College AlgebraMath 123-01 Chapter R

Review Elem. College Algebra

Instructor: Dr. Chekad Sarami

Course Number & Name:Course Number & Name: MATH 123 College AlgebraMATH 123 College Algebra Semester Hours of Credit:Semester Hours of Credit: 33 Days/Time Class Meets: MWFDays/Time Class Meets: MWF Room/Bldg. SBE 108Room/Bldg. SBE 108 Instructors Name: Instructors Name: Dr. Chekad SaramiDr. Chekad Sarami Office Location: Office Location: SBE 334SBE 334 Office Telephone:Office Telephone: 672-1129672-1129 E-mail address: csarami@uncfus.eduE-mail address: csarami@uncfus.edu Office Office Hours: Hours: MM2-3 pm, TR 2:00-3:30, F 2-3 pm and 4-5 2-3 pm, TR 2:00-3:30, F 2-3 pm and 4-5

pmpm Final : Wednesday, December 7, Time: 6-8pmFinal : Wednesday, December 7, Time: 6-8pm

COURSE DESCRIPTION: Mathematics 123 is a college level algebra Mathematics 123 is a college level algebra

course containing topics as follows: Sets, course containing topics as follows: Sets, the real number system, exponents, radicals, the real number system, exponents, radicals, polynomials, equations, inequalities, polynomials, equations, inequalities, relations and functions, graphing, relations and functions, graphing, exponential and logarithmic functions. exponential and logarithmic functions. PrerequisitesPrerequisites: High School Algebra I, II, : High School Algebra I, II, and Plane Geometry or equivalent, and and Plane Geometry or equivalent, and satisfactory placement score.satisfactory placement score.

Graphical Calculator is Required!Graphical Calculator is Required!

GRADING SCALE:

HW and Three Take-Home Tests (HW and Three Take-Home Tests (lowest lowest chapter test score is dropped)chapter test score is dropped) 65%65%

Group Projects: 10%Group Projects: 10% Instructor Option: 5%Instructor Option: 5% Final Exam (Cumulative) 20%Final Exam (Cumulative) 20%Total: 100Total: 100

Student Tips I. Prior to the beginning of the semester: Begin each course with a positive attitude and open mind.Begin each course with a positive attitude and open mind.

Determine why you are taking this course; graduation requirement, prerequisite for Determine why you are taking this course; graduation requirement, prerequisite for another course, your job demands this course, to improve your GPA. These are all-another course, your job demands this course, to improve your GPA. These are all-important reasons—remind yourself of this throughout the course.important reasons—remind yourself of this throughout the course.

Plan your school, work, and recreation schedule to allow plenty of long blocks of time to Plan your school, work, and recreation schedule to allow plenty of long blocks of time to study.study.

Plan or designate a study area that is quiet and where you won’t be interrupted.Plan or designate a study area that is quiet and where you won’t be interrupted.

Purchase your textbook, calculator and any supplies before the semester begins or on the Purchase your textbook, calculator and any supplies before the semester begins or on the first day of class.first day of class.

II. During the semester:II. During the semester:

Make an exaggerated effort the first couple of weeks—get off to a fast start.Make an exaggerated effort the first couple of weeks—get off to a fast start.

Attend all classes—your teacher needs you there in order to "teach you."Attend all classes—your teacher needs you there in order to "teach you."

Work all assignments as they are assigned. Mathematics courses are usually building courses, Work all assignments as they are assigned. Mathematics courses are usually building courses, meaning each section builds on the concept that you have worked and understood all of the material meaning each section builds on the concept that you have worked and understood all of the material in the previous section(s).in the previous section(s).

Complete only one assignment at a time (in a block of time). Allow time for each assignment to "soak Complete only one assignment at a time (in a block of time). Allow time for each assignment to "soak in" before attempting the next assignment.in" before attempting the next assignment.

Ask questions. Prepare a list of questions to ask in class, if the teacher allows, or in the teacher’s Ask questions. Prepare a list of questions to ask in class, if the teacher allows, or in the teacher’s office or in a math help center.office or in a math help center.

Get help outside of class. You many want to go your teacher’s office or you may prefer to go to a Get help outside of class. You many want to go your teacher’s office or you may prefer to go to a help center.help center.

