emati
csRo
ots
of R
eal N
umbe
rs
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-5
5-5
©G
lenc
oe/M
cGra
w-Hi
ll26
7G
lenc
oe A
lgeb
ra 2
Lesson 5-5
Pre-
Act
ivit
yH
ow d
o sq
uar
e ro
ots
app
ly t
o oc
ean
ogra
ph
y?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 5-
5 at
the
top
of
page
245
in y
our
text
book
.
Supp
ose
the
leng
th o
f a
wav
e is
5 f
eet.
Exp
lain
how
you
wou
ld e
stim
ate
the
spee
d of
the
wav
e to
the
nea
rest
ten
th o
f a
knot
usi
ng a
cal
cula
tor.
(Do
not
actu
ally
cal
cula
te t
he s
peed
.)Sa
mpl
e an
swer
:Usi
ng a
cal
cula
tor,
find
the
posi
tive
squa
re ro
ot o
f 5.M
ultip
ly th
is n
umbe
r by
1.34
.Th
en ro
und
the
answ
er to
the
near
est t
enth
.
Rea
din
g t
he
Less
on
1.Fo
r ea
ch r
adic
al b
elow
,ide
ntif
y th
e ra
dica
nd a
nd t
he in
dex.
a.3 #
23$ra
dica
nd:
inde
x:
b.#
15x2
$ra
dica
nd:
inde
x:
c.5 #
!34
3$
radi
cand
:in
dex:
2.C
ompl
ete
the
follo
win
g ta
ble.
(Do
not
actu
ally
fin
d an
y of
the
indi
cate
d ro
ots.
)
Num
ber
Num
ber o
f Pos
itive
Nu
mbe
r of N
egat
ive
Num
ber o
f Pos
itive
Nu
mbe
r of N
egat
ive
Squa
re R
oots
Squa
re R
oots
Cube
Roo
ts
Cube
Roo
ts
271
11
0
!16
00
01
3.St
ate
whe
ther
eac
h of
the
fol
low
ing
is t
rue
or f
alse
.
a.A
neg
ativ
e nu
mbe
r ha
s no
rea
l fou
rth
root
s.tru
e
b.-
#12
1$
repr
esen
ts b
oth
squa
re r
oots
of
121.
true
c.W
hen
you
take
the
fif
th r
oot
of x
5 ,yo
u m
ust
take
the
abs
olut
e va
lue
of x
to id
enti
fyth
e pr
inci
pal f
ifth
roo
t.fa
lse
Hel
pin
g Y
ou
Rem
emb
er
4.W
hat
is a
n ea
sy w
ay t
o re
mem
ber
that
a n
egat
ive
num
ber
has
no r
eal s
quar
e ro
ots
but
has
one
real
cub
e ro
ot?
Sam
ple
answ
er:T
he s
quar
e of
a p
ositi
ve o
r neg
ativ
enu
mbe
r is
posi
tive,
so th
ere
is n
o re
al n
umbe
r who
se s
quar
e is
neg
ativ
e.Ho
wev
er,t
he c
ube
of a
neg
ativ
e nu
mbe
r is
nega
tive,
so a
neg
ativ
enu
mbe
r has
one
real
cub
e ro
ot,w
hich
is a
neg
ativ
e nu
mbe
r.
5!
343
215
x23
23
©G
lenc
oe/M
cGra
w-Hi
ll26
8G
lenc
oe A
lgeb
ra 2
Appr
oxim
atin
g Sq
uare
Roo
tsC
onsi
der
the
follo
win
g ex
pans
ion.
(a%& 2b a&
)2"
a2%
&2 2a ab &%
& 4b a2 2&
"a2
%b
%& 4b a2 2&
Thi
nk w
hat
happ
ens
if a
is v
ery
grea
t in
com
pari
son
to b
.The
ter
m & 4b a2 2&
is v
ery
smal
l and
can
be
disr
egar
ded
in a
n ap
prox
imat
ion.
(a%& 2b a&
)2)
a2
%b
a%
& 2b a&)
#a2
%b
$
Supp
ose
a nu
mbe
r ca
n be
exp
ress
ed a
s a2
%b,
a ,
b.T
hen
an a
ppro
xim
ate
valu
e
of t
he s
quar
e ro
ot is
a%
& 2b a&.Y
ou s
houl
d al
so s
ee t
hat
a!
& 2b a&)
#a2
!b
$.
Use
th
e fo
rmu
la !
a2
("
b"#
a(
" 2b a"to
ap
pro
xim
ate
!10
1"
and
!62
2"
.
a.#
101
$"
#10
0%
$1$
"#
102
%$
1$b.
#62
2$
"#
625
!$
3$"
#25
2!
$3$
Let
a"
10 a
nd b
"1.
Let
a"
25 a
nd b
"3.
#10
1$
) 1
0 %
& 2(1 10
)&
#62
2$
) 2
5 !
& 2(3 25
)&
) 1
0.05
) 2
4.94
Use
th
e fo
rmu
la t
o fi
nd
an
ap
pro
xim
atio
n f
or e
ach
squ
are
root
to
the
nea
rest
hu
nd
red
th.C
hec
k y
our
wor
k w
ith
a c
alcu
lato
r.
1.#
626
$25
.02
2.#
99$9.
953.
#40
2$
20.0
5
4.#
1604
$40
.05
5.#
223
$14
.93
6.#
80$8.
94
7.#
4890
$69
.93
8.#
2505
$50
.05
9.#
3575
$59
.79
10.
#1,
441
$,1
00$
1200
.42
11.
#29
0$
17.0
312
.#
260
$16
.12
13.
Show
tha
t a
!& 2b a&
) #
a2!
b$
for
a ,
b.
$a!
" 2b a"'2
&a2
!b
$" 4b a2 2"
;
disr
egar
d " 4b a2 2"
; $a!
" 2b a"'2
#a2
!b;
a!
" 2b a"#
!a2
!b
"
En
rich
men
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-5
5-5
Exam
ple
Exam
ple
© Glencoe/McGraw-Hill A17 Glencoe Algebra 2
An
swer
s
Answers (Lesson 5-6)
Stu
dy
Gu
ide
and I
nte
rven
tion
Radi
cal E
xpre
ssio
ns
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-6
5-6
©G
lenc
oe/M
cGra
w-Hi
ll26
9G
lenc
oe A
lgeb
ra 2
Lesson 5-6
Sim
plif
y R
adic
al E
xpre
ssio
ns
For a
ny re
al n
umbe
rs a
and
b, a
nd a
ny in
tege
r n,
1:Pr
oduc
t Pro
pert
y of
Rad
ical
s1.
if n
is ev
en a
nd a
and
bar
e bo
th n
onne
gativ
e, th
en n #
ab$"
n #a$
$n #
b$.2.
if n
is od
d, th
en n #
ab$"
n #a$
$n #
b$.
To s
impl
ify
a sq
uare
roo
t,fo
llow
the
se s
teps
:1.
Fact
or t
he r
adic
and
into
as
man
y sq
uare
s as
pos
sibl
e.2.
Use
the
Pro
duct
Pro
pert
y to
isol
ate
the
perf
ect
squa
res.
3.Si
mpl
ify
each
rad
ical
.
Quo
tient
Pro
pert
y of
Rad
ical
sFo
r any
real
num
bers
aan
d b
#0,
and
any
inte
ger n
,1,
n '(
", i
f all
root
s ar
e de
fined
.
To e
limin
ate
radi
cals
fro
m a
den
omin
ator
or
frac
tion
s fr
om a
rad
ican
d,m
ulti
ply
the
num
erat
or a
nd d
enom
inat
or b
y a
quan
tity
so
that
the
rad
ican
d ha
s an
exa
ct r
oot.
n #a$
& n #b$
a & b
Sim
pli
fy 3 !
!16
a"
5 b7
".
3 #!
16a
$5 b
7$
"3 #
(!2)
3$
$2
$a
$3
$a2
$$
(b2 )
3$
$b
$"
!2a
b23 #
2a2 b
$
Sim
pli
fy %&
.
'("
Quo
tient
Pro
perty
"Fa
ctor
into
squ
ares
.
"Pr
oduc
t Pro
perty
"Si
mpl
ify.
"$
Ratio
naliz
e th
ede
nom
inat
or.
"Si
mpl
ify.
2|x|
#10
xy$
&&
15y3
#5y$
& #5y$
2|x|
#2x$
& 3y2 #
5y$
2|x|
#2x$
& 3y2 #
5y$
#(2
x)2
$$
#2x$
&&
#(3
y2 )2
$$
#5y$
#(2
x)2
$$
2x$&
&#
(3y2 )
2$
$5y
$
#8x
3$
& #45
y5$
8x3
& 45y5
8x3
" 45y5
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Sim
pli
fy.
1.5 #
54$15
!6"
2.4 #
32a9
b$
20 $2a
2 | b5 |
4 !2a"
3.#
75x4
y$
7 $5x
2 y3 !
5y"
4.'(
5.'(
6.3 '(
pq3 !
5p2
""
" 10p5 q
3&40
| a3 |
b!2b"
"" 14
a6 b3
&98
6!5"
"25
36 & 125
©G
lenc
oe/M
cGra
w-Hi
ll27
0G
lenc
oe A
lgeb
ra 2
Op
erat
ion
s w
ith
Rad
ical
sW
hen
you
add
expr
essi
ons
cont
aini
ng r
adic
als,
you
can
add
only
like
ter
ms
or l
ike
rad
ical
exp
ress
ion
s.T
wo
radi
cal e
xpre
ssio
ns a
re c
alle
d li
kera
dica
l ex
pres
sion
sif
bot
h th
e in
dice
s an
d th
e ra
dica
nds
are
alik
e.
