1

UNIT 9 (Chapter 10) Polynomials and Factoring Name:_____________________ The calendar and all assignments are subject to change. Students will be notified of any changes during class, so it is their responsibility to pay attention and make any necessary changes. All assignments are due

the following class period unless indicated otherwise. Monday Tuesday Wednesday Thursday Friday

March 16

Review

17

Unit 8 Test

18 Pretest & Section 10.1

19 Section 10.2 (F.O.I.L)

20 Section 10.3

23 Section 10.4 (Zero Product)

24 Section 10.5A (GCF Factoring)

25

Section 10.5B

26

Section 10.5C & Review

27

Quiz 10.1-10.5

End of 3rd MP

30 Section 10.6a (Box Method)

31 Section 10.6b

April 1 Section 10.6c

2 Section 10.7

3 No School-

Spring Break

13 Section 10.8

14 Review Ch. 10

15 Review Ch. 10

16

Chapter 10 Test

17

Section Page Assignment

10.1 p. 580 #20-58 even, 63-64

10.2 p. 587 #18-46 even, 54-55

10.3 p. 593 #16-46 even, 47, 58

10.4 p. 600 #20-32 even, 34-43 all

10.5 A (GCF) Worksheet in packet

10.5 B p. 607 #12-23, 28-40 even, 52-58 (remember to check odd answers)

10.5C & Review --- Worksheets in packet (10.5C and Review of 10.1-10.3)

10.6 A (Box) --- Worksheet in packet

10.6 B p. 614 #18-46 even

10.6 C --- Worksheet in packet

10.7 p. 622 #18-44 even, 51-58

10.8 p. 629 #16-34 even, 37-39, 41, 43, 50

Ch 10 Review --- Worksheet at the end of the packet

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Lesson 10.1– Adding and Subtracting Polynomials Algebra 1

Essential Question How can you add and subtract polynomials?

Warm-Up Exercise

(a) 2452 xx (b) 15)4(3 xx (c) )8(9 x

Core Concepts

Polynomials

A polynomial is a monomial or a sum of monomials. Each monomial is called a term of the

polynomial. A polynomial with two terms is a binomial. A polynomial with three terms is a

trinomial.

Binomial Trinomial

5 2x 2 5 2x x

The degree of a polynomial is the greatest degree of its terms. A polynomial in one

variable is in standard form when the exponents of the terms decrease from left to

right. When you write a polynomial in standard form, the coefficient of the first term is

the leading coefficient.

1. Examining Polynomials

Decide whether or not it is a polynomial. If it is, then write it in standard form, list its degree,

leading coefficient, and constant term.

(a) (b) (c) 8 (d)

Classifying Polynomials Polynomials can be classified according to their degree and by the number of terms. Fill out the chart below.

Polynomial Degree Classified by Degree Classified by # of terms

A.

B.

C.

D.

E.

F.

3

2. Classifying Polynomials

Classify each polynomial by its degree and by its number of terms.

(a) x4

(b) 32 7xx

(c) 37 2 xx

3. Adding Polynomials

Add the polynomials using a vertical format in

part (a) and a horizontal format in part (b).

(a)

(b)

4. Subtracting Polynomials

Subtract the polynomials using a vertical format in

part (a) and a horizontal format in part (b).

(a)

(b)

4

Lesson 10.2 – Multiplying Polynomials Algebra 1

Essential Question How can you multiply two polynomials?

Warm-Up Exercise

Simplify the Expression.

(a) )6()3( xx (b) )6(3 xx (c) 107)312( x (d) )(332 xx

1. Investigating Binomial Multiplication

Use the diagram at the right and determine the area of the

entire region. Then complete the statement below.

____________ ____________ = __________________________

2. Multiplying Binomials Using the Distributive Property

Find the product using the distributive property.

(a) )3)(2( xx (b) )35)(2( 2 xxx

(c) )15)(32( xx (d) )52)(134( 2 xxx

x

x

x

1 1

1

5

3. Multiplying Binomials Using F.O.I.L. Pattern

Find the product of binomials using the F.O.I.L. method.

(a) )15)(32( xx

(b) )12)(43( xx (c) )45)(23( xx

In Exercises 4–9, use a table to find the product.

4. 3 2x x 5. 1 6y y 6. 3 7q q

7. 2 5 3w w 8. )52)(246( 2 xxx 9. )53)(25159( 2 hhh

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Lesson 10.3 – Special Product of Polynomials Algebra 1

Essential Question What are the patterns in the special products

( a + b)2 and (a + b)( a – b)?

