chapter 8 answer key

Prentice Hall Algebra 1 • Teaching ResourcesCopyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

1

Name Class Date

8-1 Additional Vocabulary SupportAdding and Subtracting Polynomials

Concept List

binomial constant cubic

degree fourth degree linear

monomial quadratic trinomial

Choose the concept or concepts from the list above that best represent the item in each box.

1. 2x3 1 5 2. 5x 1 4x2 3. 8

4. 9 5. 3x2 1 6x 1 4 6. 3x2 1 6x

7. 4x4 1 6x3 1 2x2 8. 7x2 1 x 9. 5x4

2

constant/monomial

constant/monomial

fourth degree/trinomial fourth degree/monomial

quadratic/trinomial

degree

quadratic/binomial

cubic/binomial binomial/quadratic

Name Class Date


2

8-1 Think About a PlanAdding and Subtracting Polynomials

Geometry Th e perimeter of a trapezoid is 39a 2 7. Th ree sides have the following lengths: 9a, 5a 1 1, and 17a 2 6. What is the length of the fourth side?

Understanding the Problem

1. What is the perimeter of the trapezoid?

2. What are the lengths of the sides you are given? , ,

3. How many sides does a trapezoid have?

4. How do you fi nd the perimeter of a trapezoid?

5. What is the problem asking you to determine?

Planning the Solution

6. Draw a diagram of the trapezoid and label the information you know.

7. Write an equation that can be used to determine the length of the fourth side.

Getting an Answer

8. Solve your equation to fi nd the length of the fourth side of the trapezoid.

39a 2 7

9a

4 sides

add the side lengths

the length of the fourth side

s 5 (39a 2 7) 2 (9a 1 5a 1 1 1 17a 2 6)

8a 2 2

s

5a 1 1

17a 2 6

9a

5a 1 1 17a 2 6

Name Class Date

Prentice Hall Gold Algebra 1 • Teaching ResourcesCopyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

3

8-1 Practice Form G

Adding and Subtracting Polynomials

Find the degree of each monomial.

1. 2b2c2 2. 5x 3. 7y5 4. 19ab

5. 12 6. 12 z2 7. t 8. 4d4e

Simplify.

9. 2a3b 1 4a3b 10. 5x3 2 4x3 11. 3m6n3 2 5m6n3

12. 26ab 1 3ab 13. 4c2d6 2 7c2d6 14. 315x2 2 30x2

Write each polynomial in standard form. Th en name each polynomial based on its degree and number of terms.

15. 15x 2 x3 1 3 16. 5x 1 2x2 2 x 1 3x4 17. 9x3

18. 7b2 1 4b 19. 23x2 1 11 1 10x 20. 12t2 1 1 2 3x 1 8 2 2x

Simplify.

21. 8z 2 12

1 6z 1 90 22.

9x3 1 31 4x3 1 7

23. 6j2 2 2j 1 5

1 3j2 1 4j 2 6

24. (3k2 1 5) 1 (16x2 1 7) 25. (g4 2 4g2 1 11) 1 (2g3 1 8g)

26. A local deli kept track of the sandwiches it sold for three months. Th e polynomials below model the number of sandwiches sold, where s represents days.

Ham and Cheese: 4s3 2 28s2 1 33s 1 250 Pastrami: 27.4s2 1 32s 1 180

Write a polynomial that models the total number of these sandwiches that were sold.

4

0

6a3b

2x3 1 15x 1 3; cubic trinomial

7b2 1 4b; quadratic binomial

14z 2 3

3k2 1 16x2 1 12

4s3 2 35.4s2 1 65s 1 430

g4 2 g3 2 4g2 1 8g 1 11

13x3 1 10 9j2 1 2j 2 1

23x2 1 10x 1 11; quadratic trinomial

12t2 2 5x 1 9; quadratic trinomial

3x4 1 2x2 1 4x ; fourth degree trinomial

9x3; cubic monomial

x3 22m6n3

23ab 23c2d6 285x2

2 1 5

51 2

Name Class Date


4

Simplify.

27. 11n 2 4

2 (5n 1 2) 28. 7x4 1 9

2 (8x4 1 2) 29. 3d2 1 8d 2 2

2 (2d2 2 7d 1 6)

30. (28e3 1 3e2) 1 (19e3 1 e2) 31. (212h4 1 h) 2 (26h4 1 3h2 2 4h)

32. A small town wants to compare the number of students enrolled in public and private schools. Th e polynomials below show the enrollment for each:

Public School: 219c2 1 980c 1 48,989

Private School: 40c 1 4046

Write a polynomial for how many more students are enrolled in public school than private school.

Simplify. Write each answer in standard form.

33. (3a2 1 a 1 5) 2 (2a 2 5) 34. (6d 2 10d3 1 3d2) 2 (5d3 1 3d 2 4)

35. (24s3 1 2s 2 3) 1 (22s2 1 s 1 7) 36. (8p3 2 6p 1 2p2) 1 (9p2 2 5p 2 11)

37. Th e fence around a quadrilateral-shaped pasture is 3a2 1 15a 1 9 long. Th ree sides of the fence have the following lengths: 5a, 10a 2 2, a2 2 7. What is the length of the fourth side of the fence?

38. Error Analysis Describe and correct the error in simplifying the sum shown at the right.

39. Open-Ended Write three diff erent examples of the sum of a quadratic trinomial and a cubic monomial.

8-1 Practice (continued) Form G


a2 75a

?

10a 2

6x3 + 4x – 10+ (–3x2 + 2x + 8)

3x3 + 6x – 2

6n 2 6

47e3 1 4e2 26h4 2 3h2 1 5h

219c2 1 940c 1 44,943

3a2 2 a 1 10

24s3 2 2s2 1 3s 1 4

2a2 1 18

two unlike terms, 6x3 and 23x2, were added; 6x3 2 3x2 1 6x 2 2

Answers may vary. Sample: (x2 1 2x 1 1) 1 x3; (2x2 1 5x 1 6) 1 3x3; (r2 1 r 1 1) 1 8r3

8p3 1 11p2 2 11p 2 11

215d3 1 3d2 1 3d 1 4

2x4 1 7 d2 1 15d 2 8

Name Class Date

Prentice Hall Foundations Algebra 1 • Teaching ResourcesCopyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

5

8-1 Practice Form K



1. 3s3t3 2. 3n 3. 5xy

4. 7 5. 14k5 6. d

Simplify.

7. 3mn4 1 6mn4 8. 12g2 2 7g2

9. 211c4d 1 12c4d 10. 42z3 2 15z3

Write each polynomial in standard form. Th en name each polynomial based on its degree and number of terms.

11. 7a 1 4 2 a2 12. 5b2 1 2n

13. 211d4 14. 2x3 2 9 1 2x 1 8 2 4x

15. A pizza shop owner is monitoring the amount of cheese he uses each week. Th e polynomials below model the pounds of cheese ordered in the past, where p represents pounds.

Mozzarella: 3p3 2 6p2 1 14p 1 125

Cheddar: 12.5p2 1 18p 1 75

Write a polynomial that models the total number of pounds of cheese that were ordered.

6 1 2

0 5 1

9mn4 5g2

c4d 27z3

2a2 1 7a 1 4; quadratic trinomial 5b2 1 2n; quadratic binomial

211d4; 4th degree monomial 2x3 2 2x 2 1; cubic trinomial

3p3 1 6.5p2 1 32p 1 200

Name Class Date


6

8-1 Practice (continued) Form K


Simplify.

16. 3r 1 5

1 7r 1 3 17. (t4 2 4t2 1 9) 1 (2t3 1 3t)

18. 7b2 1 6

1 4b2 1 5 19.

4z 1 72 (6z 1 1)

20. (26k3 1 3k) 2 (25k3 1 3k2 2 8k) 21. 3p4 1 1

2 (9p4 1 5)

22. A city wants to compare the number of people who own their own home and who rent their home. Th e polynomials below show expressions for each. In each polynomial, p 5 0 corresponds to the fi rst year.

Own: 4p2 1 360p 1 22,178

Rent: 6p2 1 125p 1 5286

Write a polynomial for how many more own their home than rent their home.

23. Th e wallpaper border that runs all the way around a room is 5f 21 19f 1 11 long. Th ree sides of the room have the following lengths of border: 6f, 5f 2 7, 2f 21 2. What is the length of the fourth side of the room?

24. Open-Ended Write two diff erent polynomials with a diff erence of 23x2 1 5x 2 7.

10r 1 8 t4 2 t3 2 4t2 1 3t 1 9

11b2 1 11 22z 1 6

2k3 2 3k2 1 11k 26p4 2 4

22p2 1 235p 1 16,892

3f 2 1 8f 1 16

Answers may vary. Sample: (21x2 1 6x 2 4) 2 (2x2 1 x 1 3) and

(24x2 1 7x 2 5) 2 (2x2 1 2x 1 2)

Name Class Date


7

8-1 Standardized Test PrepAdding and Subtracting Polynomials

Multiple Choice

For Exercises 1–6, choose the correct letter.

1. What is the degree of the monomial 3x2y3?

A. 2 B. 3 C. 5 D. 6

2. What is the simplifi ed form of 8b3c2 1 4b3c2?

F. 12bc G. 12b3c2 H. 12b6c4 I. 12b9c4

3. How is 6d 2 8 1 4d2 written in standard form?

A. 4d2 1 6d 2 8 B. 4d2 1 6d 1 8 C. 4d2 2 6d 2 8 D. 4d2 2 6d 1 8

4. What is the simplifi ed form of (4j2 1 6) 1 (2j2 2 3)?

F. 6j2 2 3 G. 6j2 1 3 H. 6j2 1 9 I. 4j4 1 3

5. What is the diff erence of the following polynomials?

6x3 2 2x2 1 4

2 (2x3 1 4x2 2 5)

A. 4x3 2 2x2 2 1 B. 8x3 1 6x2 2 1 C. 4x3 2 2x2 1 1 D. 4x3 2 6x2 1 9

6. What is the simplifi ed form of (3x2 2 4x 1 6x) 1 (5x3 1 2x2 2 3x) in

standard form?

F. 5x3 1 10x2 2 x G. 8x3 2 2x2 1 3x H. 5x3 1 10x2 2 5x I. 5x3 1 5x2 2 x

Short Response

7. Suppose you have been given this polynomial. 5b 1 4b2 2 3b4 1 3

a. How can you write this polynomial in standard form?

b. What is the degree of this polynomial? Explain.

C

G

G

D

I

A

23b4 1 4b2 1 5b 1 3

4; b4 is the term with the greatest degree

[2] Both parts answered correctly with full explanations

[1] One part answered correctly or both parts answered correctly with incomplete explanations

[0] Neither part answered correctly

Name Class Date


8

8-1 EnrichmentAdding and Subtracting Polynomials

Packing boxes and packing sheets in diff erent sizes are given by the expressions below. To fi nd the number of packing boxes and sheets that will fi t in a larger shipping box, add or subtract the polynomials. Tell the total number of boxes and sheets. Th en tell how many medium and large boxes and sheets you could fi t into the shipping box. Th e fi rst one has been started for you.

Boxesa3 5 small box b3 5 medium box c3 5 large box

6 small boxes 5 1 medium box 4 medium boxes 5 1 large box

Sheetsa2 5 small sheet b2 5 medium sheet c2 5 large sheet

4 small sheets 5 1 medium sheet 8 medium sheets 5 1 large sheet

1. 7a3 1 5b3 1 5a3 2 3b3 5 12a3 2 2b3

5 12 small boxes and 2 medium boxes

5 4 medium boxes

5 large box(es)

2. 6a2 1 3b2 2 8c2 1 12b2 2 2a2 1 10c2 5

5 small sheets, medium sheets, and large sheets

5 medium sheets and large sheets

5 large sheets

3. (8a3 2 3b3 1 6c3) 2 (2a3 2 14b3 1 2c3) 5

5 small boxes, medium boxes, and large boxes

5 medium boxes and large boxes

5 large boxes

4. (15c2 1 12a3 2 9b2) 1 (214c2 1 6a3 1 5b3 1 25b2) 5

5

5 medium boxes medium sheets, and large sheets

5 large boxes and large sheets

1

4

16

4

4

6

7

11

12

16 18

2 3

4

5 medium boxes, 16 medium sheets, and 1 large sheet 18a3 1 5b3 1 16b2 1 c2

2

15 2

6a3 1 11b3 1 4c3

4a2 1 15b2 1 2c2

18 small boxes,

Name Class Date


9

You can add and subtract polynomials by lining up like terms and then adding or subtracting each part separately.

Problem

What is the simplifi ed form of (3x2 2 4x 1 5) 1 (5x2 1 2x 2 8)?

Write the problem vertically, lining up the like terms. Th en add each pair of like terms.

Solve Add the x2 terms. Add the x terms. Add the constant terms.

3x2 1 5x2 5 8x2 24x 1 2x 5 22x 5 1 (28) 5 23

3x2 2 4x 1 5

1 5x2 1 2x 2 8

8x2 2 2x 2 3

Add the sums.

Check Check your solution using subtraction.

8x2 2 5x2 5 3x2 22x 2 2x 5 24x 23 2 (28) 5 5

Solution: (3x2 2 4x 1 5) 1 (5x2 1 2x 2 8) 5 8x2 2 2x 2 3

Exercises

Simplify.

1. 5b2 1 3b

1 2b2 2 5b 2.

3c2 1 3c1 4c2 1 2c

3. 4d2 2 3d 1 6

1 2d2 1 5d 2 3

4. 23e2 2 5e 1 2

1 e2 1 2e 2 7 5.

4f 31 2f 2 1 5f1 2f 32 4f 22 3f

6. 5g3 2 2g2 1 3g

1 2g3 1 5g2 2 2g

7. (3h2 1 5) 1 (25h2 2 3) 8. (2j2 1 4j 2 6) 1 (4j2 2 3j 2 3)

8-1 ReteachingAdding and Subtracting Polynomials

3x2 2 4x 1 5

1 5x2 1 2x 2 8

7b2 2 2b

22e2 2 3e 2 5

22h2 1 2 6j2 1 j 2 9

6f 32 2f 21 2f 7g3 1 3g2 1 g

7c2 1 5c 6d2 1 2d 1 3

Name Class Date


10

To subtract polynomials, follow the same steps as in addition.