To m
ulti
ply
radi
cals
,use
the
Pro
duct
and
Quo
tien
t P
rope
rtie
s.Fo
r pr
oduc
ts o
f th
e fo
rm
(a#
b$%
c#d$)
$(e#
f$%
g#h$)
,use
the
FO
IL m
etho
d.To
rat
iona
lize
deno
min
ator
s,us
e co
nju
gate
s.N
umbe
rs o
f th
e fo
rm a
#b$
%c#
d$an
d a#
b$!
c#d$,
whe
re a
,b,c
,and
dar
era
tion
al n
umbe
rs,a
re c
alle
d co
njug
ates
.The
pro
duct
of
conj
ugat
es is
alw
ays
a ra
tion
alnu
mbe
r.
Sim
pli
fy 2
!50"
$4!
500
"!
6!12
5"
.
2#50$
%4#
500
$!
6#12
5$
"2#
52$
2$
%4#
102
$5
$!
6#52
$5
$Fa
ctor
usin
g sq
uare
s.
"2
$5
$#
2$%
4 $
10 $
#5$
!6
$5
$#
5$Si
mpl
ify s
quar
e ro
ots.
"10
#2$
%40
#5$
!30
#5$
Mul
tiply.
"10
#2$
%10
#5$
Com
bine
like
radi
cals.
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Radi
cal E
xpre
ssio
ns
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-6
5-6
Exam
ple1
Exam
ple1
Sim
pli
fy (2
!3"
!4!
2")(!
3"$
2!2")
.
(2#
3$!
4#2$)
(#3$
%2#
2$)"
2#3$
$#
3$%
2#3$
$2#
2$!
4#2$
$#
3$!
4#2$
$2#
2$"
6 %
4#6$
!4#
6$!
16"
!10
Sim
pli
fy
.
"$
" " "11
!5#
5$&
& 4
6 !
5#5$
%5
&&
9 !
5
6 !
2 #5$
!3#
5$%
(#5$)
2&
&&
32!
(#5$ )
2
3 !
#5$
& 3 !
#5$
2 !
#5$
& 3 %
#5$
2 !
#5$
& 3 %
#5$
2 !
!5"
" 3 $
!5"
Exam
ple2
Exam
ple2
Exam
ple3
Exam
ple3
Exer
cises
Exer
cises
Sim
pli
fy.
1.3 #
2$%
#50$
!4#
8$2.
#20$
%#
125
$!
#45$
3.#
300
$!
#27$
!#
75$
04!
5"2!
3"
4.3 #
81$$
3 #24$
5.3 #
2$(3 #
4$%
3 #12$
)6.
2#3$(
#15$
%#
60$)
63 !
9"2
$2
3 !3"
18!
5"
7.(2
%3#
7$)(4
%#
7$)8.
(6#
3$!
4#2$)
(3#
3$%
#2$)
9.(4
#2$
!3#
5$)(2
#20
%$
5$)
29 $
14!
7"46
!6!
6"40
!2"
!30
!5"
10.
511
.5
$3!
2"12
.13
!3"
!23
"" 11
5 !
3#3$
&&
1 %
2#3$
4 %
#2$
& 2 !
#2$
5#48$
%#
75$&
&5#
3$
© Glencoe/McGraw-Hill A18 Glencoe Algebra 2
Answers (Lesson 5-6)
Skil
ls P
ract
ice
Radi
cal E
xpre
ssio
ns
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-6
5-6
©G
lenc
oe/M
cGra
w-Hi
ll27
1G
lenc
oe A
lgeb
ra 2
Lesson 5-6
Sim
pli
fy.
1.#
24$2!
6"2.
#75$
5!3"
3.3 #
16$2
3 !2"
4.!
4 #48$
!2
4 !3"
5.4#
50x5
$20
x2!
2x"6.
4 #64
a4b
$4 $
2| ab|
4 !4"
7.3 '(!
d2 f
5!
f3 !
d2 f
2"
8.'(
s2t
| s| !
t"
9.!'(
!10
.3 '(
11. '(&2 5g z3 &
12.(
3 #3$)
(5#
3$)45
13.(
4 #12$
)(3#
20$)
48!
15"14
.#2$
%#
8$%
#50$
8!2"
15.#
12$!
2#3$
%#
108
$6!
3"16
.8#
5$!
#45$
!#
80$!
5"
17.2
#48$
!#
75$!
#12$
!3"
18.(
2 %
#3$)
(6 !
#2$)
12 !
2!2"
$6!
3"!
!6"
19.(
1 !
#5$)
(1 %
#5$)
!4
20.(
3 !
#7$)
(5 %
#2$)
15 $
3!2"
!5!
7"!
!14"
21.(
#2$
!#
6$)2
8 !
4!3"
22.
23.
24.
40 $
5!6"
"" 58
5& 8
!#
6$12
!4!
2""
" 74
& 3 %
#2$
21 $
3!2"
"" 47
3& 7
!#
2$
g!10
gz"
"" 5z
3 !6 "
"3
2 & 9!
21""
73 & 7
5 " 625 & 36
1 " 21 & 8
©G
lenc
oe/M
cGra
w-Hi
ll27
2G
lenc
oe A
lgeb
ra 2
Sim
pli
fy.
1.#
540
$6!
15"2.
3 #!
432
$!
63 !
2"3.
3 #12
8$
43 !
2"
4.!
4 #40
5$
!3
4 !5"
5.3 #
!50
0$
0$!
103 !
5"6.
5 #!
121
$5$
!3
5 !5"
7.3 #
125t
6 w$
2 $5t
23 !
w2
"8.
4 #48
v8z
$13 $
2v2 z
34 !
3z"9.
3 #8g
3 k8
$2g
k23 !
k2 "
10.#
45x3
y$
8 $3x
y4!
5x"11
. '(12
.3 '(
3 !9"
13. '(
c4d
7c2
d3 !
2d"14
. '(15
.4 '(
16.(
3#15$
)(!4#
45$)
17.(
2#24$
)(7#
18$)
18.#
810
$%
#24
0$
!#
250
$!
180!
3"16
8!3"
4!10"
$4!
15"
19.6
#20$
%8#
5$!
5#45$
20.8
#48$
!6#
75$%
7#80$
21.(
3#2$
%2#
3$)2
5!5"
2!3" $
28!
5"30
$12
!6"
22.(
3 !
#7$)
223
.(#
5$!
#6$)
(#5$
%#
2$)24
.(#
2$%
#10$
)(#2$
!#
10$)
16 !
6!7"
5 $
!10"
!!
30"!
2!3"
!8
25.(
1 %
#6$)
(5 !
#7$)
26.(
#3$
%4#
7$)2
27.(
#10
8$
!6#
3$)2
5 !
!7"
$5!
6"!
!42"
115
$8!
21"0
28.
!15"
$2!
3"29
.6!
2"$
630
.
31.
32.
27 $
11!
6"33
.
34.B
RA
KIN
GT
he f
orm
ula
s"
2#5!$
esti
mat
es t
he s
peed
sin
mile
s pe
r ho
ur o
f a
car
whe
nit
leav
es s
kid
mar
ks !
feet
long
.Use
the
for
mul
a to
wri
te a
sim
plif
ied
expr
essi
on f
or s
if
!"
85.T
hen
eval
uate
sto
the
nea
rest
mile
per
hou
r.10
!17"
;41
mi/h
35.P
YTH
AG
OR
EAN
TH
EOR
EMT
he m
easu
res
of t
he le
gs o
f a
righ
t tr
iang
le c
an b
ere
pres
ente
d by
the
exp
ress
ions
6x2
yan
d 9x
2 y.U
se t
he P
ytha
gore
an T
heor
em t
o fi
nd a
sim
plif
ied
expr
essi
on f
or t
he m
easu
re o
f th
e hy
pote
nuse
.3x
2 | y| !
13"
6 $
5!x"
$x
""
4 !
x3
%#
x$& 2
!#
x$3
%#
6$&
&5
!#
24$8
$5!
2""
" 23
%#
2$& 2
!#
2$
17 !
!3"
"" 13
5 %
#3$
& 4 %
#3$
6& #
2$!
1#
3$& #
5$!
2
4 !72
a"
"3a
8& 9a
33a
2 !a"
"8b
29a
5& 64
b41 " 16
1& 12
8
216
& 24!
11""
311 & 9
Pra
ctic
e (A
vera
ge)
Radi
cal E
xpre
ssio
ns
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-6
5-6
© Glencoe/McGraw-Hill A19 Glencoe Algebra 2
An
swer
s
Answers (Lesson 5-6)
Rea
din
g t
o L
earn
Math
emati
csRa
dica
l Exp
ress
ions
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-6
5-6
©G
lenc
oe/M
cGra
w-Hi
ll27
3G
lenc
oe A
lgeb
ra 2
Lesson 5-6
Pre-
Act
ivit
yH
ow d
o ra
dic
al e
xpre
ssio
ns
app
ly t
o fa
llin
g ob
ject
s?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 5-
6 at
the
top
of
page
250
in y
our
text
book
.
Des
crib
e ho
w y
ou c
ould
use
the
for
mul
a gi
ven
in y
our
text
book
and
aca
lcul
ator
to
find
the
tim
e,to
the
nea
rest
ten
th o
f a
seco
nd,t
hat
it w
ould
take
for
the
wat
er b
allo
ons
to d
rop
22 f
eet.
(Do
not
actu
ally
cal
cula
te t
heti
me.