Warm-Up Exercise

Simplify the Expression.

(a) 2)2

1( x (b) 23 )6( m (c) 22 )

5

2( y (d) 2)2( xx

1. Using Sum and Difference Binomial Patterns

Find the product.

(a) )53)(53( bb (b) )25)(25( dd

2. Special Product: Squaring Binomials

Find the product.

(a) 2)43( n (b) 2)72( yx

(c) 2)43

1( a

7

Lesson 10.4 – The Zero Product Property Algebra 1

Essential Question How can you solve a polynomial equation?

Work with a partner. Substitute 1, 2, 3, 4, 5, and 6 for x in each equation and determine whether the

equation is true. Organize your results in the table. Write a conjecture describing what you discovered.

Equation x = 1 x = 2 x = 3 x = 4 x = 5 x = 6

a. 1 2 0x x

b. 2 3 0x x

c. 3 4 0x x

d. 4 5 0x x

e. 5 6 0x x

f. 6 1 0x x

Work with a partner. The numbers 0 and 1 have special properties that are shared by no other

numbers. For each of the following, decide whether the property is true for 0, 1, both, or neither.

a. When you add ____ to a number n, you get n.

b. If the product of two numbers is ____, then at least one of the numbers is 0.

c. The square of ____ is equal to itself.

d. When you multiply a number n by ____, you get n.

e. When you multiply a number by n by ____, you get 0.

f. The opposite of ____ is equal to itself.

One of the properties in Exploration 3 is called the Zero-

Product Property. It is one of the most important properties in

all of algebra. Which property is it? Why do you think it is

called the Zero-Product Property? Explain how it is used in

algebra and why it so important.

1 EXPLORATION: Writing a Conjecture

2 EXPLORATION: Special Properties of 0 and 1

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1. Using the Zero Product Property

Use the Zero Product Property to solve the

equations written in factored form.

(a) 0)9)(4( xx

(b) 0)23)(12( xx

(c) 0)1)(5(43

21 xx (d) 0)8(2 xx

2. Zero Product Property with Special Products

Use the Zero Product Property to solve the equations written in factored form.

(a) 0)7( 2 x (b) 0)6( 2

41 x

(c) 0)10)(3)((543 xxx (d) 0)1)(11()53(

852 xxx

Zero Product Property

Factored Form

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Lesson 10.5a– Factoring the Greatest Common Factor (GCF) Algebra 1

Warm-up Exercise – Use the Distributive Property.

(a) )964(5 2 xx (b) )487(2 2 xxx (c) )5(7 yxxy

In Exercises 3- 5, solve the equation.

3. 26 0k k 4. 235 49 0n n 5. 24 52 0z z

6. A boy kicks a ball in the air. The height y (in feet) above the ground of the ball is

modeled by the equation 216 80 ,y x x where x is the time (in seconds) since

the ball was kicked. Find the roots of the equation when y = 0 . Explain what the

roots mean in this situation.

Common Monomial Factoring

You can think of common monomial factoring as

“reversing” the distributive property. Our goal here

is to factor, or pull out, the greatest common

monomial factor. We call this factoring the GCF.

Ex.

1. Factor the GCF out of each polynomial.

(a)

(b)

2. Factor the GCF out of each polynomial.

(a) (b)

(c) (d)

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10.5A WORKSHEET: Factoring the Greatest Common Factor Name:______________

In each of the polynomials, factor out the GCF. If there is no GCF, then just write “No GCF.”

1. xx 102 2 2. xx 189 4 3. xxx 4128 25

4. vv 186 3 5. qq 124 4 6. 293 xx

7. 35 624 tt 8. 235 284 aaa 9. ddd 3618 26

10. 10x5+24x3-16x2 11. 36x8+90x7+48x2 12. 18x8+10x6-10x5

13. 36x10-28x5-20x 14. 40x5-20x3+15x2 15. 50x8-45x3-30x2

16. 16x8+14x6-14x7 17. 30x6-40x4-15x8 18. 15x7+21x5-90x

11

Lesson 10.5b– Factoring x2 + bx + c Algebra 1

Essential Question How can you use algebra tiles to factor the trinomial

2x bx c into the product of two binomials?

Factoring a Quadratic Trinomial (x2 + bx + c)

To factor a quadratic expression means to write it as the

product of 2 linear expressions. You can think of factoring

trinomials of the form x2 + bx + c as “reverse FOILing.”