Problem

What is the simplifi ed form of (6x3 1 4x2 2 3x) 2 (2x3 1 3x2 2 5x)?

Write the problem vertically, lining up the like terms. Th en subtract each pair of like terms.

Solve

Subtract the x3 terms. Subtract the x2 terms. Subtract the x terms.

6x3 2 2x3 5 4x3 4x2 2 3x2 5 x2 23x 2 (25x) 5 2x

6x3 1 4x2 2 3x2 (2x3 1 3x2 2 5x)

4x3 1 x2 1 2x

Add the differences.

Check Check your solution using subtraction.

4x3 1 2x3 5 6x3 x2 1 3x2 5 4x2 2x 1 (25x) 5 23x

Solution: (6x3 1 4x2 2 3x) 2 (2x3 1 3x2 2 5x) 5 4x3 1 x2 1 2x

Exercises

Simplify.

9. 4k2 1 5k

2 (3k2 1 2k) 10.

5m2 2 4m2 (2m2 1 3m)

11. 7n2 1 4n 1 9

2 (4n2 1 3n 1 5)

12. 5p2 1 6p 1 4

2 (7p2 1 4p 1 8) 13. 3q3 1 2q2 1 7q

2 (6q3 2 4q2 2 5q) 14.

2r3 2 2r2 1 5r2 (4r3 1 5r2 1 3r)

15. (6s2 2 5s) 2 (22s2 1 3s) 16. (3w2 1 6w 2 5) 2 (5w2 2 4w 1 2)

8-1 Reteaching (continued)


6x3 1 4x2 2 3x

2 (2x3 1 3x2 2 5x)

k2 1 3k

22p2 1 2p 2 4

8s2 2 8s 22w2 1 10w 2 7

3m2 2 7m

23q3 1 6q2 1 12q 22r3 2 7r2 1 2r

3n2 1 n 1 4


11

Name Class Date

8-2 Additional Vocabulary SupportMultiplying and Factoring

There are two sets of note cards below that show how Brittany factors the polynomial 5x5 1 15x3 1 4x2. The set on the left explains the thinking. The set on the right shows the steps. Write the thinking and the steps in the correct order.

Think Cards Write Cards

Think Write

Factor each term of the polynomial.

x2(5x3 1 15x 1 4)

The GCF is x ? x, or x2.

5x5 1 15x3 1 4x2 5

x2(5x3) 1 x2(15x) 1 x2(4)

5x5 5 5 ? x ? x ? x ? x ? x15x3 5 3 ? 5 ? x ? x ? x4x2 5 2 ? 2 ? x ? x

Simplify.

Find the GCF of the three terms.

Factor out the GCF from each term.

Step 1

Step 2

Step 3

Step 4

First, she should factor each term of the polynomial.

5x5 5 5 ? x ? x ? x ? x ? x

15x3 5 3 ? 5 ? x ? x ? x

4x2 5 2 ? 2 ? x ? x

The GCF is x ? x , or x2.

5x5 1 15x3 1 4x2 5

x2(5x3) 1 x2(15) 1 x2(4)

x2(5x3 1 15x 1 4)

Second, she should find the GCF of the three terms.

Next, she should factor out the GCF from each term. Then factor it out of the polynomial.

Finally, she should simplify.


12

Name Class Date

a. Factor n2 1 n.

b. Writing Suppose n is an integer. Is n2 1 n always, sometimes, or never an even integer? Justify your answer.

1. Factor out n from the expression.

nau 1ub

2. What are the two factors? ,

3. What is an integer?

4. Are n and n + 1 consecutive integers? Explain.

5. What do you know about the product of odd and even integers?

EVEN 3 EVEN 5

ODD 3 ODD 5

EVEN 3 ODD 5

ODD 3 EVEN 5

6. Out of two consecutive integers, how many are odd?

7. Is the product of consecutive integers odd or even? Explain.

8. n2 1 n is an even integer because

.

8-2 Think About a PlanMultiplying and Factoring

n

n

positive and negative whole numbers and zero

yes; the next number after n is n 1 1

even

even

even

always

it is the product of two consecutive integers

1

even; Two consecutive integers

odd

n 1 1

1

must be an odd integer and an even integer. If 1 factor is even, the product will be even.

Name Class Date


13

Simplify each product.

1. 2x(x 1 8) 2. (n 1 7)5n 3. 6h2(7 1 h)

4. 2b2(b 2 10) 5. 23c(8 1 2c 2 c3) 6. y(2y2 2 3y 1 6)

7. 4t(t2 2 6t 1 2) 8. 2m(4m3 2 8m2 1 m) 9. 7j(22j2 2 8j 2 3)

10. 2t2(2t4 1 4t 2 8) 11. 2k(23k3 1 k2 2 10) 12. 8a2(2a7 1 7a 2 7)

13. 4v3(2v2 2 3v 1 5) 14. 5d(2d3 1 2d2 2 3d) 15. 11w(w2 1 2w 1 6)

Find the GCF of the terms of each polynomial.

16. 15x 1 27 17. 6w3 2 14w 18. 63s 1 45

19. 72y5 1 18y2 20. 218q3 2 6q2 21. 108 f3 2 54

22. b3 1 5b2 2 20b 23. 9m3 1 30m 2 24 24. 4p3 1 12p2 2 18p

25. 2e2 1 12e 2 22 26. 14b3 1 21b2 2 42b 27. 212x3 1 24x2 2 16x

28. 8a4 1 24a3 2 40a2 29. 36j3 2 3j2 2 15j 30. 12j8 1 30j4 2 6j3

Factor each polynomial.

31. 12x 2 9 32. 18s2 1 54 33. 108t2 2 60t

34. 220w2 1 16w 35. 32y3 1 8y2 36. 300d2 2 175d

37. 12n3 2 36n2 1 18 38. 40t3 1 25t2 1 80t 39. 42x4 2 56x3 1 28x2

40. 15c4 1 24c3 2 6c2 1 12c 41. 8m3 1 14m2 1 6m 42. 10x2 1 50x 2 25

43. 36p4 1 14p3 1 35p2 44. 9a5 1 27a4 1 63a2 45. 4b4 1 20b3 1 12b

46. x6 2 x4 1 x2 47. 34g3 1 51g2 1 17g 48. 18h4 2 27h2 1 18h

8-2 Practice Form G

Multiplying and Factoring

2x2 1 16x

2b3 1 10b2

4t3 2 24t2 1 8t

22t6 2 4t3 1 8t2

8v5 2 12v4 1 20v3

3

18y2

b

2

3(4x 2 3)

24w(5w 2 4)

6(2n3 2 6n2 1 3)

3c(5c3 1 8c2 2 2c 1 4)

p2(36p2 1 14p 1 35)

x2(x4 2 x2 1 1)

2m(4m2 1 7m 1 3)

9a2(a3 1 3a2 1 7)

17g(2g2 1 3g 1 1)

8a2

2w

26q2

3

7b

18(s2 1 3)

8y2(4y 1 1)

5t(8t2 1 5t 1 16)

25d(12d 2 7)

14x2(3x2 2 4x 1 2)

5(2x2 1 10x 2 5)

4b(b3 1 5b2 1 3)

9h(2h3 2 3h 1 2)

3j

9

54

2p

4x

12t(9t 2 5)

6j3

26k4 1 2k3 2 20k

25d4 1 10d3 2 15d2

28a9 1 56a3 2 56a2

11w3 1 22w2 1 66w

24m4 1 8m3 2 m2 214j3 2 56j2 2 21j

3c4 2 6c2 2 24c

5n2 1 35n 6h3 1 42h2

2y3 2 3y2 1 6y

Name Class Date


14

49. A circular hedge surrounds a sculpture on a square base. Th e radius of the hedge is 6x. Th e side length of the square sculpture base is 4x. What is the area of the hedge? Write your answer in factored form.

50. Suppose you are making a giant chocolate chip cookie for a raffl e. You roll out a square slab of cookie dough. Th en you use a circular plate that touches the edges of the square slab of cookie dough and cut the cookie out of the dough. What is the area of the extra dough? Write your answer in factored form.

Simplify. Write in standard form.

51. 23x(4x2 2 6x 1 12) 52. 27y2(24y3 1 6y) 53. 9a(23a2 1 a 2 5)

54. p(p 1 4)22p(p 2 8) 55. t(t 1 4)2 t(4t2 2 2) 56. 6c(2c2 2 4)2 c(8c)

57. 25m(2m3 2 7m2 1 m) 58. 2q(q 1 1)2q(q 2 1) 59. 2n2(26n2 1 2n)


60. 15xy4 1 60x2y3 61. 8m3n4 1 32mn2 62. 26a5b2 1 51a4

63. 36j2k4 1 24j4k2 64. 12w4x3 2 42wx2 65. 54c2d3 2 36c3d2

66. 12st4 1 46s3t4 67. 9v6w3 1 33v4w5 68. 11e3f3 1 132e2f 4

69. Error Analysis A student factored the polynomial at the right. Describe and correct the error made in factoring.

70. Reasoning Th e GCF of two numbers j and k is 8. What is the GCF of 2j and 2k? Justify your answer.

71. A cylinder has a radius of 3m2n and a height of 7mn. Th e formula for the

volume of a cylinder is V 5 pr2h, where r is the radius and h is the height. What is the volume of the cylinder? Simplify your answer.



6x

4x

63x4 – 14x3 + 35x2

= 7x(9x3 – 2x2 + 5x)

4x2(9π 2 4)

r2(4 2 π)

212x3 1 18x2 2 36x

2p2 1 20p

210m4 1 35m3 2 5m2

15xy3(y 1 4x)

12j2k2(3k2 1 2j2)

2st4(6 1 23s2)

The student did not fi nd the correct GCF. 7x2(9x2 2 2x 1 5)

16 The GCF will be the product of 2 and 8.

63πm5n3

28y5 2 42y3

24t3 1 t2 1 6t

q2 1 3q

8mn2(m2n2 1 4)

6wx2(2w3x 2 7)

3v4w3(3v2 1 11w2)

227a3 1 9a2 2 45a

12c3 2 8c2 2 24c

6n4 2 2n3

a4(26ab2 1 51)

18c2d2(3d 2 2c)

11e2f 3(e 1 12f )

Name Class Date


15

8-2 Practice Form K



1. 3w(w 1 2) 2. (z 1 5)2z 3. 3m2(4 1 m)

4. 2p(p2 2 6p 1 1) 5. 2y(5y3 2 3y2 1 2y) 6. 3a(23a2 1 2a 2 7)

7. 6x3(3x2 2 x 1 10) 8. 24h(2h3 2 8h2 1 2h) 9. 4n(n2 1 5n 1 6)


10. 16q 1 32 11. 4t3 2 24t 12. 32y 2 24

13. x3 1 3x2 1 5x 14. 5d3 1 20d 2 35 15. 2m3 1 10m2 1 12m

16. 7g4 1 21g3 2 14g2 17. 15z3 1 3z2 2 27z 18. 33w7 1 55w5 2 22w3


19. 9t 2 3 20. 12j3 1 28 21. 72x2 2 63x

22. 12k3 2 9k2 1 6 23. 30n3 1 18n2 1 54n 24. 32z4 2 80z3 1 112z2

25. 12n4 1 16n3 1 20n2 26. 24y6 1 36y4 1 42y2 27. 7q5 1 21q3 2 49q

16 4t 8

x 5 2m

7g2 3z 11w3

3(3t 2 1) 4(3j3 1 7) 9x(8x 2 7)

3(4k3 2 3k2 1 2) 6n(5n2 1 3n 1 9) 16z2(2z2 2 5z 1 7)

4n2(3n2 1 4n 1 5) 6y2(4y4 1 6y2 1 7) 7q(q4 1 3q2 2 7)

3w2 1 6w 2z2 1 10z 12m2 1 3m3

2p3 2 12p2 1 2p 25y4 1 3y3 2 2y2 29a3 1 6a2 2 21a

18x5 2 6x4 1 60x3 4h4 1 32h3 2 8h2 4n3 1 20n2 1 24n

Name Class Date


16



28. You are painting a rectangular wall with length 5x2 ft and width 12x ft. Th ere is a rectangular door that measures x ft by 2x ft that will not be painted. What is the area of the wall that is to be painted? Write your answer in factored form.

Simplify. Write in standard form.

29. 23m(2m2 2 5m 1 10) 30. 25t2(26t3 1 12t) 31. 10x(24x2 1 x 2 3)

32. 22v(3v3 2 6v2 1 2v) 33. 5y(y 1 2) 2 y(y 2 3) 34. 22b2(24b2 1 3b)


35. 13cd3 1 39c2d2 36. 5x3y4 2 25xy2 37. 42m5n 1 28m4

38. 36f g2 1 54f 2g4 39. 8s8t4 1 20s4t3 40. 12a2b5 1 156a2b3

41. Open-Ended Write a quadratic monomial and a cubic trinomial. Th en fi nd their product and write it in standard form.

42. A rectangle has a length of 6x3y2 2 1 and a width of 3xy 1 2. Th e formula for the perimeter of a rectangle is P 5 2l 1 2w, where l is the length and w is the width. What is the perimeter of the rectangle? Simplify your answer.

2x2(30x 2 1)

26m3 1 15m2 2 30m 30t5 2 60t3 240x3 1 10x2 2 30x

26v4 1 12v3 2 4v2 4y2 1 13y 8b4 2 6b3

13cd2(d 1 3c) 5xy2(x2y2 2 5) 14m4(3mn 1 2)

18fg2(2 1 3fg2) 4s4t3(2s4t 1 5) 12a2b3(b2 1 13)

Answers may vary. Sample: x2 and 2x3 1 x2 1 x; 2x5 1 x4 1 x3

12x3y2 1 6xy 1 2


17

Name Class Date

Multiple Choice


1. How can this product be simplifi ed?

5x2(2x 2 3)

A. 5x2 1 2x 2 3 B. 10x3 2 15x2 C. 25x2 D. 7x3 2 15x2

2. What is the GCF of the terms of 8c3 1 12c2 1 10c?

F. 2 G. 4 H. 2c I. 4c

3. How can the polynomial 6d4 1 9d3 2 12d2 be factored?

A. 3d2(2d2 1 3d 2 4)

B. 3d2(3d2 1 6d 2 9)

C. 3d(d3 1 3d2 2 4)

D. 6d2(d2 1 3d3 2 6)

4. Th ere is a circular garden in the middle of a square yard. Th e radius of the circle is 4x. Th e side length of the yard is 20x. What is the area of the part of the yard that is not covered by the circle?