)Sa
mpl
e an
swer
:Mul
tiply
22
by 2
(giv
ing
44) a
nd d
ivid
eby
32.
Use
the
calc
ulat
or to
find
the
squa
re ro
ot o
f the
resu
lt.Ro
und
this
squ
are
root
to th
e ne
ares
t ten
th.
Rea
din
g t
he
Less
on
1.C
ompl
ete
the
cond
itio
ns t
hat
mus
t be
met
for
a ra
dica
l exp
ress
ion
to b
e in
sim
plifi
ed fo
rm.
•T
he
nis
as
as p
ossi
ble.
•T
he
cont
ains
no
(oth
er t
han
1) t
hat
are
nth
pow
ers
of a
(n)
or p
olyn
omia
l.
•T
he r
adic
and
cont
ains
no
.
•N
o ap
pear
in t
he
.
2.a.
Wha
t ar
e co
njug
ates
of
radi
cal e
xpre
ssio
ns u
sed
for?
to ra
tiona
lize
bino
mia
lde
nom
inat
ors
b.H
ow w
ould
you
use
a c
onju
gate
to
sim
plif
y th
e ra
dica
l exp
ress
ion
?
Mul
tiply
num
erat
or a
nd d
enom
inat
or b
y 3
$!
2".c.
In o
rder
to
sim
plif
y th
e ra
dica
l exp
ress
ion
in p
art
b,tw
o m
ulti
plic
atio
ns a
re
nece
ssar
y.T
he m
ulti
plic
atio
n in
the
num
erat
or w
ould
be
done
by
the
met
hod,
and
the
mul
tipl
icat
ion
in t
he d
enom
inat
or w
ould
be
done
by
find
ing
the
of
.
Hel
pin
g Y
ou
Rem
emb
er
3.O
ne w
ay t
o re
mem
ber
som
ethi
ng is
to
expl
ain
it t
o an
othe
r pe
rson
.Whe
n ra
tion
aliz
ing
the
deno
min
ator
in t
he e
xpre
ssio
n ,m
any
stud
ents
thi
nk t
hey
shou
ld m
ulti
ply
num
erat
or
and
deno
min
ator
by
.How
wou
ld y
ou e
xpla
in t
o a
clas
smat
e w
hy t
his
is in
corr
ect
and
wha
t he
sho
uld
do in
stea
d.Sa
mpl
e an
swer
:Bec
ause
you
are
wor
king
with
cube
root
s,no
t squ
are
root
s,yo
u ne
ed to
mak
e th
e ra
dica
nd in
the
deno
min
ator
a p
erfe
ct c
ube,
not a
per
fect
squ
are.
Mul
tiply
num
erat
or a
nd
deno
min
ator
by
to
mak
e th
e de
nom
inat
or 3 !
8",w
hich
equ
als
2.3 !
4"" 3 !
4 "
3 #2$
& 3 #2$
1& 3 #
2$
squa
res
two
diffe
renc
e
FOIL
1 %
#2$
& 3 !
#2$
deno
min
ator
radi
cals
fract
ions
inte
ger
fact
ors
radi
cand
smal
lin
dex
©G
lenc
oe/M
cGra
w-Hi
ll27
4G
lenc
oe A
lgeb
ra 2
Spec
ial P
rodu
cts
with
Rad
ical
sN
otic
e th
at (#
3$)(#
3$)"
3,or
(#3$)
2"
3.
In g
ener
al,(
#x$)
2"
xw
hen
x .
0.
Als
o,no
tice
tha
t (#
9$)(#
4$)"
#36$
.
In g
ener
al,(
#x$)
(#y$)
"#
xy$w
hen
xan
d y
are
not
nega
tive
.
You
can
use
thes
e id
eas
to f
ind
the
spec
ial p
rodu
cts
belo
w.
(#a$
%#
b$)(#
a$!
#b$)
"(#
a$)2
!(#
b$)2
"a
!b
(#a$
%#
b$)2
"(#
a$)2
%2#
ab$%
(#b$)
2"
a%
2#ab$
%b
(#a$
!#
b$)2
"(#
a$)2
!2#
ab$%
(#b$)
2"
a!
2#ab$
%b
Fin
d t
he
pro
du
ct:(
!2"
$!
5")( !
2"!
!5")
.
(#2$
%#
5$)(#
2$!
#5$)
"(#
2$)2
!(#
5$)2
"2
!5
"!
3
Eva
luat
e ( !
2"$
!8")
2 .
(#2$
%#
8$)2
"(#
2$)2
%2#
2$#8$
%(#
8$)2
"2
%2#
16$%
8 "
2 %
2(4)
%8
"2
%8
%8
"18
Mu
ltip
ly.
1.(#
3$!
#7$)
(#3$
%#
7$)!
42.
(#10$
%#
2$)(#
10$!
#2$)
8
3.(#
2x$!
#6$)
(#2x$
!#
6$)2x
!6
4.(#
3$!
27)2
12
5.(#
1000
$%
#10$
)212
106.
(#y$
%#
5$)(#
y$!
#5$)
y!
5
7.(#
50$!
#x$)
250
!10
!2x"
$x
8.(#
x$%
20)2
x$
4!5x"
$20
You
can
exte
nd t
hese
idea
s to
pat
tern
s fo
r su
ms
and
diff
eren
ces
of c
ubes
.St
udy
the
patt
ern
belo
w.
(3 #8$
! 3 #
x$)(3 #
82 $%
3 #8x$
%3 #
x2 $)"
3 #83 $
! 3 #
x3 $"
8 !
x
Mu
ltip
ly.
9.(3 #
2$!
3 #5$)
(3 #22 $
%3 #
10$%
3 #52 $
)!
3
10.(
3 #y$
%3 #
w$)(
3 #y2 $
!3 #
yw$%
3 #w
2 $)
y$
w
11.
(3 #7$
%3 #
20$)(
3 #72 $
!3 #
140
$%
3 #20
2$
)27
12.(
3 #11$
!3 #
8$)(3 #
112
$%
3 #88$
%3 #
82 $)
3
En
rich
men
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-6
5-6
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
© Glencoe/McGraw-Hill A20 Glencoe Algebra 2
Answers (Lesson 5-7)
Stu
dy
Gu
ide
and I
nte
rven
tion
Ratio
nal E
xpon
ents
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-7
5-7
©G
lenc
oe/M
cGra
w-Hi
ll27
5G
lenc
oe A
lgeb
ra 2
Lesson 5-7
Rat
ion
al E
xpo
nen
ts a
nd
Rad
ical
s
Defin
ition
of b
" n1 "Fo
r any
real
num
ber b
and
any
posit
ive in
tege
r n,
b&1 n&"
n #b$,
exc
ept w
hen
b)
0 an
d n
is ev
en.
Defin
ition
of b
"m n"Fo
r any
non
zero
real
num
ber b
, and
any
inte
gers
man
d n,
with
n,
1,
b&m n&"
n #bm $
"(n #
b$)m, e
xcep
t whe
n b
)0
and
nis
even
.
Wri
te 2
8"1 2"in
rad
ical
for
m.
Not
ice
that
28
,0.
28&1 2&
"#
28$"
#22
$7
$"
#22 $
$#
7$"
2#7$
Eva
luat
e $
'"1 3" .
Not
ice
that
!8
)0,
!12
5 )
0,an
d 3
is o
dd.
!"&1 3&
" " "2 & 5!
2& !
53 #!
8$
& 3 #!
125
$!
8& !
125
!8
" !12
5Ex
ampl
e1Ex
ampl
e1Ex
ampl
e2Ex
ampl
e2
Exer
cises
Exer
cises
Wri
te e
ach
exp
ress
ion
in
rad
ical
for
m.
1.11
&1 7&2.
15&1 3&
3.30
0&3 2&
7 !11"
3 !15"
!30
03"
Wri
te e
ach
rad
ical
usi
ng
rati
onal
exp
onen
ts.
4.#
47$5.
3 #3a
5 b2
$6.
4 #16
2p5
$
47"1 2"
3"1 3" a"5 3" b"2 3"3
*2"1 4"
*p"5 4"
Eva
luat
e ea
ch e
xpre
ssio
n.
7.!
27&2 3&
8.9.
(0.0
004)
&1 2&
90.
02
10.8
&2 3&$
4&3 2&11
.12
.
321 " 2
1 " 4
16!
&1 2&
& (0.2
5)&1 2&
144!
&1 2&
& 27!
&1 3&
1 " 105!&1 2&
& 2#5$
©G
lenc
oe/M
cGra
w-Hi
ll27
6G
lenc
oe A
lgeb
ra 2
Sim
plif
y Ex
pre
ssio
ns
All
the
prop
erti
es o
f po
wer
s fr
om L
esso
n 5-
1 ap
ply
to r
atio
nal
expo
nent
s.W
hen
you
sim
plif
y ex
pres
sion
s w
ith
rati
onal
exp
onen
ts,l
eave
the
exp
onen
t in
rati
onal
for
m,a
nd w
rite
the
exp
ress
ion
wit
h al
l pos
itiv
e ex
pone
nts.
Any
exp
onen
ts in
the
deno
min
ator
mus
t be
pos
itiv
e in
tege
rs
Whe
n yo
u si
mpl
ify
radi
cal e
xpre
ssio
ns,y
ou m
ay u
se r
atio
nal e
xpon
ents
to
sim
plif
y,bu
t yo
uran
swer
sho
uld
be in
rad
ical
for
m.U
se t
he s
mal
lest
inde
x po
ssib
le.