Form study groups with 3 or 4 other students from your class.Form study groups with 3 or 4 other students from your class.

III. Prior to a test:•Plan ahead—set aside plenty of study time, beginning several days before the test date.

•Go through the material presented/homework assignments. Select 2 or 3 problems of each type presented and write yourself a "practice test."

•For any areas that you feel are weak, select additional practice exercises and/or get help on these areas.

•1-2 hours prior to taking the test, work a few problems—get your mind thinking mathematics.

IV. After a test:

•Go over your test.

•Study any questions which you missed—learn how to answer (work) these questions immediately.

•Keep your old tests in a safe place, as you will want to study them again prior to the final exam.

V. How to approach homework assignments:

Read back through your class notes, including reworking all examples, before you attempt your homework assignments and read through the reading material in the text, working the examples in the text.

As you are working through your homework, check your answers. Remember that you can check your answers to the odd problems from the back of the text.

If you miss a problem, try reworking it rather than trying to find your mistake. Then compare your work. If you still miss it, look for a similar example in the reading material in that section. Using your own pencil and paper, work through that example and compare to the example in the text. Ask for HELP!

Properties of Real Numbers

Rules of Signs

a(-b) = -(ab) = (-a)b (-a)(-b) = ab - ( -a) = a

b

a

b

a

b

a

b

a

b

a

Properties of Real Numbers

Cancellation Properties

Zero – Product Property

If ab = 0, then a = 0 or b = 0 or, both.

0,0 cbifb

a

bc

ac

0 cifbaimpliesbcac

Properties of Real Numbers

Arithmetic of Quotients

0,0 if

dbbd

bcad

d

c

b

a

ab

cd

acbd

b d if 0 0,

abcd

ab

dc

adbc

b c d if 0 0 0, ,

Arithmetic of Quotients: Example 1

35

23

35

23

35

23

35

33

23

55

915

1015

9 1015( ) 1

15

Arithmetic of Quotients: Example 2

1253

20

125

203

12 205 3

4 3 5 45 3

4 4

16

The Real Number Line

The negative real numbers are the coordinates of points to the left of the origin 0.

The real number zero is the coordinate of the origin O.

The positive real numbers are the coordinates of points to the right of the origin O.

Example of Domain

Find the domain of the variable in

13the expression

3

z

z+

Domain: z z 3

The result is read “The set of all real numbers z such that z is not equal to –3”

2 2 2a b c+ =

Example: The Pythagorean Theorem

Show that a triangle whose sides are of lengths 6, 8, and 10 is a right triangle.

We square the length of the sides:

2 2 26 36 8 64 10 100= = =

Notice that the sum of the first two squares (36 and 64) equals the third square (100). Hence the triangle is a right triangle, since it satisfies the Pythagorean Theorem.

For a rectangle of length L and width W:

2 2Area lw Perimeter l w= = +

For a triangle with base b and altitude (height) h:

Geometry Formulas

1

2Area bh=

For a circle of radius r (diameter d = 2r)2 2Area r Circumference r dp p p= = =

Geometry Formulas

For a rectangular box of length L, width W, and height H:

V o lu m e lw h=

For a sphere of radius r:

3 244

3Volume r Surface Area rp p= =

For a right circular cylinder of height h and radius r:

2Volume r hp=

Exponents: Basic Definitions

If a is a real number and n is a positive integer,

a a a an

n

facto rs

a a0 1 0 if

aa

ann

10 if

Examples:

4 4 4 43

6 10

41

43

3

Laws of Exponents

a a a a a ab a b

a

aa

aa

ab

a

bb

m n m n m n mn n n n

m

nm n

n m

n n

n

if

if

10

0

ab

ba

a bn n

if 0 0,

Example:

Write so that all exponents are positive.x y

x y

3 2

1 4

x y

x y

x

x

y

y

3 2

1 4

3

1

2

4

x y3 1 2 4( )

x y4 6 x

y

4

6

Example:

Simplify the expression. Express the answer so only positive exponents occur.