F. 4x(5) G. 8x2(5 2 p) H. 16x(25 1 p) I. 16x2(25 2 p)

5. What is the simplifi ed form of 23z2(z 1 2)2 4(z2 1 1)?

A. 27z2 1 1

B. 23z3 2 4z2 2 6z 2 4

C. 23z3 2 2z2 2 4

D. 23z3 2 10z2 2 4

Short Response

6. A rectangular blacktop with a length of 5x and a width of 3x has been erected inside a rectangular fi eld that has a length of 12x and a width of 7x.

a. What is the area of the part of the fi eld that is not blacktop?

b. Th ere is a circular fountain in the rectangular fi eld that has a radius of 3x. What is the area of the part of the fi eld that does not include the blacktop or the fountain? Factor your answer.

8-2 Standardized Test PrepMultiplying and Factoring

H

A

B

I

D

[2] Both parts answered correctly

[1] One part answered correctly

[0] Neither part answered correctly

69x2

3x2(23 2 3π)


18

Name Class Date

To fi nd the area of irregular fi gures, split the fi gure into simple fi gures and then add the areas of each fi gure.

1. What is the area of the fi gure to the right?

2. What is the perimeter of the fi gure?

A circle is inscribed in a square as shown.

3. What is the area of the circle?

4. What is the area of the square?

5. What is the area of the shaded region?

6. Th e area of a right triangle is 10y3 1 5y2 1 37.5y. Th e length of base of the triangle is a monomial with a whole number coeffi cient. Th e length of the height is a trinomial. Factor the polynomial to fi nd the base and height of the

triangle. (Remember to multiply the area by 2 fi rst because Atriangle 5b 3 h

2 .)

Base 5

Height 5

8-2 EnrichmentMultiplying and Factoring

24x 1 5

24x

23x

23x

10z

37x2 2 35x

228x 1 10

25πz2

100z2

100z2 2 25πz2, or 25z2(4 2 π)

5y

4y2 1 2y 1 15


19

Name Class Date

You can multiply a monomial and a trinomial by solving simpler problems. You can use the Distributive Property to make three simpler multiplication problems.

Problem

What is the simplifi ed form of 3x(2x2 1 4x 2 1)?

Use the Distributive Property to rewrite the problem as three separate multiplication problems.

3x(2x2 1 4x 2 1) 5 (3x ? 2x2) 1 (3x ? 4x) 1 (3x ? (21))

Remember that when you multiply same-base terms containing exponents, you

add the exponents.

Solve 3x ? 2x2 5 6x3 Multiply inside the fi rst pair of parentheses.

3x ? 4x 5 12x2 Multiply inside the second pair of parentheses.

3x ? (21) 5 23x Multiply inside the third pair of parentheses.

6x3 1 12x2 2 3x Add the products.

Check 6x3 4 2x2 5 3x Check your solution using division.

12x2 4 4x 5 3x

23x 4 (21) 5 3x

Solution: 3x(2x2 1 4x 2 1) 5 6x3 1 12x2 2 3x

Exercises


1. 4x(2x 2 7) 2. 3y(3y 1 4) 3. 2z2(2z 2 3)

4. 3a(24a 2 6) 5. 6b(3b2 1 2b 2 4) 6. 3c2(2c2 2 4c 1 3)

7. 22d(4d2 1 3d 2 2) 8. 5e2(23e2 2 2e 2 3) 9. 4f(23f3 1 2f2 1 6)

8-2 ReteachingMultiplying and Factoring

8x2 2 28x

212a2 2 18a

28d3 2 6d2 1 4d

9y2 1 12y

18b3 1 12b2 2 24b

215e4 2 10e3 2 15e2

4z3 2 6z2

6c4 2 12c3 1 9c2

212f 4 1 8f 31 24f


20

Name Class Date

To factor a polynomial, fi nd the greatest common factor (GCF) of the coeffi cients and constants and also the GCF of the variables.

Problem

What is the factored form of 8x4 1 12x2 2 16x?

Solve Find the GCF of the coeffi cients. Use prime factorization.

8 5 2 ? 2 ? 2

12 5 2 ? 2 ? 3

16 5 2 ? 2 ? 2 ? 2

Th e GCF of the numbers is 4.

Each term has a variable. Remember, x 5 x1. Th e GCF is the least exponent. Th e GCF of the variables is x. Th e GCF is 4x. Combine the GCFs.

Factor out the GCF of each term.

4(2 1 3 2 4) Factor the coeffi cients.

4x(2x3 1 3x 2 4) Insert the variables.

Check 4x(2x3 1 3x 2 4) 5 8x4 1 12x2 2 16x Check by multiplying.

Solution: Th e factored form of 8x4 1 12x2 2 16x is 4x(2x3 1 3x 2 4).

Exercises


10. 12x2 2 6x 11. 4y2 1 12y 1 8 12. 6z3 1 15z2 2 9z


13. 8a 1 10 14. 12b2 2 18b 15. 9c3 1 12c2

16. 5d3 2 10d2 1 20d 17. 6e2 1 10e 2 8 18. 8g3 2 24g2 1 16g



6x

2(4a 1 5)

5d(d2 2 2d 1 4)

4

6b(2b 2 3)

2(3e2 1 5e 2 4)

3z

3c2(3c 1 4)

8g(g2 2 3g 1 2)


21

Name Class Date

8-3 Additional Vocabulary SupportMultiplying Binomials

Use the Distributive Property to find the simplified form of (3x 1 2)(4x 2 3).

(3x 1 2)(4x 2 3) Write the problem.

3x(4x 2 3) 1 2(4x 2 3) Distribute the second factor, 4x 2 3.

12x2 2 9x 1 2(4x 2 3) Distribute 3x.

12x2 2 9x 1 8x 2 6 Distribute 2.

12x2 2 x 2 6 Combine like terms.

Exercises


(5x 1 6)(2x 2 4)

5x(2x 2 4) 1 6(2x 2 4)

10x2 2 20x 1 6(2x 2 4)

10x2 2 20x 1 12x 2 24

10x2 2 8x 2 24


(7x 2 3)(4x 1 6)

7x(4x 1 6) 2 3(4x 1 6)

28x2 1 42x 2 3(4x 1 6)

28x2 1 42x 2 12x 2 18

Write the problem.

Write the problem.

Distribute the second factor, 2x 2 4.

Distribute the second factor, 4x 1 6.

Distribute 5x.

Distribute 7x.

Distribute 6.

Distribute 23.

Combine like terms.

Combine like terms.28x2 1 30x 2 18


22

Name Class Date

Geometry Th e dimensions of a rectangular prism are n, n 1 7, and n 1 8. Use the formula V 5 lwh to write a polynomial in standard form for the volume of the prism.

Know

1. What are the dimensions of the rectangular prism? , ,

2. What is the formula for the volume of a rectangular prism?

3. In the volume formula, what do l, w, and h represent? , ,

4. Explain how to write a polynomial in standard form.

Need

5. To solve the problem you need to fi nd

.

Plan

6. Draw a diagram of the rectangular prism and label the information you know.

7. Write an expression for the volume of the rectangular prism.

8. Write the volume of the rectangular prism as a polynomial in standard form.

8-3 Think About a PlanMultiplying Binomials

n n 1 7 n 1 8

V 5 lwh

length

a polynomial in standard form that represents

The terms are arranged in order of

width height

degree, with the highest degree fi rst.

n 1 7n

n 1 8

V 5 n(n 1 7)(n 1 8)

n3 1 15n2 1 56n

the volume of the rectangular prism

Name Class Date


23

Simplify each product using the Distributive Property.

1. (x 1 3)(x 1 8) 2. (y 2 4)(y 1 7) 3. (m 1 9)(m 2 3)

4. (c 2 6)(c 2 4) 5. (2r 2 5)(r 1 3) 6. (3x 1 1)(5x 2 3)

7. (d 1 2)(4d 2 3) 8. (5t 2 1)(3t 2 2) 9. (a 1 11)(11a 1 1)

Simplify each product using a table.

10. (x 1 3)(x 2 5) 11. (a 2 2)(a 2 13) 12. (w 2 4)(w 1 8)

13. (5h 2 3)(h 1 7) 14. (x 2 3)(2x 1 3) 15. (2p 1 1)(6p 1 4)

Simplify each product using the FOIL method.

16. (2x 2 6)(x 1 3) 17. (n 2 5)(3n 2 4) 18. (4p2 1 2)(3p 2 1)

19. (a 1 7)(a 2 3) 20. (x 1 3)(3x 2 2) 21. (k 2 9)(k 1 5)

22. (b 2 5)(b 2 11) 23. (4m 2 1)(m 1 4) 24. (7z 1 3)(4z 2 6)

25. (2h 1 6)(5h 2 3) 26. (3w 1 12)(w 1 3) 27. (6c 2 2)(9c 2 8)

8-3 Practice Form G

Multiplying Binomials

x2 2 2x 2 15

5h2 1 32h 2 21

x2 1 11x 1 24

c2 2 10c 1 24

4d2 1 5d 2 6

2x2 2 18

a2 1 4a 2 21

b2 2 16b 1 55

10h2 1 24h 2 18

a2 2 15a 1 26

2x2 2 3x 2 9

y2 1 3y 2 28

2r2 1 r 2 15

15t2 2 13t 1 2

3n2 2 19n 1 20

3x2 1 7x 2 6

4m2 1 15m 2 4

3w2 1 21w 1 36

w2 1 4w 2 32

12p2 1 14p 1 4

m2 1 6m 2 27

15x2 2 4x 2 3

11a2 1 122a 1 11

12p3 2 4p2 1 6p 2 2

k2 2 4k 2 45

28z2 2 30z 2 18

54c2 2 66c 1 16

Name Class Date


24

28. What is the surface area of the cylinder at the right? Write your answer in simplifi ed form.

29. Th e radius of a cylindrical popcorn tin is (3x 1 1) in. Th e height of the tin is three times the radius. What is the surface area of the cylinder? Write your answer in simplifi ed form.

30. Th e radius of a cylindrical tennis ball can is (2x 1 1) cm. Th e height of the tennis ball can is six times the radius. What is the surface area of the cylinder? Write your answer in simplifi ed form.


31. (x 1 3)(x2 2 2x 1 4) 32. (k2 2 5k 1 2)(k 2 5)

33. (3a2 1 a 1 4)(2a 2 6) 34. (2x2 1 2x 2 6)(3x 2 4)

35. (4g 1 5)(2g2 2 7g 1 3) 36. (m2 2 2m 1 7)(3m 1 6)

37. (2c 1 8)(2c2 2 4c 2 1) 38. (t 1 8)(3t2 1 4t 1 5)

39. A medical center’s rectangular parking lot currently has a length of 30 meters and a width of 20 meters. Th e center plans to expand both the length and the width of the parking lot by 2x meters. What polynomial in standard form represents the area of the expanded parking lot?

40. Error Analysis Describe and correct the error made in fi nding the product.

41. Multi Step Th e height of a painting is twice its width x. You want a 3 inch wide wooden frame for the painting. Th e area of the frame alone is 216 square inches.

a. Draw a diagram that represents this situation. b. Write a variable expression for the area of the frame alone.

c. What are the dimensions of the frame?



x 4

x 7

(2x – 3)(x + 7)

2x2+ 17x + 21

2x

3x32x2 14x

21

7x

4πx2 1 38πx 1 88π

72πx2 1 48πx 1 8π

56πx2 1 56πx 1 14π

x3 1 x2 2 2x 1 12

6a3 2 16a2 1 2a 2 24

8g3 2 18g2 2 23g 1 15 3m3 1 9m 1 42

4c3 1 8c2 2 34c 2 8

4x2 1 100x 1 600

length is 26; width is 1618x 1 36

In the table, the 3 should be 23. Therefore, 3x should be 23x and 21 should be 221. The answer is 2x2 1 11x 2 21.

k3 2 10k2 1 27k 2 10

6x3 2 2x2 2 26x 1 24

3t3 1 28t2 1 37t 1 40

x 1 6

x

2x 1 62x

Name Class Date


25

8-3 Practice Form K


Simplify each product using the Distributive Property.

1. (b 2 2)(b 1 1) 2. (x 1 6)(x 1 5)

3. (3n 1 1)(n 2 8) 4. (2t 2 7)(t 2 5)

5. (y 1 3)(y 1 7) 6. (b 2 6)(b 1 3)

Simplify each product using a table.

7. (x 1 1)(x 2 11) 8. (h 2 2)(3h 1 5) 9. (8w 2 3)(4w 2 7)

10. (3c 1 13)(13c 1 3) 11. (3a 1 2)(a 2 2) 12. (t 1 7)(2t 2 4)

13. (3q2 1 6)(2q 2 5) 14. (x 1 6)(x 2 7) 15. (p 2 10)(2p 1 5)

16. (j 2 12)(j 2 11) 17. (3z 2 4)(7z 2 5) 18. (2m 1 11)(6m 2 1)

19. (7h 1 6)(7h 2 6) 20. (23z 1 7)(4z 2 8) 21. (23t 1 5)(23t 2 2)

b2 2 b 2 2 x2 1 11x 1 30

3n2 2 23n 2 8 2t2 2 17t 1 35

y2 1 10y 1 21 b2 2 3b 2 18

x2 2 10x 2 11 3h2 2 h 2 10 32w2 2 68w 1 21

39c2 1 178c 1 39 3a2 2 4a 2 4 2t2 1 10t 2 28

6q3 2 15q2 1 12q 2 30 x2 2 x 2 42 2p2 2 15p 2 50

j2 2 23j 1 132 21z2 2 43z 1 20 12m2 1 64m 2 11

49h2 2 36 212z2 1 52z 2 56 9t2 2 9t 2 10

Name Class Date


26



22. Th e radius of a circle is (7x 1 3) cm. Write an expression to represent the area of the circle in simplifi ed form.

23. A rectangle has a length of (x 1 2) in. and a width of (2x 1 3) in. Find an expression that represents the area of the rectangle. Write the expression in simplifi ed form.

Simplify each product using the FOIL method.