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Ratio
nal E
xpon
ents
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-7
5-7
Sim
pli
fy y
"2 3"*
y"3 8" .
y&2 3&$
y&3 8&"
y&2 3&%
&3 8&"
y&2 25 4&
Sim
pli
fy 4 !
144x
6"
.4 #
144x
6$
"(1
44x6
)&1 4&
"(2
4$
32$
x6)&1 4&
"(2
4 )&1 4&
$(3
2 )&1 4&
$(x
6 )&1 4&
"2
$3&1 2&
$x&3 2&
"2x
$(3
x)&1 2&
"2x
#3x$
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Sim
pli
fy e
ach
exp
ress
ion
.
1.x&4 5&
$x&6 5&
2.! y&2 3& "&3 4&
3.p&4 5&
$p& 17 0&
x2y"1 2"
p"3 2"
4.! m
!&6 5& "&2 5&
5.x!
&3 8&$
x&4 3&6.
! s!&1 6& "!
&4 3&
x"2 23 4"s"2 9"
7.8.
! a&2 3& "&6 5&$! a&2 5& "3
9.
p"2 3"a2
10.
6 #12
8$
11.
4 #49$
12.
5 #28
8$
2!2"
!7"
25 !
9"
13.#
32$$
3#16$
14.
3 #25$
$#
125
$15
.6 #
16$
48!
2"25
6 !5"
3 !4"
16.
17.#
$3 #48$
18.
6 !48"
!a"
6 !b5 "
"b
x !3"
!6 !
35 ""
" 6
a3 #
b4 $& #
ab3
$x
!3 #
3$&
#12$
x"5 6"
" xx!&1 2&
& x!&1 3&
p & p&1 3&1 " m3
© Glencoe/McGraw-Hill A21 Glencoe Algebra 2
An
swer
s
Answers (Lesson 5-7)
Skil
ls P
ract
ice
Ratio
nal E
xpon
ents
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-7
5-7
©G
lenc
oe/M
cGra
w-Hi
ll27
7G
lenc
oe A
lgeb
ra 2
Lesson 5-7
Wri
te e
ach
exp
ress
ion
in
rad
ical
for
m.
1.3&1 6&
6 !3"
2.8&1 5&
5 !8"
3.12
&2 3&3 !
122
"or
(3 !12"
)24.
(s3)&3 5&
s5 !
s4 "
Wri
te e
ach
rad
ical
usi
ng
rati
onal
exp
onen
ts.
5.#
51$51
"1 2"6.
3 #37$
37"1 3"
7.4 #
153
$15
"3 4"8.
3 #6x
y2$
6"1 3" x"1 3" y"2 3"
Eva
luat
e ea
ch e
xpre
ssio
n.
9.32
&1 5&2
10.8
1&1 4&3
11.2
7!&1 3&
12.4
!&1 2&
13.1
6&3 2&64
14.(
!24
3)&4 5&
81
15.2
7&1 3&$
27&5 3&
729
16. !
"&3 2&
Sim
pli
fy e
ach
exp
ress
ion
.
17.c
&1 52 &$
c&3 5&c3
18.m
&2 9&$
m&1 96 &
m2
19.! q
&1 2& "3q"3 2"
20.p
!&1 5&
21.x
!& 16 1&
22.
x" 15 2"
23.
24.
25.12 #
64$!
2"26
.8 #
49a8
b$
2 $| a
|4 !7b"
n"2 3"
" nn&1 3&
& n&1 6&$
n&1 2&
y"1 4"
" yy!
&1 2&
& y&1 4&
x&2 3&
& x&1 4&
x" 15 1"
" x
p"4 5"
" p8 " 274 & 9
1 " 21 " 3
©G
lenc
oe/M
cGra
w-Hi
ll27
8G
lenc
oe A
lgeb
ra 2
Wri
te e
ach
exp
ress
ion
in
rad
ical
for
m.
1.5&1 3&
2.6&2 5&
3.m
&4 7&4.
(n3 )&2 5&
3 !5"
5 !62 "
or (5 !
6")27 !
m4
"or
(7 !m"
)4n
5 !n"
Wri
te e
ach
rad
ical
usi
ng
rati
onal
exp
onen
ts.
5.#
79$6.
4 #15
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7.3 #
27m
6 n$
4 $8.
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Eva
luat
e ea
ch e
xpre
ssio
n.
9.81
&1 4&3
10.1
024!
&1 5&11
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12.!
256!
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fy e
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ress
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.
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&4 7&$
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30.E
LEC
TRIC
ITY
The
am
ount
of
curr
ent
in a
mpe
res
Ith
at a
n ap
plia
nce
uses
can
be
calc
ulat
ed u
sing
the
for
mul
a I
"!
"&1 2& ,whe
re P
is t
he p
ower
in w
atts
and
Ris
the
resi
stan
ce in
ohm
s.H
ow m
uch
curr
ent
does
an
appl
ianc
e us
e if
P"
500
wat
ts a
nd
R"
10 o
hms?
Rou
nd y
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answ
er t
o th
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ares
t te
nth.
7.1
amps
31.B
USI
NES
SA
com
pany
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t pr
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Pra
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e (A
vera
ge)
Ratio
nal E
xpon
ents
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-7
5-7
© Glencoe/McGraw-Hill A22 Glencoe Algebra 2
Answers (Lesson 5-7)
Rea
din
g t
o L
earn
Math
emati
csRa
tiona
l Exp
onen
ts
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-7
5-7
©G
lenc
oe/M
cGra
w-Hi
ll27
9G
lenc
oe A
lgeb
ra 2
Lesson 5-7
Pre-
Act
ivit
yH
ow d
o ra
tion
al e
xpon
ents
ap
ply
to
astr
onom
y?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 5-
7 at
the
top
of
page
257
in y
our
text
book
.
The
form
ula
in t
he in
trod
ucti
on c
onta
ins
the
expo
nent
.W
hat
do y
ou t
hink
it
mig
ht m
ean
to r
aise
a n
umbe
r to
the
po
wer
?
Sam
ple
answ
er:T
ake
the
fifth
root
of t
he n
umbe
r and
squ
are
it.
Rea
din
g t
he
Less
on
1.C
ompl
ete
the
follo
win
g de
fini
tion
s of
rat
iona
l exp
onen
ts.
•Fo
r an
y re
al n
umbe
r b
and
for
any
posi
tive
inte
ger
n,b& n1 &
"ex
cept
whe
n b
and
nis
.
•Fo
r an
y no
nzer
o re
al n
umbe
r b,
and
any
inte
gers
man
d n,
wit
h n
,
b&m n&"
"
,exc
ept
whe
n b
and
nis
.
2.C
ompl
ete
the
cond
itio
ns t
hat
mus
t be
met
in o
rder
for
an
expr
essi
on w
ith
rati
onal
expo
nent
s to
be
sim
plif
ied.
•It
has
no
expo
nent
s.
•It
has
no
expo
nent
s in
the
.
•It
is n
ot a
fr
acti
on.
•T
he
of a
ny r
emai
ning
is
the
nu
mbe
r po
ssib
le.
3.M
arga
rita
and
Pie
rre
wer
e w
orki
ng t
oget
her
on t
heir
alg
ebra
hom
ewor
k.O
ne e
xerc
ise
aske
d th
em t
o ev
alua
te t
he e
xpre
ssio
n 27
&4 3& .Mar
gari
ta t
houg
ht t
hat
they
sho
uld
rais
e 27
to
the
four
th p
ower
fir
st a
nd t
hen
take
the
cub
e ro
ot o
f th
e re
sult
.Pie
rre
thou
ght
that
they
sho
uld
take
the
cub
e ro
ot o
f 27
fir
st a
nd t
hen
rais
e th
e re
sult
to
the
four
th p
ower
.W
hose
met
hod
is c
orre
ct?
Both
met
hods
are
cor
rect
.
Hel
pin
g Y
ou
Rem
emb
er
4.So
me
stud
ents
hav
e tr
oubl
e re
mem
beri
ng w
hich
par
t of
the
fra
ctio
n in
a r
atio
nal
expo
nent
giv
es t
he p
ower
and
whi
ch p
art
give
s th
e ro
ot.H
ow c
an y
our
know
ledg
e of
inte
ger
expo
nent
s he
lp y
ou t
o ke
ep t
his
stra
ight
?Sa
mpl
e an
swer
:An
inte
ger
expo
nent
can
be
writ
ten
as a
ratio
nal e
xpon
ent.
For e
xam
ple,
23&
2"3 1" .Yo
u kn
ow th
at th
is m
eans
that
2 is
rais
ed to
the
third
pow
er,s
o th
enu
mer
ator
mus
t giv
e th
e po
wer
,and
,the
refo
re,t
he d
enom
inat
or m
ust
give
the
root
.
leas
tra
dica
lin
dexco
mpl
exde
nom
inat
orfra
ctio
nal
nega
tive
even
+0
(n !b")
mn !
bm ",
1ev
en+
0
n !b "
2 & 5
2 & 5
©G
lenc
oe/M
cGra
w-Hi
ll28
0G
lenc
oe A
lgeb
ra 2
Less
er-K
now
n G
eom
etric
For
mul
asM
any
geom
etri
c fo
rmul
as in
volv
e ra
dica
l exp
ress
ions
.
Mak
e a
dra
win
g to
ill
ust
rate
eac
h o
f th
e fo
rmu
las
give
n o
n t
his
pag
e.T
hen
eva
luat
e th
e fo
rmu
la f
or t
he
give
n v
alu
e of
th
e va
riab
le.R
oun
d
answ
ers
to t
he
nea
rest
hu
nd
red
th.