(x + p)(x + q) =

(x + p)(x + q) =

In order to factor x2 + bx + c, you must find p and q

such that: (p + q) = ___ AND pq = ___

Example: x2 + 6x + 8 = (x + ? )(x + ? )

1. Factor the following

(a) x2 + 3x + 2

(b) x2 - 8x + 15

(c) x2 + 11x + 10

(d) x2 - 8x – 9

(e) x2 + 3x – 18

(f) x2 - 17x + 60

(g) y2 – 2y - 48

Solving Quadratic Equations by

Factoring

Example: x2 – 3x = 10

a. Rewrite

b. Factor

c. Use

d. Solve

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2. Solve the Equation by Factoring

(a) z2 + 11z = 26 (b) x2 + 11x +18 = 0 (c) x2 + 16x = -15

3. Factoring Trinomials with a G.C.F.

Completely factor the trinomial.

(a) 45243 2 xx (b) xxx 24306 23

(c) 345 11012111 xxx (d) yxyxyx 8910 56147

4. Solving Equations by Factoring Trinomials and a G.C.F.

Solve each equation using the Zero Product Property.

(a) 090639 2 xx (b) 090213 23 yyy

(c) 4234 2963 xxxx (d) 147707 2 yy

5. The area of a right triangle is 32 square miles. One leg of the triangle is 4 miles longer than

the other leg. Find the length of each leg.

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Name ________________________________________Date _____________

10.5C Worksheet – Factoring the GCF, Factoring Trinomials, and Solving Equations

Factor the GCF for each polynomial.

1. xx 255 2 2. 57 1512 xx 3. xxx 4124 26

4. vv 1821 3 5. 39 336 qq 6. 63620 xx

Factor each trinomial of the form cbxx 2

7. 862 xx 8. 432 xx 9. 232 xx

10. 822 xx 11. 1272 xx 12. 562 xx

13. 202 xx 14. 1682 xx 15. 24102 xx

Factor the GCF first, and then factor the remaining trinomial.

16. 234 6xxx 17. 345 158 xxx 18. xxx 30162 23

14

19. xxx 16204 23 20. 234 8422 xxx 21. 345 64244 xxx

22. 345 128322 xxx 23. xxx 108393 23 24. 234 108453 xxx

Solve the following equations by factoring.

25. 04032 xx 26. 063162 xx 27. xx 672

28. 052 23 xx 29. 018122 23 xxx 30. 2234 1224123 xxxx

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10.1-10.3 Review Worksheet Name _______________________

Identify the leading coefficient and constant term; then classify the polynomial by degree and

by number of terms.

1. 5x – 2 2. 8x3 – 3x + 45 3. – 6x + 5x4 - 3x

Add or subtract .

4. ( 5x3- 3x) – (7x2 – 3x + 1) 5. ( 2x3+ 4) – (- x2+ 3x)

6. ( m + 3m3- 4m5) + (2m3 + 5m5 - 4) 7. (x2+ 1) + (-3x2 – 7) - ( x2 + 3x)

8. (5x2+ 4) - (3x + 7) + ( 2x2 -1) 9. 2(x2- 4x + 5) – ( x2 + 6x – 1)

Find the product.

10. -5x ( 2x – 5) 11. ( 4x2 – 2x)( 8)

16

12. ( 3x - 5)(4x + 1) 13. ( 2x – 1)( 3x3 – 2x + 7)

14. 4(x – 5)( 3x + 4) 15. ( 5x – ½ ) ( 4x - 3)

16. ( 3x3 + 1)( 4x - 7) 17. ( 2x3 + 3x2 – 1) (3x3 – 3x + 2)

Write the square of the binomial as a trinomial.

18. ( x + 5)2 19. ( 3y – 4)2

Find the product.

20. ( x + 7)(x – 7) 21. (3m – 5) (3m + 5)

22. ( ½ x + 4)( ½ x – 4)

17

Lesson 10.6(a) – Factoring Quadratic Trinomials:

ax2 bx c Algebra 1

Essential Question How can you use algebra tiles to factor the trinomial2 + +ax bx c into the product of two binomials?

Warm-Up Exercise

Factor each expression completely.

(a)

x2 6x 40 (b)

2x2 12x 10

Find each product using the distributive property.