24. (x 1 4)(x 1 6) 25. (a 2 5)(2a 2 6) 26. (6d2 1 4)(8d 2 3)

27. (t 2 4)(t 2 9) 28. (n 1 8)(2n 2 7) 29. (f 2 7)(f 1 3)


30. (c 1 4)(c2 2 3c 1 5) 31. (p2 2 2p 1 5)(p 2 7)

32. (4x2 1 2x 1 3)(3x 2 8) 33. (5t2 1 3t 2 11)(6t 2 1)

34. A community center is expanding the size of its rectangular meeting hall. Th e hall is currently 300 ft long and 150 ft wide. Th e center plans to expand both the length and the width of the meeting hall by 3x ft. What polynomial in standard form represents the area of the expanded meeting hall?

35. Open-Ended Write a cubic monomial and a fourth-degree trinomial. Th en fi nd their product and write it in standard form.

49πx2 1 42πx 1 9π cm2

2x2 1 7x 1 6 in.2

x2 1 10x 1 24 2a2 2 16a 1 30 48d3 2 18d2 1 32d 2 12

t2 2 13t 1 36 2n2 1 9n 2 56 f 2 2 4f 2 21

c3 1 c2 2 7c 1 20 p3 2 9p2 1 19p 2 35

12x3 2 26x2 2 7x 2 24 30t3 1 13t2 2 69t 1 11

9x2 1 1350x 1 45,000 ft2

Answers may vary. Sample: 2x3 and x4 1 2x 1 3; 2x7 1 4x4 1 6x3


27

Name Class Date

Multiple Choice


1. What is the simplifi ed form of (x 2 2)(2x 1 3)? Use the Distributive Property.

A. 2x2 2 x 2 6 B. 2x2 2 6 C. 2x2 2 7x 2 6 D. 2x2 1 x 2 6

2. What is the simplifi ed form of (3x 1 2)(4x 2 3)? Use a table.

F. 12x2 1 18x 1 6 G. 12x2 1 x 2 6 H. 12x2 1 18x 2 6 I. 12x2 2 x 2 6

3. What is the simplifi ed form of (4p 2 2)(p 2 4)?

A. 4p2 1 6p 2 16 B. 4p2 2 18p 1 8 C. 4p2 2 14p 2 6 D. 4p2 2 6p 1 16

4. Th e radius of a cylinder is 3x 2 2 cm. Th e height of the cylinder is x 1 3 cm. What is the surface area of the cylinder?

F. 2p(3x2 1 10x 2 8)

G. 2p(12x2 1 7x 2 2)

H. 2p(12x2 2 2x 1 13)

I. 2p(12x2 2 5x 2 2)

5. What is the simplifi ed form of (2x2 1 4x 2 3)(3x 1 1)?

A. 6x3 1 10x2 2 5x 1 3

B. 6x3 1 14x2 1 5x 2 3

C. 6x3 1 14x2 2 5x 2 3

D. 6x3 2 10x2 2 5x 2 3

Short Response

6. A soup can that is a cylinder has a radius of 2x 2 1 and a height of 3x. What is the surface area of the soup can? Show your work.

8-3 Standardized Test PrepMultiplying Binomials

A

I

B

I

C

[2] Correct polynomial written with all work shown[1] Polynomial written with minor calculation error or inadequate work shown[0] No correct work shown

20πx2 2 14πx 1 2π


28

Name Class Date

You can fi nd the volume of irregular fi gures by dividing the fi gure into smaller rectangular prisms, fi nding the volume of each separate fi gure, and then adding them together. Th e fi gure to the right can be divided into two rectangular prisms.

V1 5 (x 1 1)(x 1 1)(x 2 1)

5 (x2 1 2x 1 1)(x 2 1)

5 x3 1 x2 2 x 2 1

Subtract to fi nd the length of Prism 2.

(2x 1 3) 2 (x 1 1) 5 x 1 2

V2 5 (x 1 2)(x 2 1)(2x 2 2)

5 (x2 1 x 2 2)(2x 2 2)

5 2x3 2 6x 1 4

VTotal 5 (x3 1 x2 2 x 2 1) 1 (2x3 2 6x 1 4)

5 3x3 1 x2 2 7x 1 3

You can also fi nd the volume of an irregular fi gure by fi nding the volume of the

whole fi gure, as if no pieces were cut away. Next, fi nd the volume of the cut away

piece, and then subtract that volume from the whole. Prism 2 is x 2 3 taller than

Prism 1.

VWhole 5 (x 2 1)(2x 1 3)(2x 2 2) 5 (2x2 1 x 2 3)(2x 2 2) 5 4x3 2 2x2 2 8x 1 6

VPiece 5 (x 2 1)(x 1 1)(x 2 3) 5 (x2 2 1)(x 2 3) 5 x3 2 3x2 2 x 1 3

VTotal 5 (4x3 2 2x2 2 8x 1 6) 2 (x3 2 3x2 2 x 1 3) 5 3x3 1 x2 2 7x 1 3

What is the volume of each fi gure? Write your answer as a polynomial in standard form.

1. 2.

8-3 EnrichmentMultiplying Binomials

2x 2

2x 3

x 1

x 1

x 1

PRISM 1PRISM 2

?

2x 2

2x 3

x 1

x 1

x 1

3x 2

5x 2

x 2

x 2x 3

x 3x 1

x 4

4x 1

3x 5

2x 7

x 1

10x3 1 54x2 1 54x 2 8 10x3 1 25x2 2 14x 2 48


29

Name Class Date

You can multiply binomials by using the FOIL method. FOIL stands for First, Outer, Inner, and Last.

Problem

What is the simplifi ed form of (4x 1 3)(2x 1 6)?

Use the FOIL method to simplify the binomial.

Solve 4x ? 2x 5 8x2 Multiply the First terms.

4x ? 6 5 24x Multiply the Outer terms.

3 ? 2x 5 6x Multiply the Inner terms.

3 ? 6 5 18 Multiply the Last terms.

8x2 1 24x 1 6x 1 18 Add the products.

8x2 1 30x 1 18 Add the like terms.

Check Substitute any number for x. Try x 5 2. If the two sides of the equation are equal the simplifi cation may be correct.

(4x 1 3)(2x 1 6) 0 8x2 1 30x 1 18

(4 ? 2 1 3)(2 ? 2 1 6) 0 (8 ? 22) 1 (30 ? 2) 1 18

(11)(10) 0 32 1 60 1 18

110 5 110 3

Solution: Th e simplifi ed form of (4x 1 3)(2x 1 6) is 8x2 1 30x 1 18.

Exercises


1. (a 1 6)(a 2 3) 2. (b 2 4)(b 1 5) 3. (c 1 3)(c 1 7)

4. (2d 1 4)(3d 2 2) 5. (4e 2 5)(3e 1 3) 6. (3f 2 2)(2f 2 4)

7. (5g 1 3)(g 2 3) 8. (4h 1 4)(2h 1 5) 9. (3j 2 5)(4j 2 3)

8-3 ReteachingMultiplying Binomials

a2 1 3a 2 18

6d2 1 8d 2 8

5g2 2 12g 2 9

b2 1 b 2 20

12e2 2 3e 2 15

8h2 1 28h 1 20

c2 1 10c 1 21

6f 22 16f 1 8

12j2 2 29j 1 15


30

Name Class Date

To multiply a trinomial by a binomial, use the same steps as you would to multiply a 3-digit number by a 2-digit number. Find the partial products for each term of the binomial and then add the like terms of the partial products.

Problem

What is the simplifi ed form of (2x2 1 3x 2 4)(3x 1 2)?

Solve Start by arranging the polynomials vertically.

Multiply each part of the trinomial by 2.

2x2 1 3x 2 42x2 1 3x 1 24x2 1 6x 2 8

Multiply each part of the trinomial by 3x.

6x3 1 2x2 1 3x 2 46x2 1 4x2 1 3x 1 2

6x3 1 4x2 1 6x 2 8

6x3 1 9x2 2 12x 2 8

2x2 ? 3x 5 6x3

3x ? 3x 5 9x2

24 ? 3x 5 212x

Add the partial products.

6x3 1 4x2 1 6x 2 8

6x3 1 9x2 2 12x 2 8

6x3 1 13x2 2 6x 2 8

Check Substitute any number for x. Try x 5 2. If the two sides of the equation are equal, the simplifi cation may be correct.

(2x2 1 3x 2 4)(3x 1 2) 0 6x3 1 13x2 2 6x 2 8

(8 1 6 2 4)(6 1 2) 0 48 1 52 2 12 2 8

80 5 80 3

Solution: Th e simplifi ed form of (2x2 1 3x 2 4)(3x 1 2) is 6x3 1 13x2 2 6x 2 8.

ExercisesSimplify each product.

10. (w2 1 3w 2 4)(2w 1 3) 11. (x2 2 8x 1 6)(3x 2 4)

12. (2y2 1 4y 2 5)(4y 1 2) 13. (3z2 2 6z 1 4)(4z 1 1)



2x2 ? 2 5 4x2

3x ? 2 5 6x24 ? 2 5 28

2w3 1 9w2 1 w 2 12

8y3 1 20y2 2 12y 2 10

3x3 2 28x2 1 50x 2 24

12z3 2 21z2 1 10z 1 4


31

Name Class Date

8-4 Additional Vocabulary SupportMultiplying Special Cases

Use the list below to complete the diagram.

The square of a binomial is the square of the first term plus twice the product of the two terms plus the square of the last term.

(x 1 3)(x 2 3) 5 x2 2 32 5 x2 2 9

The product of the sum and difference of the same two terms is the difference of their squares.

(a 1 b)2 5 a2 1 2ab 1 b2 (a 2 b)2 5 a2 2 2ab 1 b2 (a 1 b)(a 2 b) 5 a2 2 b2

The Square of a Binomial

The Product of a Sum and Difference

The square of the binomial is the square of the first term plus twice the product of the two terms plus the square of the last term.

(a 1 b)2 5

a2 1 2ab 1 b2

(a 1 b)(a 2 b) 5

a2 2 b2

(x 1 3)(x 2 3) 5

x2 2 32 5 x2 2 9(a 2 b)2 5

a2 2 2ab 1 b2

The product of the sum and difference of the same two terms is the difference of their squares.


32

Name Class Date

Construction A square deck has a side length of x 1 5. You are expanding the deck so that each side is four times as long as the side length of the original deck. What is the area of the new deck? Write your answer in standard form.

Understanding the Problem

1. What is the shape of the deck?

2. How long is each side of the deck?

3. Th e new deck has sides that are times longer than the original sides.

4. What is the problem asking you to fi nd?

Planning the Solution

5. Write an expression for the new side length of the deck.

6. Write an expression for the area of the new deck.

Getting an Answer

7. What is the standard form of the expression for the area of the new deck?

8-4 Think About a PlanMultiplying Special Cases

square

4

area of new deck

4(x 1 5), or 4x 1 20

(4x 1 20)2

16x2 1 160x 1 400

x 1 5

Name Class Date


33

Simplify each expression.

1. (x 1 7)2 2. (w 1 9)2 3. (h 1 3)2

4. (2s 1 4)2 5. (3s 1 1)2 6. (5s 1 2)2

7. (a 2 5)2 8. (k 2 10)2 9. (n 2 4)2

10. (3m 2 4)2 11. (6m 2 2)2 12. (4m 2 2)2

Th e fi gures below are squares. Find an expression for the area of each shaded region. Write your answers in standard form.

13. 14.

15. 16.

17. A square brown tarp has a square green patch green in the corner. Th e side length of the tarp is (x 1 8) and the side length of the patch is x. What is the area of the brown part of the tarp?

18. A square red placemat has a gold square in the center. Th e side length of the gold square is (x 2 2) inches and the width of the red region is 4 inches. What is the area of the red part of the placemat?

8-4 Practice Form G

Multiplying Special Cases

x 1 2

x 2 1

x 2 1x 1 2

x 1 5

x 1 1

x 1 1x 1 5

x 1 6

x

xx 1 6

x 1 7

x 2 2x 2 2 x 1 7

x2 1 14x 1 49

4s2 1 16s 1 16

a2 2 10a 1 25

9m2 2 24m 1 16

6x 1 3

8x 1 24

16x 1 64

2x2 1 4x 1 12 square inches

12x 1 36

18x 1 45

w2 1 18w 1 81

9s2 1 6s 1 1

k2 2 20k 1 100

36m2 2 24m 1 4

h2 1 6h 1 9

25s2 1 20s 1 4

n2 2 8n 1 16

16m2 2 16m 1 4

Name Class Date


34

Mental Math Simplify each product.

19. 482 20. 312 21. 292

22. 522 23. 632 24. 412

25. 892 26. 1992 27. 3022


28. (v 1 7)(v 2 7) 29. (b 1 2)(b 2 2) 30. (z 2 9)(z 1 9)

31. (x 1 12)(x 2 12) 32. (8 1 y)(8 2 y) 33. (t 2 15)(t 1 15)

34. (m 1 1)(m 2 1) 35. (a 1 4)(a 2 4) 36. (5 1 g)(5 2 g)

37. (p 1 20)(p 2 20) 38. (f 2 18)(f 1 18) 39. (2c 1 3)(2c 2 3)


40. 61 ? 59 41. 27 ? 33 42. 202 ? 198

43. 74 ? 66 44. 597 ? 603 45. 85 ? 75


46. (m 1 4n)2 47. (3a 1 b)2 48. (6s 2 t)2

49. (s 1 7t2)2 50. (p5 2 8q3)2 51. (e4 1 f 2)2

52. (r2 1 5s)(r2 2 5s) 53. (6p2 1 2q)(6p2 2 2q) 54. (3w4 2 z3)(3w4 1 z3)

55. Error Analysis Describe and correct the error made in simplifying the product.

56. Th e formula V 5 43pr3 gives the volume of a sphere

with radius r. Find the volume of a sphere with radius x 1 9. Write your answer in standard form.



(2x + 7)(2x – 7) = 4x2 – 28x – 49

2304

2704

7921

v2 2 49

x2 2 144

m2 2 1

p2 2 400

3599

4884

m2 1 8mn 1 16n2

s2 1 14st2 1 49t4 e8 1 2e4f 21 f 4

b2 2 4

64 2 y2

a2 2 16

f 22 324

891

359,991

9a2 1 6ab 1 b2

p10 2 16p5q3 1 64q6

36p4 2 4q2

z2 2 81

t2 2 225

25 2 g2

4c2 2 9

39,996

6375

36s2 2 12st 1 t2

r4 2 25s2

The x terms should have a sum of zero; 4x2 2 49

V 5 43πx3 1 36πx2 1 324πx 1 972π

9w8 2 z6

39,601 91,204

3969 1681

961 841

Name Class Date


35

8-4 Practice Form K



1. (y 1 1)2 2. (n 1 11)2 3. (t 1 7)2

4. (3m 1 6)2 5. (4x 1 1)2 6. (3n 1 2)2

7. (t 2 3)2 8. (7v 2 3)2 9. (6p 2 5)2

Th e fi gures below are squares. Find an expression for the area of each shaded region. Write your answers in standard form.