En
rich
men
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-7
5-7
1.T
he a
rea
of a
n is
osce
les
tria
ngle
.Tw
osi
des
have
leng
th a
;the
oth
er s
ide
has
leng
th c
.Fin
d A
whe
n a
"6
and
c"
7.
A"
& 4c & #4a
2!
$c2 $
A(
17.0
6
3.T
he a
rea
of a
reg
ular
pen
tago
n w
ith
asi
de o
f le
ngth
a.F
ind
Aw
hen
a"
4.
A"
&a 42 &#
25 %
$10
#5$
$A
(27
.53
5.T
he v
olum
e of
a r
egul
ar t
etra
hedr
onw
ith
an e
dge
of le
ngth
a.F
ind
Vw
hen
a"
2.
V"
& 1a 23 &#
2$V
(0.
94
7.H
eron
’s F
orm
ula
for
the
area
of
a tr
iang
le u
ses
the
sem
i-pe
rim
eter
s,
whe
re s
"&a
%2b
%c
&.T
he s
ides
of
the
tria
ngle
hav
e le
ngth
s a,
b,an
d c.
Fin
d A
whe
n a
"3,
b"
4,an
d c
"5.
A"
#s(
s!
$a)
(s!
$b)
(s!
$c)$
A&
6
2.T
he a
rea
of a
n eq
uila
tera
l tri
angl
e w
ith
a si
de o
f le
ngth
a.F
ind
Aw
hen
a"
8.
A"
&a 42 &#
3$A
(27
.71
4.T
he a
rea
of a
reg
ular
hex
agon
wit
h a
side
of
leng
th a
.Fin
d A
whe
n a
"9.
A"
&3 2a2 &#
3$A
(21
0.44
6.T
he a
rea
of t
he c
urve
d su
rfac
e of
a r
ight
cone
wit
h an
alt
itud
e of
han
d ra
dius
of
base
r.F
ind
S w
hen
r"
3 an
d h
"6.
S"
+r#
r2%
h$
2 $S
(63
.22
8.T
he r
adiu
s of
a c
ircl
e in
scri
bed
in a
giv
entr
iang
le a
lso
uses
the
sem
i-pe
rim
eter
.F
ind
rw
hen
a"
6,b
"7,
and
c"
9.
r" r(
1.91
rb ca
#s(
s!
$a)
(s!
$b)
(s!
$c)$
&&
&s
r
h
aa
aa
aa
a
a
a
a
b
c
aa
a
aa
a
aa
a
a
a
a
c
a
© Glencoe/McGraw-Hill A23 Glencoe Algebra 2
An
swer
s
Answers (Lesson 5-8)
Stu
dy
Gu
ide
and I
nte
rven
tion
Radi
cal E
quat
ions
and
Ineq
ualit
ies
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-8
5-8
©G
lenc
oe/M
cGra
w-Hi
ll28
1G
lenc
oe A
lgeb
ra 2
Lesson 5-8
Solv
e R
adic
al E
qu
atio
ns
The
fol
low
ing
step
s ar
e us
ed in
sol
ving
equ
atio
ns t
hat
have
vari
able
s in
the
rad
ican
d.So
me
alge
brai
c pr
oced
ures
may
be
need
ed b
efor
e yo
u us
e th
ese
step
s.
Step
1Is
olat
e th
e ra
dica
l on
one
side
of th
e eq
uatio
n.St
ep 2
To e
limin
ate
the
radi
cal,
raise
eac
h sid
e of
the
equa
tion
to a
pow
er e
qual
to th
e in
dex
of th
e ra
dica
l.St
ep 3
Solve
the
resu
lting
equa
tion.
Step
4Ch
eck
your
sol
utio
n in
the
orig
inal
equ
atio
n to
mak
e su
re th
at y
ou h
ave
not o
btai
ned
any
extra
neou
s ro
ots.
Sol
ve 2
!4x
$"
8"!
4 &
8.
2#4x
%8
$!
4 "
8O
rigin
al e
quat
ion
2#4x
%8
$"
12Ad
d 4
to e
ach
side.
#4x
%8
$"
6Is
olat
e th
e ra
dica
l.
4x%
8 "
36Sq
uare
eac
h sid
e.
4x"
28Su
btra
ct 8
from
eac
h sid
e.
x"
7Di
vide
each
sid
e by
4.
Ch
eck
2#4(
7) %
$8$
!4
!8
2#36$
!4
!8
2(6)
!4
!8
8 "
8T
he s
olut
ion
x"
7 ch
ecks
.
Sol
ve !
3x$
"1"
&!
5x"!
1.
#3x
%1
$"
#5x$
!1
Orig
inal
equ
atio
n
3x%
1"
5x!
2#5x$
%1
Squa
re e
ach
side.
2#5x$
"2x
Sim
plify
.
#5x$
"x
Isol
ate
the
radi
cal.
5x"
x2Sq
uare
eac
h sid
e.
x2!
5x"
0Su
btra
ct 5
xfro
m e
ach
side.
x(x
!5)
"0
Fact
or.
x"
0 or
x"
5C
hec
k#
3(0)
%$
1$"
1,bu
t #
5(0)
$!
1 "
!1,
so 0
isno
t a
solu
tion
.#
3(5)
%$
1$"
4,an
d #
5(5)
$!
1 "
4,so
the
solu
tion
is x
"5.
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Sol
ve e
ach
equ
atio
n.
1.3
%2x
#3$
"5
2.2#
3x%
4$
%1
"15
3.8
%#
x%
1$
"2
15no
sol
utio
n
4.#
5 !
x$
!4
"6
5.12
%#
2x!
1$
"4
6.#
12 !
$x$
"0
!95
no s
olut
ion
12
7.#
21$!
#5x
!4
$"
08.
10 !
#2x$
"5
9.#
x2%
7$
x$"
#7x
!9
$
512
.5no
sol
utio
n
10.4
3 #2x
%1
$1$
!2
"10
11.2
#x
%11
$"
#x
%2
$%
#3x
!6
$12
.#9x
!1
$1$
"x
%1
814
3,4
!3"
"3
©G
lenc
oe/M
cGra
w-Hi
ll28
2G
lenc
oe A
lgeb
ra 2
Solv
e R
adic
al In
equ
alit
ies
A r
adic
al i
neq
ual
ity
is a
n in
equa
lity
that
has
a v
aria
ble
in a
rad
ican
d.U
se t
he f
ollo
win
g st
eps
to s
olve
rad
ical
ineq
ualit
ies.
Step
1If
the
inde
x of
the
root
is e
ven,
iden
tify
the
valu
es o
f the
var
iabl
e fo
r whi
ch th
e ra
dica
nd is
non
nega
tive.
Step
2So
lve th
e in
equa
lity a
lgeb
raica
lly.
Step
3Te
st v
alue
s to
che
ck y
our s
olut
ion.
Sol
ve 5
!!
20x
$"
4"-
!3.
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Radi
cal E
quat
ions
and
Ineq
ualit
ies
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-8
5-8
Exam
ple
Exam
ple
Sinc
e th
e ra
dica
nd o
f a
squa
re r
oot
mus
t be
gre
ater
tha
n or
equ
al t
oze
ro,f
irst
sol
ve20
x%
4 .
0.20
x%
4 .
020
x.
!4
x.
!1 & 5
Now
sol
ve 5
!#
20x
%$
4$.
!3.
5 !
#20
x%
$4$
.!
3O
rigin
al in
equa
lity
#20
x%
$4$
(8
Isol
ate
the
radi
cal.
20x
%4
(64
Elim
inat
e th
e ra
dica
l by
squa
ring
each
sid
e.
20x
(60
Subt
ract
4 fr
om e
ach
side.
x(
3Di
vide
each
sid
e by
20.
It a
ppea
rs t
hat
!(
x(
3 is
the
sol
utio
n.Te
st s
ome
valu
es.
x&
!1
x&
0x
&4
#20
(!1
$) %
4$
is no
t a re
al
5 !
#20
(0)
$%
4$
"3,
so
the
5 !
#20
(4)
$%
4$
&!
4.2,
so
num
ber,
so th
e in
equa
lity is
ineq
uality
is s
atisf
ied.
th
e in
equa
lity is
not
not s
atisf
ied.
satis
fied
The
refo
re t
he s
olut
ion
!(
x(
3 ch
ecks
.
Sol
ve e
ach
in
equ
alit
y.
1.#
c!
2$
%4
.7
2.3#
2x!
1$
%6
)15
3.#
10x
%$
9$!
2 ,
5
c-
11.
x+
5x
,4
4.5
3 #x
%2
$!
8 )
25.
8 !
#3x
%4
$.
36.
#2x
%8
$!
4 ,
2
x+
6!
.x
.7
x,
14
7.9
!#
6x%
3$
.6
8.(
49.
2#5x
!6
$!
1 )
5
!.
x.
1x
-8
.x
+3
10.#
2x%
1$
2$%
4 .
1211
.#2d
%$
1$%
#d$
(5
12.4
#b
%3
$!
#b
!2
$.
10
x-
260
.d
.4
b-
6
6 " 51 " 2
20&
&#
3x%
1$4 " 3
1 " 2
1 & 5
1 & 5
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A24 Glencoe Algebra 2
Answers (Lesson 5-8)
Skil
ls P
ract
ice
Radi
cal E
quat
ions
and
Ineq
ualit
ies
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-8
5-8
©G
lenc
oe/M
cGra
w-Hi
ll28
3G
lenc
oe A
lgeb
ra 2
Lesson 5-8
Sol
ve e
ach
equ
atio
n o
r in
equ
alit
y.