(c)

2x 3 3x 1 (d)

x 5 4x 7

Example:

3x2 2x 8

Step 3:

Step 4:

Factoring: The Box Method- Steps

1. Insert the first term of the trinomial into the upper left box.

2. Insert the last term into the lower right box.

3. Find the product of the leading coefficient and the constant term.

Be sure to carry all negative signs (if necessary)

4. Find and list all factors of the product from step 3.

5. Find the pair of factors that sum to the middle term’s coefficient.

6. Insert each factor into the empty boxes as x terms.

7. Find the GCF of each row and each column

- If the front box of each row or column is negative, then the GCF is negative

- If there is nothing in common, then the GCF is 1

8. Write the GCF’s from the rows as a binomial, and write the GCF’s form the

columns as a binomial.

Factoring: The Box Method

When the leading coefficient of a trinomial is greater than 1, the factors of both the

leading coefficient and the constant play a role in determining. The box method

helps us organize the work needed to factor each expression.

This method will only work if the greatest common factor is factored out first. Steps 1&2:

Steps 4&5:

Steps 6&7

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1. Factoring Using the Box Method

Factor each expression completely.

(a)

9x2 15x 4 (b)

3x2 10x 8 (c)

12y2 17y 6

Practice Problems - Complete all problems in the space below. Show all work

Factor each expression completely.

(a)

3b2 11b 6 (b) 584 2 xx

(c) )3108(5 2 mm

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Worksheet 10.6(a) – Factoring Quadratic Trinomials:

ax2 bx c Name_________________

Factor each polynomial of the form cbxax 2 . Use the “Box” method to factor.

1. 2116 2 xx 2. 295 2 xx 3. 10116 2 xx

4. 44373 2 yy 5. 41514 2 yy 6. 706 2 xx

(Draw your own boxes)

7. 10192 2 xx 8. 12317 2 xx 9. 2032 2 xx

10. 4113 2 xx 11. 8103 2 xx 12. 672 2 xx

13. 352 2 xx 14. 130 2 xx 15. 592 2 xx

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Lesson 10.6(b) – Solving Quadratic Equations w/ Zero Product Property Algebra 1

Warm-up Exercises

Factor the following expressions

(a) 3x2 – 4x – 7 (b) –11 + 2x2 + 21x (c) 21x –11 + 2x2

1. Solving Quadratic Equations Using Zero Product Property

Solve the equation

(a) – 21x + 4x2 + 5 = 0

(b) 7 + 14n + 21n2 = 6n + 11 (c) – 10t + 5t2 = -11t2 – 2t + 48

(d) – x + x2 – 8 = 82

1. Factoring Polynomials Involving the Greatest Common

Factor

Factor each expression completely.

(a)

14x2 32x 8 (b)

72n2 6x 45 (c)

12x4 26x3 10x2

Solving Quadratic Equations 1.

2.

3.

21

2. Completely Factoring Polynomials of Higher Degrees

Factor each expression completely.

(a) 6113 2 bb (b) 584 2 xx

(c) 155040 2 mm (d) xxx 279648 23

10.6C Worksheet: Factoring the GCF and cbxax 2 Name:

In each of the polynomials, factor out the GCF and then factor the remaining

trinomial.

1. 62712 2 xx 2. 345 8103 xxx 3. xxx 2010840 23

4. 21012 2 xx 5. 256020 2 xx 6. 23 412 ddd

22

7. 324 693615 ggg 8. 456 15148 xxx 9. 5018090 2 xx

10. 48816 2 xx 11. 234 32130 xxx 12. xxx 202040 23

13. 243624 2 xx 14. 345 2032 xxx 15. 1293 2 xx

23

Solve the equations by factoring.

16. 6x2 + 13x + 5 = 0 17. 3x2 + 7x = -2

18. 10x2 = 5 = - 15x 19. 12x2 + 32x = -5

20. 6x2 – 10x – 4 = 0 21. 6x2 – 27x + 27 = 0

22. 8x2 + 10x + 3 = 0 23. 4x2 – 8x – 5 = 0

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Lesson 10.7 – Difference of Squares and Perfect Square Trinomials Algebra 1

Essential Question How can you recognize and factor special products?

Warm-Up Exercise

Find the product.

(a) )12)(12( xx (b) )12)(12( yy (c) 2)13( n (d) 2)32( n

1. Factoring the Difference of Two Squares

Factor each expression.

(a) 642 x (b) 94 2 x

(c) 236121 y (d) 16250 2 a

2. Factoring Perfect Squares Trinomial

Factor each expression and identify the pattern for each perfect square trinomial.