10. 11.

12. A fl at, square roof needs a square patch in the corner to seal a leak. Th e side length of the roof is (x 1 12) ft and the side length of the patch is x ft. What is the area of the good part of the roof?

13. A white, square quilt has a purple square in the center. Th e side length of the purple square is (x 2 5) inches and the width of the quilt is 60 inches. What is the area of the white part of the quilt?

x 8

x 8

x

x x 5

x 5

x

x

y2 1 2y 1 1 n2 1 22n 1 121 t2 1 14t 1 49

9m2 1 36m 1 36 16x2 1 8x 1 1 9n2 1 12n 1 4

t2 2 6t 1 9 49v2 2 42v 1 9

16x 1 64 10x 1 25

(24x 1 144) ft

(2x2 1 10x 1 3575) in.2

36p2 2 60p 1 25

Name Class Date


36




14. 522 15. 182 16. 1192

17. 4952 18. 722 19. 1512


20. (x 1 1)(x 2 1) 21. (m 1 5)(m 2 5) 22. (a 2 4)(a 1 4)

23. (s 2 13)(s 1 13) 24. (2z 2 3)(2z 1 3) 25. (4d 1 6)(4d 2 6)


26. 99 ? 101 27. 48 ? 52 28. 178 ? 182


29. (s 1 3t)2 30. (2x 1 y)2 31. (4a 2 b)2

32. (m2 1 3n)(m2 2 3n) 33. (9f 21 4g)(9f 22 4g) 34. (6m4 2 n3)(6m4 1 n3)

35. Th e formula V 5 pr2h gives the volume of a cylinder with radius r and height h. Find the volume of a cylinder with radius (x 1 4) cm and height 5 cm. Write your answer in standard form.

2704 324 14,161

245,025 5184 22,801

x2 2 1 m2 2 25 a2 2 16

s2 2 169 4z2 2 9 16d2 2 36

9999 2496 32,396

s2 1 6st 1 9t2 4x2 1 4xy 1 y2 16a2 2 8ab 1 b2

m4 2 9n2 81f 4 2 16g2 36m8 2 n6

(5πx2 1 40πx 1 80π) cm3


37

Name Class Date

Gridded Response

Solve each exercise and enter your answer on the grid provided.

1. What is coeffi cient of the x-term in the simplifi ed form of (2x 1 4)2?

2. What is 272? Use mental math.

3. What is constant in the simplifi ed form of (x 2 6)2?

4. What is the product of 38 and 42? Use mental math.

`5. How much greater is the product of 73 and 67 than the product of 74 and 66?

8-4 Standardized Test PrepMultiplying Special Cases

16

729

36

1596

7

1.

9876543

10

61

987654

210

9876543210

987

543210

987654321

9876543210

2

3

6

0

2

2.

9876543

10

927

987654

210

9876543210

987

543210

987654321

9876543210

2

3

6

0

2

3.

9876543

10

63

987654

210

9876543210

987

543210

987654321

9876543210

2

3

6

0

2

4.

9876543

10

951 6

987654

210

9876543210

987

543210

987654321

9876543210

2

3

6

0

2

5.

9876543

10

7

987654

210

9876543210

987

543210

987654321

9876543210

2

3

6

0

2


38

Name Class Date

Find the volume of each cube.

1. 2.

3. Find the volume of the rectangular prism.

4. How much greater is the volume of Cube B than the volume of Cube A?

8-4 EnrichmentMultiplying Special Cases

x 5

x 1

x 4

x 4

x 6

Cube A

x2 3

x 9

Cube B

x3 1 15x2 1 75x 1 125 x6 2 9x4 1 27x2 2 27

x3 2 7x2 1 8x 1 16

9x2 1 135x 1 513


39

Name Class Date

A binomial is squared when it is multiplied by itself. Th e square of a binomial is the square of the fi rst term plus the twice the product of the two terms plus the

square of the last term. Th is can be expressed as (a 1 b)2 5 a2 1 2ab 1 b2.

Problem

What is the simplifi ed form of (x 1 5)2?

Use the rules for squaring a binomial.

Solve x ? x 5 x2 Square the fi rst term.

2(5 ? x) 5 10x Multiply the product of the two terms by 2.

5 ? 5 5 25 Square the last term.

So, (x 1 5)2 5 x2 1 10x 1 25.

Check (x 1 5)2 5 (x 1 5)(x 1 5) Rewrite the binomials.

x ? x 5 x2 Multiply the First addends.

x ? 5 5 5x Multiply the Outer addends.

5 ? x 5 5x Multiply the Inner addends.

5 ? 5 5 25 Multiply the Last addends.

x2 1 5x 1 5x 1 25 Add the products.

x2 1 10x 1 25 Combine the like terms.

Solution: Th e simplifi ed form of (x 1 5)2 is x2 1 10x 1 25.

Exercises


1. (a 1 7)2 2. (b 2 4)2 3. (2c 1 3)2 4. (3d 2 5)2

5. (4e 1 1)2 6. (2f 2 6)2 7. (g 2 10)2 8. (5h 1 8)2

9. (3j 2 3)2 10. (2k 1 4)2 11. (4m 2 2)2 12. (3n 1 6)2

8-4 ReteachingMultiplying Special Cases

a2 1 14a 1 49

16e2 1 8e 1 1

9j2 2 18j 1 9

b2 2 8b 1 16

4f 2 2 24f 1 36

4k2 1 16k 1 16

4c2 1 12c 1 9

g2 2 20g 1 100

16m2 2 16m 1 4

9d2 2 30d 1 25

25h2 1 80h 1 64

9n2 1 36n 1 36


40

Name Class Date

Th e product of the sum and the diff erence of the same two terms produces a pattern

that can be expanded algebraically as (a 1 b)(a 2 b) 5 a2 2 ab 1 ab 2 b2. Th e

sum of the two ab- terms is 0. Th erefore, (a 1 b)(a 2 b) 5 a2 2 b2. Th e product is the square of the fi rst term minus the square of the last term.

Problem

What is the simplifi ed form of (2x 2 3)(2x 1 3)?

Use the rules for fi nding the product of the sum and the diff erence of the same two terms.

Solve 2x ? 2x 5 4x2 Square the fi rst term.

3 ? 3 5 9 Square the last term.

Remember, the product is the diff erence of the two squares.

Th e product is 4x2 2 9.

Check Multiply the binomials using the FOIL Method.

2x ? 2x 5 4x2 Multiply the First addends.

2x ? 3 5 6x Multiply the Outer addends.

23 ? 2x 5 26x Multiply the Inner addends.

23 ? 3 5 29 Multiply the Last addends.

4x2 1 6x 2 6x 2 9 Add the products.

4x2 2 9 Combine the like terms.

Solution: Th e simplifi ed form of (2x 2 3)(2x 1 3) is 4x2 2 9.

Exercises


13. (p 2 4)(p 1 4) 14. (q 1 5)(q 2 5) 15. (3r 1 2)(3r 2 2)

16. (4s 2 6)(4s 1 6) 17. (2t 2 1)(2t 1 1) 18. (5u 2 3)(5u 1 3)

19. (6v 2 4)(6v 1 4) 20. (3w 2 8)(3w 1 8) 21. (7x 2 9)(7x 1 9)



p2 2 16

16s2 2 36

36v2 2 16

q2 2 25

4t2 2 1

9w2 2 64

9r2 2 4

25u2 2 9

49x2 2 81


41

Name Class Date

8-5 Additional Vocabulary Support

Factoring x2 1 bx 1 c

For Exercises 1–5, draw a line from each term in Column A to its definition in Column B. The first one is done for you.

(x 1 4)(x 1 8) 5 x2 1 (8 1 4)x 1 4 ? 8 5 x2 1 12x 1 32

Column A Column B

1. 1 coefficient of trinomial’s x2 term

2. 12 binomial

3. 32 coefficient of trinomial’s x term

4. x 1 4 product of (x 1 4) and (x 1 8)

5. x2 1 12x 1 32 trinomial’s constant term

For Exercises 6–9, match the expression in Column A with its definition in Column B.

n2 2 9n 2 36 5 (n 2 12)(n 1 3)

Column A Column B

6. (n 2 12)(n 1 3) factors of 236

7. n2 2 9n 2 36 sum of 212 and 3

8. 212 and 3 trinomial

9. 29 factored form of n2 2 9n 2 36


42

Name Class Date

Recreation A rectangular skateboard park has an area of x2 1 15x 1 54. What are possible dimensions of the park? Use factoring.

Know

1. Th e area of the skateboard park is .

2. Th e dimensions of a rectangle are its and .

3. Th e of the area polynomial are possible dimensions of the skateboard park.

Need

4 To solve the problem I need to fi nd

Plan

5. Complete the table. List the pairs of factors of u.

Identify the pair that has a sum of u.

6. Write the factored polynomial.

7. What are possible dimensions of the skateboard park?

8. Justify your answer.

8-5 Think About a PlanFactoring x2 1 bx 1 c

Factorsof 54

Sum ofFactors

x2 1 15x 1 54

length

factors

the factors of x2 1 15x 1 54

54

15

(x 1 6)(x 1 9)

length: x 1 9; width x 1 6

If the length is x 1 9 and the width is x 1 6, then the area is

(x 1 9)(x 1 6) 5 x2 1 15x 1 54.

width

15

Factors Sum552921

1 and 542 and 273 and 186 and 9

Name Class Date


43

Complete.

1. k2 1 11k 1 30 5 (k 1 5)(k 1u) 2. x2 1 6x 1 9 5 (x 1 3)(x 1u)

3. t2 1 7t 1 10 5 (t 1 2)(t 1u) 4. n2 1 9n 1 14 5 (n 1 7)(n 1u)

5. w2 1 13w 1 36 5 (w 1 4)(w 1u) 6. y2 1 18y 1 65 5 (y 1 13)(y 1u)

7. s2 2 12s 1 32 5 (s 2 8)(s 2u) 8. g2 2 14g 1 45 5 (g 2 9)(g 2u)

9. v2 2 17v 1 60 5 (v 2 12)(v 2u) 10. q2 2 13q 1 42 5 (q 2 6)(q 2u)

11. d2 2 9d 1 8 5 (d 2 8)(d 2u) 12. r2 2 9r 1 20 5 (r 2 5)(r 2u)

Factor each expression. Check your answer.

13. y2 1 5y 1 6 14. t2 1 9t 1 18 15. x2 1 16x 1 63

16. n2 2 12n 1 35 17. r2 2 12r 1 27 18. q2 2 12q 1 20

19. w2 1 19w 1 60 20. b2 2 11b 1 24 21. z2 2 13z 1 12

Complete.

22. q2 1 q 2 56 5 (q 2 7)(q 1u) 23. z2 2 3z 2 18 5 (z 2 6)(z 1u)

24. n2 2 6n 2 40 5 (n 1 4)(n 2u) 25. y2 1 3y 2 4 5 (y 1 4)(y 2u)

26. v2 2 5v 2 36 5 (v 2 9)(v 1u) 27. d2 1 2d 2 15 5 (d 2 3)(d 1u)

28. m2 2 5m 2 14 5 (m 1 2)(m 2u) 29. p2 2 6p 2 16 5 (p 2 8)(p 1u)

8-5 Practice Form G


6

5

3

2

5

5

4

7

1

5

4

9

(y 1 3)(y 1 2)

(n 2 7)(n 2 5)

(w 1 15)(w 1 4)

(t 1 6)(t 1 3)

(r 2 9)(r 2 3)

(b 2 8)(b 2 3)

(x 1 9)(x 1 7)

(q 2 10)(q 2 2)

(z 2 12)(z 2 1)

8

10

4

7 2

5

1

3

Name Class Date


44


30. r2 1 3r 2 10 31. w2 1 2w 2 8 32. z2 1 3z 2 40

33. d2 2 4d 2 12 34. p2 2 7p 2 8 35. s2 2 5s 2 24

36. x2 1 5x 2 6 37. v2 1 3v 2 28 38. n2 1 2n 2 63

39. t2 2 2t 2 24 40 a2 2 7a 2 18 41. c2 2 c 2 30

42. Th e area of a rectangular door is given by the trinomial x2 2 14x 1 45. Th e door’s width is (x 2 9). What is the door’s length?

43. Th e area of a rectangular painting is given by the trinomial a2 2 6a 2 16. Th e painting’s length is (a 1 2). What is the painting’s width?

Write the correct factored form for each expression.

44. k2 1 4kn 2 96n2 45. g2 2 13gh 1 42h2 46. m2 2 4mn 2 32n2

47. x2 1 5xy 2 14y2 48. s2 1 17st 1 72t2 49. h2 1 3hj 2 88j2

50. Error Analysis Describe and correct the error made in factoring the trinomial.

51. A rectangular pool cover has an area of p2 1 9p 2 36. What are possible dimensions of the pool cover? Use factoring.



x2 + 2x – 80= (x + 8)(x – 10)

(r 1 5)(r 2 2)

(d 2 6)(d 1 2)

(x 1 6)(x 2 1)

(t 2 6)(t 1 4)

(w 1 4)(w 2 2)

(p 2 8)(p 1 1)

(v 1 7)(v 2 4)

(a 2 9)(a 1 2)

x 2 5

a 2 8

(k 1 12n)(k 2 8n)

(x 1 7y)(x 2 2y)

(g 2 6h)(g 2 7h)

(s 1 8t)(s 1 9t)

The operation signs are wrong. The answer should be (x 2 8)(x 1 10).

(p 1 12) and (p 2 3)

(m 2 8n)(m 1 4n)

(h 1 11j)(h 2 8j)

(z 1 8)(z 2 5)

(s 2 8)(s 1 3)

(n 1 9)(n 2 7)

(c 2 6)(c 1 5)

Name Class Date


45

8-5 Practice Form K


Complete.