1.#
x$"
525
2.#
x$%
3 "
716
3.5#
j$"1
4.v&1 2&
%1
"0
no s
olut
ion
5.18
!3y
&1 2&"
25no
sol
utio
n6.
3 #2w$
"4
32
7.#
b!
5$
"4
218.
#3n
%$
1$"
58
9.3 #
3r!
6$
"3
1110
.2 %
#3p
%$
7$"
63
11.#
k!
4$
!1
"5
4012
.(2
d%
3)&1 3&
"2
13.(
t!
3)&1 3&
"2
1114
.4 !
(1 !
7u)&1 3&
"0
!9
15.#
3z!
2$
"#
z!
4$
no s
olut
ion
16.#
g%
1$
"#
2g!
$7$
8
17.#
x!
1$
"4#
x%
1$
no s
olut
ion
18.5
%#
s!
3$
(6
3 .
s.
4
19.!
2 %
#3x
%3
$)
7!
1 +
x+
2620
.!#
2a%
$4$
.!
6!
2 .
a.
16
21.2
#4r
!3
$,
10r,
722
.4 !
#3x
%1
$,
3!
+x
+0
23.#
y%
4$
!3
.3
y-
3224
.!3#
11r
%$
3$.
!15
!.
r.2
3 " 11
1 " 3
5 " 2
1 " 25
©G
lenc
oe/M
cGra
w-Hi
ll28
4G
lenc
oe A
lgeb
ra 2
Sol
ve e
ach
equ
atio
n o
r in
equ
alit
y.
1.#
x$"
864
2.4
!#
x$"
31
3.#
2p$%
3 "
104.
4#3h$
!2
"0
5.c&1 2&
%6
"9
96.
18 %
7h&1 2&
"12
no s
olut
ion
7.3 #
d%
2$
"7
341
8.5 #
w!
7$
"1
8
9.6
%3 #
q!
4$
"9
3110
.4 #
y!
9$
%4
"0
no s
olut
ion
11.#
2m!
$6$
!16
"0
131
12.
3 #4m
%$
1$!
2 "
2
13. #
8n!
$5$
!1
"2
14.#
1 !
4$
t$!8
"!
6!
15.#
2t!
5$
!3
"3
16.(
7v!
2)&1 4&
%12
"7
no s
olut
ion
17.(
3g%
1)&1 2&
!6
"4
3318
.(6u
!5)
&1 3&%
2 "
!3
!20
19.#
2d!
$5$
"#
d!
1$
420
.#4r
!6
$"
#r$
2
21.#
6x!
4$
"#
2x%
1$
0$22
.#2x
%5
$"
#2x
%1
$no
sol
utio
n
23.3
#a$
.12
a-
1624
.#z
%5
$%
4 (
13!
5 .
z.
76
25.8
%#
2q$(
5no
sol
utio
n26
.#2a
!$
3$)
5+
a+
14
27.9
!#
c%
4$
(6
c-
528
.3 #
x!
1$
)!
2x
+!
7
29.S
TATI
STIC
SSt
atis
tici
ans
use
the
form
ula
!"
#v$
to c
alcu
late
a s
tand
ard
devi
atio
n !
,w
here
vis
the
var
ianc
e of
a d
ata
set.
Fin
d th
e va
rian
ce w
hen
the
stan
dard
dev
iati
on
is 1
5.22
5
30.G
RA
VIT
ATI
ON
Hel
ena
drop
s a
ball
from
25
feet
abo
ve a
lake
.The
for
mul
a
t"
#25
!$
h$de
scri
bes
the
tim
e t
in s
econ
ds t
hat
the
ball
is h
feet
abo
ve t
he w
ater
.
How
man
y fe
et a
bove
the
wat
er w
ill t
he b
all b
e af
ter
1 se
cond
?9
ft
1 & 4
3 " 2
7 " 2
41 " 2
3 " 47 " 4
63 " 4
1 " 1249 " 2
Pra
ctic
e (A
vera
ge)
Radi
cal E
quat
ions
and
Ineq
ualit
ies
NAM
E__
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____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-8
5-8
© Glencoe/McGraw-Hill A25 Glencoe Algebra 2
An
swer
s
Answers (Lesson 5-8)
Rea
din
g t
o L
earn
Math
emati
csRa
dica
l Equ
atio
ns a
nd In
equa
litie
s
NAM
E__
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____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-8
5-8
©G
lenc
oe/M
cGra
w-Hi
ll28
5G
lenc
oe A
lgeb
ra 2
Lesson 5-8
Pre-
Act
ivit
yH
ow d
o ra
dic
al e
quat
ion
s ap
ply
to
man
ufa
ctu
rin
g?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 5-
8 at
the
top
of
page
263
in y
our
text
book
.
Exp
lain
how
you
wou
ld u
se t
he f
orm
ula
in y
our
text
book
to
find
the
cos
t of
prod
ucin
g 12
5,00
0 co
mpu
ter
chip
s.(D
escr
ibe
the
step
s of
the
cal
cula
tion
in t
heor
der
in w
hich
you
wou
ld p
erfo
rm t
hem
,but
do
not
actu
ally
do
the
calc
ulat
ion.
)
Sam
ple
answ
er:R
aise
125
,000
to th
e po
wer
by
taki
ng th
e cu
be ro
ot o
f 125
,000
and
squ
arin
g th
e re
sult
(or r
aise
125
,000
to
the
pow
er b
y en
terin
g 12
5,00
0 ^
(2/3
) on
a ca
lcul
ator
).M
ultip
ly th
e nu
mbe
r you
get
by
10 a
nd th
en a
dd 1
500.
Rea
din
g t
he
Less
on
1.a.
Wha
t is
an
extr
aneo
us s
olut
ion
of a
rad
ical
equ
atio
n?Sa
mpl
e an
swer
:a n
umbe
rth
at s
atis
fies
an e
quat
ion
obta
ined
by
rais
ing
both
sid
es o
f the
orig
inal
equa
tion
to a
hig
her p
ower
but
doe
s no
t sat
isfy
the
orig
inal
equ
atio
n
b.D
escr
ibe
two
way
s yo
u ca
n ch
eck
the
prop
osed
sol
utio
ns o
f a
radi
cal e
quat
ion
in o
rder
to d
eter
min
e w
heth
er a
ny o
f th
em a
re e
xtra
neou
s so
luti
ons.
Sam
ple
answ
er:O
new
ay is
to c
heck
eac
h pr
opos
ed s
olut
ion
by s
ubst
itutin
g it
into
the
orig
inal
equ
atio
n.An
othe
r way
is to
use
a g
raph
ing
calc
ulat
or to
gra
phbo
th s
ides
of t
he o
rigin
al e
quat
ion.
See
whe
re th
e gr
aphs
inte
rsec
t.Th
is c
an h
elp
you
iden
tify
solu
tions
that
may
be
extra
neou
s.
2.C
ompl
ete
the
step
s th
at s
houl
d be
fol
low
ed in
ord
er t
o so
lve
a ra
dica
l ine
qual
ity.
Step
1If
the
of
the
roo
t is
,i
dent
ify
the
valu
es o
f
the
vari
able
for
whi
ch t
he
is
.
Step
2So
lve
the
alge
brai
cally
.
Step
3Te
st
to c
heck
you
r so
luti
on.
Hel
pin
g Y
ou
Rem
emb
er
3.O
ne w
ay t
o re
mem
ber
som
ethi
ng is
to
expl
ain
it t
o an
othe
r pe
rson
.Sup
pose
tha
t yo
urfr
iend
Leo
ra t
hink
s th
at s
he d
oes
not
need
to
chec
k he
r so
luti
ons
to r
adic
al e
quat
ions
by
subs
titu
tion
bec
ause
she
kno
ws
she
is v
ery
care
ful a
nd s
eldo
m m
akes
mis
take
s in
her
wor
k.H
ow c
an y
ou e
xpla
in t
o he
r th
at s
he s
houl
d ne
vert
hele
ss c
heck
eve
ry p
ropo
sed
solu
tion
in t
he o
rigi
nal e
quat
ion?
Sam
ple
answ
er:S
quar
ing
both
sid
es o
f an
equa
tion
can
prod
uce
an e
quat
ion
that
is n
ot e
quiv
alen
t to
the
orig
inal
one.
For e
xam
ple,
the
only
sol
utio
n of
x&
5 is
5,b
ut th
e sq
uare
deq
uatio
n x2
&25
has
two
solu
tions
,5 a
nd !
5.
valu
esin
equa
lity
nonn
egat
ive
radi
cand
even
inde
x
2 " 3
2 " 3
©G
lenc
oe/M
cGra
w-Hi
ll28
6G
lenc
oe A
lgeb
ra 2
Trut
h Ta
bles
In m
athe
mat
ics,
the
basi
c op
erat
ions
are
add
itio
n,su
btra
ctio
n,m
ulti
plic
atio
n,di
visi
on,f
indi
ng a
roo
t,an
d ra
isin
g to
a p
ower
.In
logi
c,th
e ba
sic
oper
atio
ns
are
the
follo
win
g:no
t(*
),an
d (+
),or
(,
),an
dim
plie
s(→
).
If P
and
Qar
e st
atem
ents
,the
n *
Pm
eans
not
P;*
Qm
eans
not
Q;P
+ Q
mea
ns P
and
Q;P
,Q
mea
ns P
or Q
;and
P →
Qm
eans
Pim
plie
s Q
.The
op
erat
ions
are
def
ined
by
trut
h ta
bles
.On
the
left
bel
ow is
the
tru
th t
able
for
th
e st
atem
ent
*P
.Not
ice
that
the
re a
re t
wo
poss
ible
con
diti
ons
for
P,t
rue
(T)
or f
alse
(F).