(a) 962 xx (b) 24246 2 xx

(c) 100020010 2 yy (d) 92416 2 yy

3. Solving Equations Involving Special Product Factorization

Solve the equation.

(a) 025 2 x (b) 221812 xx

Difference of Two Squares

Since the special product of two binomials has the

following property:

The difference of two squares reverses the process by

factoring –

________________

Perfect Square Trinomial

Since the special product of two binomials has the

following property:

By reversing the process –

________________

________________

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Lesson 10.8 – Factoring a Polynomial Completely Algebra 1

Essential Question How can you factor a polynomial completely?

Work with a partner. Match the standard form of the polynomial with the equivalent factored form on

the next page. Explain your strategy.

a. 3 2x x b. 3x x c. 3 2 2x x x

d. 3 24 4x x x e. 3 22 3x x x f. 3 22x x x

g. 3 4x x h. 3 22x x i. 3 2x x

j. 3 23 2x x x k. 3 22 3x x x l. 3 24 3x x x

m. 3 22x x n. 3 24 4x x x o. 3 22x x x

Factoring a Polynomial Completely

A polynomial is completely factored if it can be written as

the product of monomials and prime polynomials.

Ex.

Prime Polynomials

1 EXPLORATION: Matching Standard and Factored Forms

A. B. C.

D. E. F.

G. H. I.

J. K. L.

M. N. O.

26

1. Factor the Polynomials Completely.

If it is prime, then say so.

(a) 25 12133 xx (b) 94 2 x (c) xxx 30255 23 (d) 24 375 xx

(e) 132 xx

2. Factoring By Grouping

Factor the polynomials

(a) 2464 23 xxx (b) 72362 23 xxx

3. Solving Equations By Factoring Completely

Find all solutions to each equation.

(a) 010016 3 xx (b) 0252523 xxx

(c ) 23 21 30 0x x (d) 25 5 30 0y y ( e ) 4 281 0c c

Factoring By Grouping You can use the distributive property to factor

some polynomials that have FOUR terms.

27

Unit 9 (Chapter 10) Test Review

I. Identifying Polynomials

Name each polynomial by its degree, number of terms, and leading coefficient

1. 5x – 2 2. 336 x 3. 725 432 xxxx

II. Polynomial Operations

Perform the indicated operation for each polynomial expression below.

4. 13735 23 xxxx 5. 541 22 xx 6. 5353 5243 mmmmm

7. 47735 222 xxx 8. 753283 22 tttt 9. 13125 44 xxxx

10. 3625 2 xxx 11. 482

14 32

xx 12. 4725 xx

13. 5143 2 xxx 14.

2

12864 2 xxx 15. 5252 xx

16. 274 x 17. xxxx 4343 22 18. 2

115 x

28

Guidelines for Factoring Polynomials Completely

To factor a polynomial completely, you should try each of these steps.

1. Factor out the greatest common monomial factor. 23 6 3 2x x x x

___________________________________________________________________

2. Look for a difference of two squares or a perfect

square trinomial. 22 4 4 2x x x

___________________________________________________________________

3. Factor a trinomial of the form 2 ax bx c into a product

of binomial factors. 23 5 2 3 1 2x x x x

___________________________________________________________________

4. Factor a polynomial with four terms by grouping.

3 2 24 4 1 4x x x x x

___________________________________________________________________

I. Factoring Polynomials.

Factor each expression completely

1. 35 1824 tt 2. 34 213 ww 3. 1012 644 xx

4. 457 3312111 xxx 5. 6910 121212 yyy 6. 7212 x

7. 72172 xx 8. 100202 xx 9. 72382 xx

10. 21544 xx 11. xx 902 12. xxx 23010

29

13. yyy 12102 23 14. 234 6022 ddd 15. mmm 21287 23

16. 212 2 xx 17. 169 2 xx 18. 734 2 xx

19. 144921 2 xx 20. 106424 2 xx 21. xxx 162024 23

22. 259 2 x 23. 964 2 x 24. 3162200 xx

25. 162362 2 xx 26. 2164836 xx 27. 100609 2 xx

30

28. 4423 xxx 29. 1234 23 xxx 30. 153102 23 ddd

Solve each equation. Be sure to find all solutions.

31. 064122 xx 32. 2832 xx 33. 014322 xx

34. 010113 2 xx 35. 81012 2 xx 36. xx 1356 2

37. 0483 3 xx 38. xx 20254 2 39. 01892 23 ddd