1. n2 1 9n 1 18 5 (n 1 3)(n 1u) 2. t2 1 9t 1 14 5 (t 1 2)(t 1u)

3. d2 1 11d 1 30 5 (d 1 5)(d 1u) 4. v2 1 2v 1 1 5 (v 1 1)(v 1u)

5. m2 2 8m 1 15 5 (m 2 5)(m 2u) 6. a2 2 13a 1 22 5 (a 2 2)(a 2u)

7. z2 2 17z 1 72 5 (z 2 8)(z 2u) 8. w2 2 7w 1 12 5 (w 2 3)(w 2u)


9. g2 1 6g 1 8 10. y2 1 10y 1 24 11. r2 1 12r 1 35

12. k2 1 9k 1 8 13. x2 2 16x 1 60 14. h2 2 19h 1 78

Complete.

15. g2 1 5g 2 24 5 (g 2 3)(g 1u) 16. b2 2 6b 2 7 5 (b 2 7)(b 1u)

17. y2 1 4y 2 45 5 (y 1 9)(y 2u) 18. k2 1 4k 2 12 5 (k 1 6)(k 2u)

19. p2 2 7p 2 60 5 (p 1 5)(p 2u) 20. n2 2 6n 2 40 5 (n 2 10)(n 1u)

8 1

5 2

12 4

(g 1 2)(g 1 4) (y 1 6)(y 1 4) (r 1 5)(r 1 7)

(k 1 1)(k 1 8) (x 2 10)(x 2 6) (h 2 13)(h 2 6)

7

1

3 11

9 4

6

6

Name Class Date


46




21. x2 2 4x 2 5 22. t2 1 t 2 20 23. z2 2 z 2 72

24. m2 2 6m 2 27 25. a2 1 4a 2 21 26. v2 2 4v 2 12

27. c2 2 7c 2 44 28. r2 1 6r 2 16 29. f 21 f 2 6

30. j2 2 6j 2 55 31. y2 1 3y 2 54 32. n2 2 10n 2 11

33. Th e area of a rectangular window is given by the trinomial x2 2 14x 1 48. Th e window’s length is (x 2 8). What is the window’s width?

34. Th e area of a rectangular area rug is given by the trinomial f 2 2 4f 2 77. Th e length of the rug is (f 1 7). What is the width of the rug?

35. Reasoning Write possible expressions for the length and the width of a

rectangle with area x2 1 13x 1 42.

36. A rectangular tabletop has an area of t2 1 2t 2 99. What are possible dimensions of the tabletop? Use factoring.

(x 1 1)(x 2 5) (t 1 5)(t 2 4) (z 1 8)(z 2 9)

(m 1 3)(m 2 9) (a 1 7)(a 2 3) (v 1 2)(v 2 6)

(c 1 4)(c 2 11) (r 1 8)(r 2 2) (f 1 3)(f 2 2)

(j 1 5)(j 2 11) (y 1 9)(y 2 6) (n 1 1)(n 2 11)

(x 2 6)

(f 2 11)

(x 1 6); (x 1 7)

t 1 11 and t 2 9


47

Name Class Date

Multiple Choice


1. Which number makes this equation true?

v2 1 10v 1 16 5 (v 1 8)(v 1 u )

A. 2 B. 4 C. 6 D. 8

2. What is the factored form of x2 1 6x 1 8?

F. (x 1 5)(x 1 3) G. (x 1 4)(x 1 2) H. (x 1 7)(x 1 1) I. (x 1 3)(x 1 3)

3. What is the factored form of x2 2 7x 1 12? A. (x 2 5)(x 2 3) B. (x 2 6)(x 2 1) C. (x 2 2)(x 2 5) D. (x 2 4)(x 2 3)

4. Which number makes this equation true?

q2 1 3q 2 18 5 (q 1 6)(q 2 u )

F. 1 G. 2 H. 3 I. 12

5. What is the factored form of x2 1 3x 2 10? A. (x 1 5)(x 2 2) C. (x 2 2)(x 2 5)

B. (x 2 5)(x 1 2) D. (x 1 5)(x 1 2)

6. Th e area of a garden is given by the trinomial g2 2 2g 2 24. Th e garden’s length is g 1 4. What is the garden’s width?

F. g 2 2 G. g 2 6 H. g 2 8 I. g 1 2

7. What is the factored form of x2 1 3xy 2 28y2?

A. (x 1 14y)(x 2 2y) B. (x 1 2y)(x 2 14y) C. (x 1 4y)(x 2 7y) D. (x 2 4y)(x 1 7y)

Short Response

8. Th e area of a rectangular backyard is given by the trinomial b2 1 5b 2 24. What are possible dimensions of the backyard? Show why your answer is correct.

8-5 Standardized Test PrepFactoring x2 1 bx 1 c

A

G

D

H

A

G

D

[2] Both length and width calculated correctly with all work shown[1] Correct answer with minor calculation error or inadequate work shown[0] No correct work shown

length: (b 1 8); width: (b 2 3); (b 1 8)(b 2 3) 5 b2 1 5b 2 24


48

Name Class Date

To factor a trinomial of the form x2 1 bx 1 c as the product of binomials, you must fi nd factor pairs that have a sum of b and a product of c. Examine what happens to c as you increase b when c is greater than zero.

If b = 2, the factor pair is 1, 1 and the product is 1. (x 1 1)(x 1 1) 5 x2 1 2x 1 1

If b = 3, the factor pair is 1, 2 and the product is 2. (x 1 1)(x 1 2) 5 x2 1 3x 1 2If b = 4, the factor pairs are 1, 3 and 2, 2. Th e products are 3 and 4.

(x 1 1)(x 1 3) 5 x2 1 4x 1 3 (x 1 2)(x 1 2) 5 x2 1 4x 1 4If b = 5, the factor pairs are 1, 4 and 2, 3. Th e products are 4 and 6.

(x 1 1)(x 1 4) 5 x2 1 5x 1 4 (x 1 2)(x 1 3) 5 x2 1 5x 1 6

1. What are the factor pairs and products (values of c) for the following values

of b, for x2 1 bx 1 c if c . 0?b 5 6 b 5 7 b 5 8 b 5 9 b 5 10

2. What pattern do you see in the number of factor pairs (and thus values for c) as you increase the value of b?

3. Describe at least one pattern you see in the value of c in terms of b.

Now examine what happens to the value of b when the value of c changes, when c . 0.

If c = 1, the factor pair is 1, 1, and the sum is 2. (x 1 1)(x 1 1) 5 x2 1 2x 1 1

If c = 2, the factor pair is 1, 2 and the sum is 3. (x 1 1)(x 1 2) 5 x2 1 3x 1 2

If c = 3, the factor pair is 1, 3 and the sum is 4. (x 1 1)(x 1 3) 5 x2 1 4x 1 3If c = 4, the factors pairs are 1, 4 and 2, 2. Th e sums are 5 and 4.

(x 1 1)(x 1 4) 5 x2 1 5x 1 4 (x 1 2)(x 1 2) 5 x2 1 4x 1 4

4. What are the factor pairs and sums (values of b) for the following values of c,

for x2 1 bx 1 c if c . 0?c 5 5 c 5 6 c 5 7 c 5 8

5. Describe at least one pattern you see in the value of b in terms of c. Explain why this might be.

8-5 EnrichmentFactoring x2 1 bx 1 c

b 5 6; pairs: 1,5; 2, 4; 3, 3; products: 5, 8, 9 b 5 7; pairs: 1, 6; 2, 5; 3, 4; products: 6, 10, 12b 5 8; pairs: 1,7; 2, 6; 3, 5; 4, 4; products: 7, 12, 15, 16 b 5 9; pairs: 1, 8; 2, 7; 3, 6; 4, 5; products: 8, 14, 18, 20 b 5 10; pairs: 1, 9; 2, 8; 3, 7; 4, 6; 5, 5; products: 9, 16, 21, 24, 25

The number of factor pairs increases as b increases

Answers may vary. Sample: If b is even then b2 is the number of c values. If b is

odd, then (b 2 1)2 is the number of c values.

c 5 5; pairs: 1, 5; sums: 6 c 5 6; pairs: 1, 6; 2, 3; sums: 7, 5c 5 7; pairs: 1, 7; sums: 8 c 5 8; pairs: 1, 8; 2, 4; sums: 9, 6

Answers may vary. Sample: Prime numbers have only one pair of factors because the factors of a prime number are the number and 1.


49

Name Class Date

8-5 ReteachingFactoring x2 1 bx 1 c

If a trinomial of the form x2 1 bx 1 c can be written as the product of two binomials, then:

• Th e coeffi cient of the x-term in the trinomial is the sum of the constants in the binomials.

• Th e trinomial’s constant term is the product of the constants in the binomials.

Problem

What is the factored form of x2 1 12x 1 32?

To write the factored form, you are looking for two factors of 32 that have a sum of 12.

Solve Make a table showing the factors of 32.

x2 1 12x 1 32 5 (x 1 4)(x 1 8)

Check (x 1 4)(x 1 8)

x2 1 8x 1 4x 1 32 Use FOIL Method.

x2 1 12x 1 32 Combine the like terms.

Solution: Th e factored form of x2 1 12x 1 32 is (x 1 4)(x 1 8).

Exercises

Factor each expression.

1. x2 1 9x 1 20 2. y2 1 12y 1 35 3. z2 1 8z 1 15

4. a2 1 11a 1 28 5. b2 1 10b 1 16 6. c2 1 12c 1 27

7. d2 1 6d 1 5 8. e2 1 15e 1 54 9. f 2 1 11f 1 24

Factors of 32 Sum of Factors

33

18

12

1 and 32

2 and 16

4 and 8

(x 1 5)(x 1 4) (y 1 7)(y 1 5) (z 1 5)(z 1 3)

(c 1 9)(c 1 3)

(f 1 8)(f 1 3)

(b 1 8)(b 1 2)

(e 1 9)(e 1 6)

(a 1 4)(a 1 7)

(d 1 5)(d 1 1)


50

Name Class Date



Some factorable trinomials in the form of x2 1 bx 1 c will have negative coeffi cients. Th e rules for factoring are the same as when the x-term and the constant are positive.

• Th e coeffi cient of the x-term of the trinomial is the sum of the constants in the binomials.

• Th e trinomial’s constant term is the product of the constants in the binomials.

However, one or both constants in the binomial factors will be negative.

Problem

What is the factored form of x2 2 3x 2 40?

To write the factored form, you are looking for two factors of 240 that have a sum of 23. Th e negative constant will have a greater absolute value than the positive constant.

Solve Make a table showing the factors of 240.

x2 2 3x 2 40 5 (x 2 8)(x 1 5)

Check (x 2 8)(x 1 5)

x2 1 5x 2 8x 2 40 Use FOIL Method.

x2 1 (23x) 2 40 Combine the like terms.

Solution: Th e factored form of x2 2 3x 2 40 is (x 2 8)(x 1 5).

Exercises


10. s2 1 2s 2 35 11. t2 2 4t 2 32 12. u2 1 6u 2 27

13. v2 2 2v 1 48 14. w2 2 8w 2 9 15. x2 1 3x 2 18

3

Factors of ]40 Sum of Factors39186

1 and 402 and 204 and 105 and 8

(s 1 7)(s 2 5)

(v 2 8)(v 1 6)

(t 2 8)(t 1 4)

(w 2 9)(w 1 1)

(u 1 9)(u 2 3)

(x 1 6)(x 2 3)


51

Name Class Date

8-6 Additional Vocabulary Support

Factoring ax2 1 bx 1 c

A student is trying to factor 3x2 1 13x 1 4. She wrote these steps to solve the problem on note cards, but they got mixed up.

Use the note cards to complete the steps below.

1. First,

2. Second,

3. Third,

4. Then,

5. Finally,

Find factors of ac that have sum b.

Since ac 5 12 and b 5 13, find positive factors of 12 that have sum 13.(3x 1 1)(x 1 4)

To factor the trinomial, use the factors you found to rewrite bx as 1x 1 12x.

Make a table.Factors of 12

2, 6 3, 4 1, 12

Sum of factors

8 7 13 3

find factors of ac that have sum b.

Factors of 12 2, 6 3, 4 1, 12Sum of factors 8 7 13 3

since ac 5 12 and b 5 13, find positive factors of 12 that have sum 13.

(3x 1 1) (x 1 4)

to factor the trinomial, use the factors you found to rewrite bx.


52

Name Class Date

Carpentry Th e top of a rectangular table has an area of 18x2 1 69x 1 60. Th e width of the table is 3x 1 4. What is the length of the table?

Know

1. Th e area of the table top is .

2. Th e width of the table top is .

3. Some quadratic trinomials can be written as the product of two .

4. One of the factors of the polynomial 18x2 1 69x 1 60 is .

Need

5. To solve the problem I need to fi nd

.

Plan

6. Find the missing factor.

What can you multiply by 3x to get 18x2? 3x ?u 5 18x2

What can you multiply by 4 to get 60? 4 ?u 5 60

7. What is the factored form of 18x2 1 69x 1 60?

8. What is the length of the table? Check your answer.

8-6 Think About a PlanFactoring ax2 1 bx 1 c

18x2 1 69x 1 60

3x 1 4

binomials

3x 1 4

the other factor

6x

15

(3x 1 4)(6x 1 15)

length: (6x 1 15)

Check: (3x 1 4)(6x 1 15) 5 18x2 1 69x 1 60

Name Class Date


53


1. 2w2 1 13w 1 15 2. 3d2 1 20d 1 12 3. 4n2 1 62n 2 32

4. 3p2 2 7p 2 40 5. 6r2 2 10r 2 24 6. 5z2 2 17z 1 14

7. 14k2 2 67k 1 63 8. 2m2 2 m 2 15 9. 3x2 1 9x 2 84

10. 4y2 1 26y 1 30 11. 5t2 2 24t 2 5 12. 7c2 2 2c 2 9

13. 8k2 2 42k 1 27 14. 6g2 2 2g 2 20 15. 2c2 2 23c 1 11

16. Th e area of a rectangular computer screen is 4x2 1 20x 1 16. Th e width of the screen is 2x 1 8. What is the length of the screen?

17. Th e area of a rectangular granite countertop is 12x2 1 10x 2 12. Th e width of the countertop is 2x 1 3. What is the length of the countertop?

18. Th e area of a rectangular book cover is 4x2 2 6x 2 40. Th e width of the book cover is 2x 2 8. What is the length of the book cover?