If P
is t
rue,
*P
is f
alse
;if P
is f
alse
,*P
is t
rue.
Als
o sh
own
are
the
trut
h ta
bles
for
P +
Q,P
,Q
,and
P →
Q.
P*
PP
QP
+ Q
PQ
P ,
QP
QP
→ Q
TF
TT
TT
TT
TT
TF
TT
FF
TF
TT
FF
FT
FF
TT
FT
TF
FF
FF
FF
FT
You
can
use
this
info
rmat
ion
to f
ind
out
unde
r w
hat
cond
itio
ns a
com
plex
st
atem
ent
is t
rue. U
nd
er w
hat
con
dit
ion
s is
)P
*Q
tru
e?
Cre
ate
the
trut
h ta
ble
for
the
stat
emen
t.U
se t
he in
form
atio
n fr
om t
he t
ruth
ta
ble
abov
e fo
r P
*Q
to c
ompl
ete
the
last
col
umn.
PQ
*P
*P
,Q
TT
FT
Whe
n on
e st
atem
ent i
s T
FF
Ftru
e an
d on
e is
fals
e,F
TT
Tth
e co
njun
ctio
n is
true
.F
FT
T
The
tru
th t
able
indi
cate
s th
at *
P ,
Qis
tru
e in
all
case
s ex
cept
whe
re P
is t
rue
and
Qis
fals
e.
Use
tru
th t
able
s to
det
erm
ine
the
con
dit
ion
s u
nd
er w
hic
h e
ach
sta
tem
ent
is t
rue.
1.*
P ,
*Q
2.*
P →
(P →
Q)
all e
xcep
t whe
re b
oth
all
Pan
d Q
are
true
3.(P
,Q
),(*
P +
*Q
)4.
(P →
Q),
(Q →
P)
all
all
5.(P
→ Q
)+ (Q
→ P
)6.
(*P
+ *
Q)→
*(P
,Q
)bo
th P
and
Qar
e tru
e;al
lbo
th P
and
Qar
e fa
lse
En
rich
men
t
NAM
E__
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____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-8
5-8
Exam
ple
Exam
ple
© Glencoe/McGraw-Hill A26 Glencoe Algebra 2
Answers (Lesson 5-9)
To s
olve
a q
uadr
atic
equ
atio
n th
at d
oes
not
have
rea
l sol
utio
ns,y
ou c
an u
se t
he f
act
that
i2"
!1
to f
ind
com
plex
sol
utio
ns.
Sol
ve 2
x2$
24 &
0.
2x2
%24
"0
Orig
inal
equ
atio
n
2x2
"!
24Su
btra
ct 2
4 fro
m e
ach
side.
x2"
!12
Divid
e ea
ch s
ide
by 2
.
x"
-#
!12
$Ta
ke th
e sq
uare
root
of e
ach
side.
x"
-2i
#3$
#!
12$
"#
4$$
#!
1$
$#
3$
Sim
pli
fy.
1.(!
4 %
2i) %
(6 !
3i)
2.(5
!i)
!(3
!2i
)3.
(6 !
3i) %
(4 !
2i)
2 !
i2
$i
10 !
5i
4.(!
11 %
4i) !
(1 !
5i)
5.(8
%4i
) %(8
!4i
)6.
(5 %
2i) !
(!6
!3i
)!
12 $
9i16
11 $
5i
7.(1
2 !
5i) !
(4 %
3i)
8.(9
%2i
) %(!
2 %
5i)
9.(1
5 !
12i)
%(1
1 !
13i)
8 !
8i7
$7i
26 !
25i
10.i
411
.i6
12.i
15
1!
1!
i
Sol
ve e
ach
equ
atio
n.
13.5
x2%
45 "
014
.4x2
%24
"0
15.!
9x2
"9
(3i
(i!
6"(
i
Stu
dy
Gu
ide
and I
nte
rven
tion
Com
plex
Num
bers
NAM
E__
____
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____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-9
5-9
©G
lenc
oe/M
cGra
w-Hi
ll28
7G
lenc
oe A
lgeb
ra 2
Lesson 5-9
Ad
d a
nd
Su
btr
act
Co
mp
lex
Nu
mb
ers
Aco
mpl
ex n
umbe
r is
any
num
ber t
hat c
an b
e wr
itten
in th
e fo
rm a
%bi
, Co
mpl
ex N
umbe
rwh
ere
aan
d b
are
real
num
bers
and
iis
the
imag
inar
y un
it (i
2"
!1)
.a
is ca
lled
the
real
par
t, an
d b
is ca
lled
the
imag
inar
y pa
rt.
Addi
tion
and
Com
bine
like
term
s.Su
btra
ctio
n of
(a
%bi
) %(c
%di
) "(a
%c)
%(b
%d
)iCo
mpl
ex N
umbe
rs(a
%bi
) !(c
%di
) "(a
!c)
%(b
!d
)i
Sim
pli
fy (
6 $
i) $
(4 !
5i).
(6 %
i) %
(4 !
5i)
"(6
%4)
%(1
!5)
i"
10 !
4i
Sim
pli
fy (
8 $
3i)
!(6
!2i
).
(8 %
3i) !
(6 !
2i)
"(8
!6)
%[3
!(!
2)]i
"2
%5i
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exam
ple
3Ex
ampl
e 3
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-Hi
ll28
8G
lenc
oe A
lgeb
ra 2
Mu
ltip
ly a
nd
Div
ide
Co
mp
lex
Nu
mb
ers
Mul
tiplic
atio
n of
Com
plex
Num
bers
Use
the
defin
ition
of i2
and
the
FOIL
met
hod:
(a%
bi)(c
%di
) "(a
c!
bd) %
(ad
%bc
)i
To d
ivid
e by
a c
ompl
ex n
umbe
r,fi
rst
mul
tipl
y th
e di
vide
nd a
nd d
ivis
or b
y th
e co
mp
lex
con
juga
teof
the
div
isor
.
Com
plex
Con
juga
tea
%bi
and
a!
biar
e co
mpl
ex c
onju
gate
s. T
he p
rodu
ct o
f com
plex
con
juga
tes
is al
ways
a re
al n
umbe
r.
Sim
pli
fy (
2 !
5i)
*(!
4 $
2i).
(2 !
5i) $
(!4
%2i
)"
2(!
4) %
2(2i
) %(!
5i)(
!4)
%(!
5i)(
2i)
FOIL
"!
8 %
4i%
20i
!10
i2M
ultip
ly.
"!
8 %
24i
!10
(!1)
Sim
plify
.
"2
%24
iSt
anda
rd fo
rm
Sim
pli
fy
.
"$
Use
the
com
plex
con
juga
te o
f the
divi
sor.
"M
ultip
ly.
"i2
"!
1
"!
iSt
anda
rd fo
rm
Sim
pli
fy.
1.(2
%i)
(3 !
i)7
$i
2.(5
!2i
)(4
!i)
18 !
13i
3.(4
!2i
)(1
!2i
)!
10i
4.(4
!6i
)(2
%3i
)26
5.(2
%i)
(5 !
i)11
$3i
6.(5
!3i
)(!
1 !
i)!
8 !
2i
7.(1
!i)
(2 %
2i)(
3 !
3i)
8.(4
!i)
(3 !
2i)(
2 %
i)9.
(5 !
2i)(
1 !
i)(3
%i)
12 !
12i
31 !
12i
16 !
18i
10.
!i
11.
!!
i12
.!
!2i
13.
1 !
i14
.!
!2i
15.
!$
i
16.
$17
.!
1 !
2i!
2"18
.$
2 i!
6 ""
3!
3 ""
3#
6$%
i#3$
&&
#2$
!i
4 !
i#2$
&&
i#2$
3 i!
5 ""
72 " 7
3 %
i#5$
&&
3 !
i#5$
31 " 418 " 41
3 %
4i& 4
!5i
1 " 2!
5 !
3i& 2
!2i
4 !
2i& 3
%i
5 " 36
!5i
&3i
7 " 213 " 2
7 !
13i
&2i
1 " 23 " 2
5& 3
%i
11 & 133 & 133
!11
i&
13
6 !
9i!
2i%
3i2
&&
&4
!9i
2
2 !
3i& 2
!3i
3 !
i& 2
%3i
3 !
i& 2
%3i
3 !
i" 2
%3i
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Com
plex
Num
bers
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-9
5-9
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A27 Glencoe Algebra 2
An
swer
s
Answers (Lesson 5-9)
Skil
ls P
ract
ice
Com
plex
Num
bers
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-9
5-9
©G
lenc
oe/M
cGra
w-Hi
ll28
9G
lenc
oe A
lgeb
ra 2
Lesson 5-9
Sim
pli
fy.
1.#
!36
$6i
2.#
!19
6$
14i
3.#
!81
x6$
9|x3
| i4.
#!
23$
$#
!46
$!
23!
2"
5.(3
i)(!
2i)(
5i)
30i
6.i1
1!
i
7.i6
5i
8.(7
!8i
) %(!
12 !
4i)
!5
!12
i
9.(!
3 %
5i) %
(18
!7i
)15
!2i
10.(
10 !
4i) !
(7 %
3i)
3 !
7i
11.(
2 %
i)(2
%3i
)1
$8i
12.(
2 %
i)(3
!5i
)11
!7i
13.(
7 !
6i)(
2 !
3i)
!4
!33
i14
.(3
%4i
)(3
!4i
)25
15.
16.