3 2 4

3

2x y

x y

3 2 3 4 1 2

x y 3 1 3 2

x y

3 2 1 2 3 2

x y 3 2 2 6x y x

y

2

69

Using your calculator

For Scientific Calculators:

Evaluate: 4(3.4)

Keystrokes: 3.4 yx 3.4 =

133.6336

For Graphing Calculators:

Evaluate: 4(3.4)

Keystrokes: 3.4 4 =133.6336

^

Monomial Coefficient Degree

3

2

9

4x

x

3 4

2 1

-9 0

Determine the coefficients and degree

of 2 3 54 2x x x .

Coefficients: 2, 0, -3, 1, -5

Degree: 4

2 8 1 3 5 23 2 3x x x x x

2 3 8 5 1 23 3 2x x x x x

5 3 13 2x x x

2 8 1 3 5 23 2 3x x x x x

2 8 1 3 5 23 2 3x x x x x

2 3 8 5 1 23 3 2x x x x x

x x x3 2 13 3

3 2 4 32x x x

3 3 4 3 3 2 2 4 2 32 2x x x x x x x

3 12 9 2 8 63 2 2x x x x x

3 14 17 63 2x x x

The process of expressing a polynomial as a product of other polynomials is called factoring.

ExampleMultiply: ( )23 2 4x x x- -

23 ( ) 3 (2 ) 3 (4)x x x x x= - -

3 23 6 12x x x= - -

Factoring is the same process in reverse

Factor: 3 23 6 12x x x- -

Notice that each term in this trinomial has a greatest common factor of 3x.

3 23 6 12x x x- -

23 ( ) 3 (2 ) 3 (4)x x x x x= - -

( )23 2 4x x x= - -

Special FormulasWhen you factor a polynomial, first check whether you can use one of the special formulas shown in the previous section.

Difference of Two Squares:

Perfect Squares:

Sum of Two Cubes:

Difference of Two Cubes:

2 2 ( )( )x a x a x a- = - +

( )22 22x ax a x a+ + = +

( )22 22x ax a x a- + = -

( )( )3 3 2 2x a x a x ax a+ = + - +

( )( )3 3 2 2x a x a x ax a- = - + +

Example:

9 64 3 82 2 2x x

3 8 3 8x x

Factor Completely: 29 64x -

Factor Completely: x 4 1

x 4 1 x x2 21 1

x x x2 1 1 1

Factor the trinomial: x x2 11 18

Look for factors of 18 whose sum is 11.

9 2 18

9 2 11

x x x x2 11 18 2 9

Factor the trinomial: x x2 3 10

Factors of -10

10, -1

5, -2

-5, 2

-10, 1

Sum

9

3

-3

-9

x x x x2 3 10 5 2

Factor completely: 3 7 62x x

3 7 6 32x x x x

3 7 6

3 1 6

3 6 1

3 2 3

3 3 2

2x x

x x

x x

x x

x x

3 7 6

3 1 6

3 1 6

3 6 1

3 6 1

3 2 3

3 2 3

3 3 2

3 3 2

2x x

x x

x x

x x

x x

x x

x x

x x

x x

+

2 10 3 153 2x x x

2 10 3 153 2x x x

2 5 3 52x x x

x x5 2 32

Factor By Grouping:

2

In general, if is a nonnegative real number,

the nonnegative number such that is

the of and is denoted

by .

a

b b a

a

b a

=

=

principal square root

a a2

Absolute Value is needed here, since the principal square root produces a positive value.

Example: 2( 4)- 16= 4= 4= -

Product Property of Square Roots

a b ab

18 9 2 9 2 3 2 Example:

Example: 50 3x 25 22x x

25 22x x

5 2x x

Rationalize the denominator in each expression

32

32

22

3 22

aa

33

aa

aa

33

33

a

a

3

9

2

The principal nth root of a real number a, symbolized by is defined as follows:an

a b a bn n means where a > 0 and b > 0 if n is even and a, b are any real numbers if n is odd

a a n

a a n

nn

nn

,

,

if is odd

if is even

Examples: 81 9 9 812 because

27 3 273 3 because - 3

Examples: 12 1255

12 1255

5 566

5 5 566

z z88

Properties of Radicals

ab a b

ab

ab

a a

a a

n n n

nn

n

mn n m

nm mn

Simplify: 32 16 254 44x x x

16 244 4x x

2 24x x

Simplify: xxx 5038 3

xxxx 225342 2

2 2 15 2x x x x= - 13 2x x=-