19. Th e area of a rectangular parking lot is 21x2 2 44x 1 15. Th e width of the parking lot is 3x 2 5. What is the length of the parking lot?

Factor each expression completely.

20. 6x2 2 10x 2 4 21. 6d2 1 21d 1 15 22. 8n2 1 68n 1 84

23. 20p2 2 115p 2 30 24. 15r2 1 141r 2 90 25. 12z2 2 14z 1 4

26. 20k2 1 110k 1 120 27. 9m2 2 66m 1 21 28. 40x2 2 136x 2 96

29. 42y2 1 28y 2 14 30. 8t2 2 16t 2 90 31. 24c2 1 96c 1 90

8-6 Practice Form G


(2w 1 3)(w 1 5)

(3p 1 8)(p 2 5)

(2k 2 7)(7k 2 9)

(5t 1 1)(t 2 5)

(4k 2 3)(2k 2 9)

(3d 1 2)(d 1 6)

2(3r 1 4)(r 2 3)

(2m 1 5)(m 2 3)

2(2y 1 3)(y 1 5)

2(3g 1 5)(g 2 2)

2x 1 2

6x 2 4

2x 1 5

7x 2 3

2(3x 1 1)(x 2 2)

5(4p 1 1)(p 2 6)

10(2k 1 3)(k 1 4)

14(3y 2 1)(y 1 1)

3(2d 1 5)(d 1 1)

3(5r 2 3)(r 1 10)

3(3m 2 1)(m 2 7)

2(2t 1 5)(2t 2 9)

4(2n 1 3)(n 1 7)

2(2z 2 1)(3z 2 2)

8(5x 1 3)(x 2 4)

6(2c 1 5)(2c 1 3)

2(2n 2 1)(n 1 16)

(5z 2 7)(z 2 2)

3(x 1 7)(x 2 4)

(7c 2 9)(c 1 1)

(2c 2 1)(c 2 11)

Name Class Date


54

Open-Ended Find two diff erent values that complete each expression so that the trinomial can be factored into the product of two binomials. Factor your trinomials.

32. 4x2 1ux 1 12 33. 6t2 2ut 2 4 34. 9m2 2um 1 8

35. 8n2 1un 2 10 36. 12v2 2uv 1 15 37. 5w2 2uw 2 24

38. Error Analysis Describe and correct the error made in factoring the expression at the right.

39. A parallelogram has an area of 4x2 1 7x 2 15. Th e base of the parallelogram is x 1 3. What is the height of the parallelogram?

a. Write the formula for the area of a parallelogram.

b. Writing Explain how factoring the trinomial helps you solve the problem.

40. A rectangular window pane has an area of 15x2 2 19x 1 6. Th e width of the window pane is 3x 2 2. What is the length of the window pane?


41. 28y2 1 43y 2 48 42. 16z2 2 54z 1 35 43. 27n2 2 54n 1 15

44. 36p2 1 63p 1 20 45. 28r2 2 20r 2 33 46. 30z2 2 53z 1 12

47. 32x3 1 28x2 1 5x 48. 25p2 1 20pq 2 12q2 49. 72g2h 2 43gh 1 6h



(6x2 + 3x – 9) = 3(2x2 + x – 3) = 3(2x2 - 3x + 2x – 3) = 3(2x2 - 3x + (2x – 3) = 3[ x (2x - 3) + 1 (2x – 3) ]= 3(x + 1) (2x – 3)

Answers may vary. Sample: 19, 16: (4x 1 3)(x 1 4);(4x 1 4)(x 1 3)

Answers may vary. Sample: 11, 211; (8n 2 5)(n 1 2); (n 2 2)(8n 1 5)

Answers may vary. Sample: 23, 25; (6t 1 1)(t 2 4); (3t 1 4)(2t 2 1)

Answers may vary. Sample: 29, 27; (4v 2 3)(3v 2 5); (4v 2 5)(3v 2 3)

Answers may vary. Sample: 73, 27; (9m 2 1)(m 2 8); (3m 2 8)(3m 2 1)

Answers may vary. Sample. 26, 14; (5w 1 4)(w 2 6); (5w 1 6)(w 2 4)

In the second step, the student wrote 21x instead of 1x. x should be written as 3x 2 2x . Answer: 3(2x 1 3)(x 2 1)

A 5 bh

Factor to fi nd h: (x 1 3)(4x 2 5) 5 4x2 1 7x 2 15; h 5 4x 2 5

5x 2 3

(4y 2 3)(7y 1 16)

(3p 1 4)(12p 1 5)

x(4x 1 1)(8x 1 5)

(8z 2 7)(2z 2 5)

(2r 2 3)(14r 1 11)

(5p 2 2q)(5p 1 6q)

3(3n 2 1)(3n 2 5)

(2z 2 3)(15z 2 4)

h(9g 2 2)(8g 2 3)

Name Class Date


55

8-6 Practice Form K



1. 3n2 2 8n 2 3 2. 5a2 2 22a 1 8 3. 2s2 1 13s 1 6

4. 6t2 1 21t 2 12 5. 9b2 2 65b 1 14 6. 5z2 1 11z 1 6

7. 7r2 2 9r 2 10 8. 2m2 1 m 2 21 9. 3g2 1 20g 1 32

10. Th e area of a rectangular driveway is 2x2 1 15x 1 25. Th e width of the driveway is x 1 5. What is the length of the driveway?

11. Th e area of a rectangular fl oor is 8x2 1 6x 2 20. Th e width of the fl oor is 2x 1 4. What is the length of the fl oor?

12. Th e area of a rectangular desktop is 6x2 2 3x 2 3. Th e width of the desktop is 2x 1 1. What is the length of the desktop?


13. 24n2 1 2n 2 12 14. 72q2 2 12q 2 40 15. 30j2 2 27j 2 21

16. 60h2 1 280h 1 45 17. 40a2 1 126a 1 44 18. 45f 2 1 24f 2 189

2(4n 1 3)(3n 2 2) 4(3q 1 2)(6q 2 5) 3(2j 1 1)(5j 2 7)

5(6h 1 1)(2h 1 9) 2(4a 1 11)(5a 1 2) 3(5f 2 9)(3f 1 7)

2x 1 5

4x 2 5

3x 2 3

(3n 1 1)(n 2 3) (5a 2 2)(a 2 4) (2s 1 1)(s 1 6)

3(2t 2 1)(t 1 4) (9b 2 2)(b 2 7) (5z 1 6)(z 1 1)

(7r 1 5)(r 2 2) (2m 1 7)(m 2 3) (3g 1 8)(g 1 4)

Name Class Date


56



Open-Ended Find two diff erent values that complete each expression so that the trinomial can be factored into the product of two binomials. Factor your trinomials.

19. 4n2 1un 2 3 20. 12r2 1u 1 6

21. 24a2 1ua 2 15 22. 18b2 1ub 1 8

23. A parallelogram has an area of 8x2 2 2x 2 45. Th e height of the parallelogram is 4x 1 9.

a. Write the formula for the area of a parallelogram.

b. What is the length of the base of the parallelogram?

c. Writing Explain how you solved the problem.

24. A rectangular athletic fi eld has an area of 40x2 1 190x 2 50. Th e width of the athletic fi eld is 8x 2 2. What is the length of the athletic fi eld?


25. 96d2 2 76d 2 77 26. 48h2 2 86h 1 35

27. 24m2 1 18m 2 15 28. 36c2 1 27c 2 55

2x 2 5

A 5 bh

Sample: You know that the product of 4x 1 9 and another factor is 8x2 2 2x 2 45. 4x times 2x is 8x2 and 9 times 25 is 245. So, 8x2 2 2x 2 45 5 (4x 1 9)(2x 2 5). Then use FOIL to check.

5x 1 25

(12d 1 7)(8d 2 11) (8h 2 5)(6h 2 7)

3(2m 2 1)(4m 1 5) (3c 1 5)(12c 2 11)

Answers may vary. Sample: 24, 11; 4n2 2 4n 2 3 5 (2n 2 3)(2n 1 1);4n2 1 11n 2 3 5 (n 1 3)(4n 2 1)

Answers may vary. Sample: 17, 38; 12r2 1 17r 1 6 5 (3r 1 2)(4r 1 3); 12r2 1 38r 1 6 5 (2r 1 6)(6r 1 1)

Answers may vary. Sample: 218, 37; 24a2 2 18a 2 15 5 (6a 1 3)(4a 2 5); 24a2 1 37a 2 15 5 (8a 1 15)(3a 2 1)

Answers may vary. Sample: 24, 74; 18b2 1 24b 1 8 5 (3b 1 2)(6b 1 4); 18b2 1 74b 1 8 5 (9b 1 1)(2b 1 8)


57

Name Class Date

Multiple Choice



A. (2x 1 4)(2x 1 3) B. (4x 1 5)(x 1 1) C. (2x 1 1)(2x 1 5) D. (4x 1 1)(x 1 5)

2. What is the factored form of 2x2 1 x 2 3 ?

F. (2x 1 3)(x 2 1) G. (2x 1 1)(x 2 3) H. (2x 2 3)(x 1 1) I. (2x 2 1)(x 1 3)

3. Th e area of a rectangular swimming pool is 10x2 2 19x 2 15. Th e length of the pool is 5x 1 3. What is the width of the pool?

A. 2x 2 18 B. 2x 2 5 C. 5x 2 5 D. 5x 2 22


F. 4(2x 2 2)(2x 1 2)

G. 4(4x 2 6)(x 1 2)

H. 4(2x 2 2)(2x 1 3)

I. 4(2x 2 3)(2x 1 1)

5. What is the factored form of 3x2 1 21x 2 24? A. 3(x 1 8)(x 2 1)

B. 3(x 1 6)(x 1 1)

C. 3(x 1 5)(x 2 3)

D. 3(x 1 7)(x 2 3)

Short Response

6. Th e perimeter around a dog’s running space is 20x2 1 28x 1 8. Th e length of the dog’s running space is 10x 1 4. What is the width of the dog’s running space? Show why your answer is correct.

8-6 Standardized Test PrepFactoring ax2 1 bx 1 c

C

F

B

I

A

[2] Correct expression written with all work shown[1] Expression written with minor calculation error or inadequate

work shown[0] No correct work shown

2x 1 2


58

Name Class Date

You can use a function to estimate the volume of an adult body based on the length of one part, such as the length of an index fi nger, x. Start by using x to calculate the volume of an index fi nger. Assume the ratio of the length to height to

width of an average index fi nger is 7 : 1 : 2. Th erefore, the volume is 249 x3.

You can then estimate that approximately 10 index fi ngers make up one hand. Multiply the volume of one index fi nger by 10 to fi nd the volume in one hand: 2049x3. Use this more convenient hand measure to fi gure out how many hands

make up each large body area.

Hand = 1 hand Arm ≈ 12 hands Head ≈ 12 hands Neck ≈ 8 handsTorso ≈ 100 hands Leg ≈ 45 hands Foot ≈ 3 hands

Add up all the parts, making sure to double the hands, arms, and legs: 2 1 24 1 12 1 8 1 100 1 90 1 6 5 242 hands Now multiply the number of hands by the volume in one hand:

242Q2049Rx3 5

484049 x3 5 9838

49 x3

Now that you have a function for the volume of a human body, you can use it to fi nd expressions for other body parts without measuring.

1. Use the function V 5 983849 x3 , the volume of an adult body to write an

expression for the length of the foot in an adult body. where the ratio of the length to height to width of the foot is 6 : 1 : 1.

2. Use the same function to write an expression for the length of an arm in an adult body where the ratio of the length to height to width of the arm is 10 : 1 : 1.

3. Measure the lengths of three people’s index fi ngers, feet, and arms. How do the results compare to your estimates?

8-6 EnrichmentFactoring ax2 1 bx 1 c

6049 x

24049 x

Check students’ work.


59

Name Class Date

You can use your knowledge of prime numbers to help you factor some trinomials as two binomials. A prime number has only 1 and itself as factors. For trinomials of the form

ax2 1 bx 1 c, if a is a prime number then you already know the fi rst term of each binomial: ax and 1x. Th en list the factors that will multiply to produce c. Use guess and check to fi nd the factor pair that will add to b.

Problem

What is the factored form of 7x2 1 31x 1 12?

7x2 1 31x 1 12 5 (7x )(1x ) a is 7, which is prime, so the factors are 7 and 1.

5 (7x )(x ) You don’t need the 1 in front of the variable, so drop it.

7x2 1 31x 1 12 5 (7x 1 )(x 1 ) The trinomial has two plus signs, so the

binomials also have plus signs.

Because c is 12, fi nd factor pairs that multiply to 12: (1 and 12), (2 and 6), (3 and 4).

Try each pair in the expression to see if the INNER and OUTER products add to b, or 31.

(7x 1 1)(x 1 12) 5 7x2 1 x 1 84x 5 7x2 1 85x 1 12 (NO)

(7x 1 2)(x 1 6) 5 7x2 1 2x 1 42x 5 7x2 1 44x 1 12 (NO)

(7x 1 3)(x 1 4) 5 7x2 1 3x 1 28x 5 7x2 1 31x 1 12 (YES)

Th e factored form of 7x2 1 31x 1 12 is (7x 1 3)(x 1 4).

Exercises


1. 3x2 1 14x 1 8 2. 5y2 1 43y 1 24 3. 2z2 1 19z 1 42

4. 11a2 1 39a 1 18 5. 13b2 1 58b 1 24 6. 23c2 1 56c 1 20

7. 7d2 1 d 2 8 8. 3e2 1 20e 2 32 9. 19f 21 10f 2 9

10. 5s2 2 18s 1 16 11. 17t2 2 12t 2 5 12. 29u2 1 48u 2 20

8-6 ReteachingFactoring ax2 1 bx 1 c

(3x 1 2)(x 1 4)

(11a 1 6)(a 1 3)

(7d 1 8)(d 2 1)

(5s 2 8)(s 2 2)

(5y 1 3)(y 1 8)

(13b 1 6)(b 1 4)

(3e 2 4)(e 1 8)

(17t 1 5)(t 2 1)

(2z 1 7)(z 1 6)

(23c 1 10)(c 1 2)

(19f 2 9)(f 1 1)

(29u 2 10)(u 1 2)


60

Name Class Date

If you are given the area and one side of a rectangle, you can fi nd the second side by factoring the trinomial. One binomial is the width and the other binomial is the length.