Sol
ve e
ach
equ
atio
n.
17.3
x2%
3 "
0(
i18
.5x2
%12
5 "
0(
5i
19.4
x2%
20 "
0(
i!5"
20.!
x2!
16 "
0(
4i
21.x
2%
18 "
0(
3i!
2"22
.8x2
%96
"0
(2i
!3"
Fin
d t
he
valu
es o
f m
and
nth
at m
ake
each
equ
atio
n t
rue.
23.2
0 !
12i
"5m
%4n
i4,
!3
24.m
!16
i"
3 !
2ni
3,8
25.(
4 %
m) %
2ni
"9
%14
i5,
726
.(3
!n)
%(7
m!
14)i
"1
%7i
3,2
3 $
6i"
103i
& 4 %
2i!
6 !
8i"
38
!6i
&3i
©G
lenc
oe/M
cGra
w-Hi
ll29
0G
lenc
oe A
lgeb
ra 2
Sim
pli
fy.
1.#
!49
$7i
2.6#
!12
$12
i!3"
3.#
!12
1$
s8 $11
s4i
4.#
!36
a$
3 b4
$5.
#!
8$
$#
!32
$6.
#!
15$
$#
!25
$6| a
| b2 i
!a"
!16
!5!
15"
7.(!
3i)(
4i)(
!5i
)8.
(7i)
2 (6i
)9.
i42
!60
i!
294i
!1
10.i
5511
.i89
12.(
5 !
2i) %
(!13
!8i
)!
ii
!8
!10
i
13.(
7 !
6i) %
(9 %
11i)
14.(
!12
%48
i) %
(15
%21
i)15
.(10
%15
i) !
(48
!30
i)16
$5i
3 $
69i
!38
$45
i
16.(
28 !
4i) !
(10
!30
i)17
.(6
!4i
)(6
%4i
)18
.(8
!11
i)(8
!11
i)18
$26
i52
!57
!17
6i
19.(
4 %
3i)(
2 !
5i)
20.(
7 %
2i)(
9 !
6i)
21.
23 !
14i
75 !
24i
22.
23.
24.
!1
!i
Sol
ve e
ach
equ
atio
n.
25.5
n2%
35 "
0(
i!7"
26.2
m2
%10
"0
(i!
5"
27.4
m2
%76
"0
(i!
19"28
.!2m
2!
6 "
0(
i!3"
29.!
5m2
!65
"0
(i!
13"30
.x2
%12
"0
(4i
Fin
d t
he
valu
es o
f m
and
nth
at m
ake
each
equ
atio
n t
rue.
31.1
5 !
28i
"3m
%4n
i5,
!7
32.(
6 !
m) %
3ni
"!
12 %
27i
18,9
33.(
3m%
4) %
(3 !
n)i
"16
!3i
4,6
34.(
7 %
n) %
(4m
!10
)i"
3 !
6i1,
!4
35.E
LEC
TRIC
ITY
The
impe
danc
e in
one
par
t of
a s
erie
s ci
rcui
t is
1 %
3joh
ms
and
the
impe
danc
e in
ano
ther
par
t of
the
cir
cuit
is 7
!5j
ohm
s.A
dd t
hese
com
plex
num
bers
to
find
the
tot
al im
peda
nce
in t
he c
ircu
it.
8 !
2joh
ms
36.E
LEC
TRIC
ITY
Usi
ng t
he f
orm
ula
E"
IZ,f
ind
the
volt
age
Ein
a c
ircu
it w
hen
the
curr
ent
Iis
3 !
jam
ps a
nd t
he im
peda
nce
Zis
3 %
2joh
ms.
11 $
3jvo
lts
3 & 4
2 !
4i& 1
%3i
7 $
i"
53
!i
& 2 !
i14
$16
i"
"11
32
& 7 !
8i
!5
$6i
"2
6 %
5i&
!2i
Pra
ctic
e (A
vera
ge)
Com
plex
Num
bers
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-9
5-9
© Glencoe/McGraw-Hill A28 Glencoe Algebra 2
Answers (Lesson 5-9)
Rea
din
g t
o L
earn
Math
emati
csCo
mpl
ex N
umbe
rs
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-9
5-9
©G
lenc
oe/M
cGra
w-Hi
ll29
1G
lenc
oe A
lgeb
ra 2
Lesson 5-9
Pre-
Act
ivit
yH
ow d
o co
mp
lex
nu
mbe
rs a
pp
ly t
o p
olyn
omia
l eq
uat
ion
s?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 5-
9 at
the
top
of
page
270
in y
our
text
book
.
Supp
ose
the
num
ber
iis
defin
ed s
uch
that
i2
"!
1.C
ompl
ete
each
equ
atio
n.
2i2
"(2
i)2
"i4
"
Rea
din
g t
he
Less
on
1.C
ompl
ete
each
sta
tem
ent.
a.T
he f
orm
a%
biis
cal
led
the
of a
com
plex
num
ber.
b.In
the
com
plex
num
ber
4 %
5i,t
he r
eal p
art
is
and
the
imag
inar
y pa
rt is
.
Thi
s is
an
exam
ple
of a
com
plex
num
ber
that
is a
lso
a(n)
nu
mbe
r.
c.In
the
com
plex
num
ber
3,th
e re
al p
art
is
and
the
imag
inar
y pa
rt is
.
Thi
s is
exa
mpl
e of
com
plex
num
ber
that
is a
lso
a(n)
nu
mbe
r.
d.In
the
com
plex
num
ber
7i,t
he r
eal p
art
is
and
the
imag
inar
y pa
rt is
.
Thi
s is
an
exam
ple
of a
com
plex
num
ber
that
is a
lso
a(n)
nu
mbe
r.
2.G
ive
the
com
plex
con
juga
te o
f ea
ch n
umbe
r.
a.3
%7i
b.2
!i
3.W
hy a
re c
ompl
ex c
onju
gate
s us
ed in
div
idin
g co
mpl
ex n
umbe
rs?
The
prod
uct o
fco
mpl
ex c
onju
gate
s is
alw
ays
a re
al n
umbe
r.
4.E
xpla
in h
ow y
ou w
ould
use
com
plex
con
juga
tes
to f
ind
(3 %
7i) *
(2 !
i).
Writ
e th
edi
visi
on in
frac
tion
form
.The
n m
ultip
ly n
umer
ator
and
den
omin
ator
by
2 $
i.
Hel
pin
g Y
ou
Rem
emb
er
5.H
ow c
an y
ou u
se w
hat
you
know
abo
ut s
impl
ifyi
ng a
n ex
pres
sion
suc
h as
to
help
you
rem
embe
r ho
w t
o si
mpl
ify
frac
tion
s w
ith
imag
inar
y nu
mbe
rs in
the
deno
min
ator
?Sa
mpl
e an
swer
:In
both
cas
es,y
ou c
an m
ultip
ly th
enu
mer
ator
and
den
omin
ator
by
the
conj
ugat
e of
the
deno
min
ator
.
1 %
#3$
& 2 !
#5$
2 $
i
3 !
7i
pure
imag
inar
y 7
0re
al0
3im
agin
ary
54
stan
dard
form
1!
4!
2
©G
lenc
oe/M
cGra
w-Hi
ll29
2G
lenc
oe A
lgeb
ra 2
Conj
ugat
es a
nd A
bsol
ute
Valu
eW
hen
stud
ying
com
plex
num
bers
,it
is o
ften
con
veni
ent
to r
epre
sent
a c
ompl
ex
num
ber
by a
sin
gle
vari
able
.For
exa
mpl
e,w
e m
ight
let
z"
x%
yi.W
e de
note
th
e co
njug
ate
of z
by z $.
Thu
s,z $
"x
!yi
.
We
can
defi
ne t
he a
bsol
ute
valu
e of
a c
ompl
ex n
umbe
r as
fol
low
s.
-z-"
-x%
yi-"
#x2
%y
$2 $
The
re a
re m
any
impo
rtan
t re
lati
onsh
ips
invo
lvin
g co
njug
ates
and
abs
olut
e va
lues
of
com
plex
num
bers
.
Sh
ow +z
+2&
zz "fo
r an
y co
mp
lex
nu
mbe
r z.
Let
z"
x%
yi.T
hen,
z"
(x%
yi)(
x!
yi)
"x2
%y2
"#
( x2%
y2$
)2 $"
-z-2
Sh
ow
is t
he
mu
ltip
lica
tive
in
vers
e fo
r an
y n
onze
ro
com
ple
x n
um
ber
z.
We
know
-z-2
"zz $.
If z
# 0
,the
n w
e ha
ve z!
""1.
Thu
s,is
the
mul
tipl
icat
ive
inve
rse
of z
.
For
eac
h o
f th
e fo
llow
ing
com
ple
x n
um
bers
,fin
d t
he
abso
lute
val
ue
and
mu
ltip
lica
tive
in
vers
e.
1.2i
2;"! 2i "
2.!
4 !
3i5;
"!4 2$ 5
3i"
3.12
!5i
13;"12
1$ 695i
"
4.5
!12
i13
;"5$ 16
912i
"5.
1 %
i!
2";"1
! 2i
"6.
#3$
!i
2;
7.%
i8.
!i
9.&1 2&
!i
;1;
$i
1;"1 2"
$i
!3"
"3
!2"
"2
!2"
"2
!3"
!i!
3""
" 2!
6""
3
#3$
&2
#2$
&2
#2$
&2
#3$
&3
#3$
&3
!3"
$i
"4
z $& -z
-2
z $& -z
-2
z "" +z
+2
En
rich
men
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
5-9
5-9
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2