Problem

Th e area of a rectangular swimming pool is 6x2 1 11x 1 3. Th e width of the pool is 2x 1 3. What is the length of the pool?

You are given the area and length of the pool. Set up an equation with what you are given and solve or length.

6x2 1 11x 1 3 5 (2x 1 3)(uuu ) Area = length × width.

6x2 1 11x 1 3 5 (2x 1 3)(3x uu ) 6x2 5 (2x)(3x), so the fi rst term of the second

binomial is 3x.

6x2 1 11x 1 3 5 (2x 1 3)(3x 1 u ) The trinomial has two plus signs, so the sign for

the second binomial must also be plus.

6x2 1 11x 1 3 5 (2x 1 3)(3x 1 1) The value of c is 3. Since 3 5 3 3 1, the second

term must be 1.

Multiply to check your answer. Use FOIL.

(2x 1 3)(3x 1 1) 5 6x2 1 2x 1 9x 1 3 5 6x2 1 11x 1 3 3

Th e length of the swimming pool is 3x 1 1.

Exercises

13. Th e area of a rectangular cookie sheet is 8x2 1 26x 1 15. Th e width of the cookie sheet is 2x 1 5. What is the length of the cookie sheet?

14. Th e area of a rectangular lobby fl oor in the new offi ce building is

15x2 1 47x 1 28. Th e length of one side of the lobby is 5x 1 4. What is the width?

15. Th e area of a rectangular school banner is 12x2 1 13x 2 90. Th e width of the banner is 3x 1 10. What is the length of the banner?

16. Th e distance a train has traveled is 6x2 2 23x 1 20. Th e train’s average speed is 3x 2 4. How long has the train been traveling?



4x 1 3

3x 1 7

4x 2 9

2x 2 5


61

Name Class Date

8-7 Additional Vocabulary SupportFactoring Special Cases

Complete the vocabulary chart by filling in the missing information.

Word or Word Phrase

Definition Picture or Example

difference of two squares

A binomial in which a perfect square monomial is subtracted from another perfect square monomial

x2 2 16

factoring a difference of two squares

1. x2 2 25 5 (x 1 5)(x 2 5)

factoring perfect-square trinomials

For every real number a and b: a2 1 2ab 1 b2 5

(a 1 b)(a 1 b) 5 (a 1 b)2 or

a2 2 2ab 1 b2 5

(a 2 b)(a 2 b) 5 (a 2 b)2.

2.

perfect-square trinomial

3. 9x2 1 24x 1 16Any trinomial of the form a2 1 2ab 1 b2 or a2 2 2ab 1 b2 is a perfect-square trinomial because it is the result of squaring a binomial.

To factor the difference of two squares a2 and b2, multiply the sum of the two factors a and b by the difference of the two factors a and b.

4x2 2 20x 1 25 5(2x 2 5)(2x 2 5)

a2 2 b2 5 (a 1 b)(a 2 b)

Name Class Date


62

8-7 Think About a PlanFactoring Special Cases

Interior Design A square rug has an area of 49x2 2 56x 1 16. A second square rug has an area of 16x2 1 24x 1 9. What is an expression that represents the diff erence of the areas of the rugs? Show two diff erent ways to fi nd the solution.

1. What are two methods you could use to solve this problem?

2. How would you fi nd the diff erence without factoring?

3. What polynomial do you get when you use this method?

4. Can you factor that polynomial?

5. How could you use factoring to solve the problem?

6. What do the shape of the rug and the polynomials tell you about how to factor

the polynomials for the area of the rugs?

7. Factor each trinomial.

49x2 2 56x 1 16 5 ( u 2 u ) ( u 2 u ) 5 ( u u u )2

16x2 1 24x 1 9 5 ( u 1 u ) ( u 1 u ) 5 ( u u u )2

8. Use your results from Exercise 7 to write an expression for the diff erence in the areas.

9. Factor the expression from Exercise 8 using the diff erence of two squares. Simplify the expressions within each set of parentheses.

10. Do the two methods give you the same result?

subtraction; factoring before subtracting

subtract the polynomials

33x2 2 80x 1 7

yes ; (11x 2 1)(3x 2 7)

(7x 2 4)(7x 2 4) 5 (7x 2 4)2

(4x 1 3)(4x 1 3) 5 (4x 2 3)2

(7x 2 4)2 2 (4x 1 3)2

f(7x 2 4) 1 (4x 1 3)gf(7x 2 4) 2 (4x 1 3)g 5 (11x 2 1)(3x 2 7)

yes

Factoring gives you a second way to

fi nd the difference. You can represent the difference in the form a2 2 b2.

The factors of each square polynomial will be

the same.

Name Class Date


63

8-7 Practice Form G

Factoring Special Cases


1. h2 1 10h 1 25 2. v2 2 14v 1 49 3. d2 2 22d 1 121

4. m2 1 4m 1 4 5. q2 1 6q 1 9 6. p2 2 24p 1 144

7. 36x2 1 60x 1 25 8. 64x2 1 48x 1 9 9. 49n2 1 14n 1 1

10. 16s2 2 72s 1 81 11. 25r2 2 80r 1 64 12. 9g2 2 24g 1 16

13. 81w2 1 144w 1 64 14. 16e2 2 88e 1 121 15. 25j2 1 100j 1 100

16. 144f 2 2 24f 1 1 17. 4a2 2 36a 1 81 18. 49d2 2 84d 1 36

Th e given expression represents the area. Find the side length of the square.

19. 20. 21.

22. 23. 24.

25. Error Analysis Describe and correct the error made in factoring the expression at the right.

64x2 + 80x + 25 9y2 - 24y + 16 4t2 + 36t + 81

36n2 + 84n + 49 100w2 + 20w + 1 16s2 + 104s + 169

175x2 - 28 = 7(25x2 - 4)

= 7(5x - 2)(5x - 2)

= 7(5x - 2)2

(h 1 5)2

(m 1 2)2

(6x 1 5)2

(4s 2 9)2

(9w 1 8)2

(12f 2 1)2

8x 1 5

6n 1 7

(25x2 2 4) factors to (5x 2 2)(5x 1 2), not (5x 2 2)2

(v 2 7)2

(q 1 3)2

(8x 1 3)2

(5r 2 8)2

(4e 2 11)2

(2a 2 9)2

3y 2 4

10w 1 1

(d 2 11)2

(p 2 12)2

(7n 1 1)2

(3g 2 4)2

(5j 1 10)2

(7d 2 6)2

2t 1 9

4s 1 13

Name Class Date


64




26. m2 2 49 27. c2 2 100 28. p2 2 16

29. 4a2 2 25 30. 64n2 2 1 31. 25x2 2 144

32. 50g2 2 8 33. 8d2 2 8 34. 27x2 2 48

35. 24e2 2 54 36. 245k2 2 20 37. 112h2 2 63

38. 48x2 1 72x 1 27 39. 8b2 1 80b 1 200 40. 48w2 1 48w 1 12

41. 45s2 2 210s 1 245 42. 45t2 2 72t 1 24 43. 100z2 2 120z 1 36

44. Writing Explain how to recognize a perfect-square trinomial.

45. a. Open-Ended Write an expression that shows the factored form of a diff erence of two squares.

b. Explain how you know that your expression is a diff erence of two squares.


46. 36s8 2 60s4 1 25 47. c10 2 30c5d2 1 225d4 48. 25n6 1 40n3 1 16

Mental Math For Exercises 49–51, fi nd a pair of factors for each number by using the diff erence of two squares.

49. 24 50. 28 51. 72

52. Reasoning Explain how reversing the rules for multiplying squares of binomials can help you factor a perfect-square trinomial.

53. Writing Th e area of a square parking lot is 49p4 2 84p2 1 36. Explain how

you would fi nd the length of the parking lot.

(m 1 7)(m 2 7)

(2a 1 5)(2a 2 5)

2(5g 1 2)(5g 2 2)

6(2e 1 3)(2e 2 3)

3(4x 1 3)2

5(3s 2 7)2

(c5 2 15d2)2 (5n3 1 4)2(6s4 2 5)2

24 5 52 2 12

5 (5 1 1)(5 2 1) 5 (6)(4)28 5 82 2 62

5 (8 2 6)(8 1 6) 5 (2)(14)72 5 92 2 32

5 (9 1 3)(9 2 3) 5 (12)(6)

When the b term in a trinomial is exactly twice the product of a and c, you can factor it as (a 1 b)2 or as (a 2 b)2.

Factor 49p4 2 84p2 1 36 to fi nd the length. You get (7p2 2 6)2 so each side has a

length of (7p2 2 6).

(c 1 10)(c 2 10)

(8n 1 1)(8n 2 1)

8(d 1 1)(d 2 1)

5(7k 1 2)(7k 2 2)

8(b 1 5)2

3(15t2 2 24t 1 8)

(p 1 4)(p 2 4)

(5x 1 12)(5x 2 12)

3(3x 1 4)(3x 2 4)

7(4h 1 3)(4h 2 3)

12(2w 1 1)2

4(5z 2 3)2

The coeffi cient of the squared term and the constant will be perfect squares. Twice the product of these numbers is the coeffi ecient of the middle term.The sign before the constant will be positive.

Answers may vary. Sample: (2x 1 3)(2x 2 3)

Answers may vary. Sample: 4x2 2 9; 4x2 and 9 are squares and they are separated by a subtraction.

Name Class Date


65

8-7 Practice Form K



1. c2 1 2c 1 1 2. d2 2 10d 1 25 3. p2 2 24p 1 144

4. w2 1 14w 1 49 5. s2 1 16s 1 64 6. 9g2 1 24g 1 16

7. 25m2 2 60m 1 36 8. 4q2 2 32q 1 64 9. 49y2 2 84y 1 36

10. 121n2 2 66n 1 9 11. 81x2 2 18x 1 1 12. 100t2 2 100t 1 25

Th e given expression represents the area. Find the side length of the square.

13. 14.

15. 16.

17. Writing How can you tell that x2 2 19x 1 90 is not a perfect square trinomial?

36w2 12w 1 81w2 72w 16

9w2 48w 64 121w2 66w 9

(c 1 1)2 (d 2 5)2 (p 2 12)2

(w 1 7)2 (s 1 8)2 (3g 1 4)2

(5m 2 6)2 4(q 2 4)2 (7y 2 6)2

(11n 2 3)2 (9x 2 1)2

6w 1 1 9w 2 4

3w 2 8 11w 2 3

Sample: 90 is not a perfect square.

25(2t 2 1)2

Name Class Date


66




18. b2 2 121 19. d2 2 81 20. f 22 625

21. 108x2 2 3 22. 50n2 2 8 23. 405z2 2 245

24. 216h2 2 150 25. 28y2 2 28 26. 50t2 1 40t 1 8

27. 12n2 2 36n 1 27 28. 180a2 2 300a 1 125 29. 250k2 2 200k 1 40

30. Writing Explain how to recognize a diff erence of two squares.

31. a. Open-Ended Write an expression that shows the factored form of a perfect-square trinomial.

b. Explain how you know your expression is a perfect-square trinomial when expanded.

Mental Math For Exercises 32–34, fi nd a pair of factors for each number by using the diff erence of two squares.

32. 84 33. 55 34. 80

35. Writing Th e area of a square painting is 225x4 1 240x2 1 64. Explain how you would fi nd a possible length of one side of the painting.

(14)(6) (11)(5) (20)(4)

Answers may vary. Sample: (5x 1 3)(5x 1 3) or (5x 1 3)2

It is in the form a2 1 2ab 1 b2.

The expression is the difference of two terms that are both perfect squares.

Since the trinomial is a perfect-square trinomial, the length of the side could be a factor of the trinomial.

(b 1 11)(b 2 11) (d 1 9)(d 2 9) (f 1 25)(f 2 25)

3(6x 1 1)(6x 2 1) 2(5n 1 2)(5n 2 2) 5(9z 1 7)(9z 2 7)

6(6h 1 5)(6h 2 5) 28(y 1 1)(y 2 1) 2(5t 1 2)(5t 1 2)

3(2n 2 3)(2n 2 3) 5(6a 2 5)(6a 2 5) 10(5k 2 2)(5k 2 2)

Name Class Date


67

Multiple Choice


1. What is the factored form of q2 2 12q 1 36?

A. (q 1 6)(q 2 6) B. (q 2 6)(q 2 6) C. (q 2 9)(q 1 4) D. (q 1 4)(q 1 9)

2. What is the factored form of 9x2 1 12x 1 4? F. (3x 1 2)2 G. (3x 1 3)2 H. (3x 2 2)2 I. (3x 2 3)2

3. What is the factored form of x2 2 196? A. (x 2 14)2 B. (x 1 14)2 C. (x 2 28)(4x 1 7) D. (x 2 14)(x 1 14)

4. What is the factored form of 9x2 2 64? F. (3x 2 8)2 G. (3x 1 8)2 H. (3x 2 8)(3x 1 8) I. (9x 2 8)(x 1 8)

5. What is the factored form of 12m2 2 75? A. 3(2m 2 5)2 B. 3(2m 1 5)(2m 2 5) C. 3(2m 1 5)2 D. (6m 2 25)(2m 1 3)

6. What is the factored form of 49x2 2 56x 1 16? F. (7x 2 4)2 G. (7x 1 4)(7x 2 4) H. (7x 1 4)2 I. (7x 2 8)2

Extended Response

7. A four-sided building has an area of 36x2 1 48x 1 16. Explain how to fi nd a possible length and width of the building. What is a possible shape of the building?

8-7 Standardized Test PrepFactoring Special Cases

B

F

D

H

B

F

(6x 1 4)2 ; The length and width could be the same, so the shape is a square.

[4] Answer correctly factors the polynomial and indicates the building could be a square with sides 6x 1 4. Complete explanation is provided.

[3] Minor calculation error in the answer or incomplete explanation[2] Polynomial correctly factored but not related to length and width of the

building[1] Some steps in solution of problem completed correctly [0] No correct work shown

Name Class Date


68

8-7 EnrichmentFactoring Special Cases

Th e surface area of a cube is determined by the formula SA 5 6s2, where s is the length of a side of the cube. You can use this formula to analyze a polynomial that represents the surface area of a cube.

Start by dividing the polynomial by 6. Th is will leave an expression for of the area of one face of the cube. You can see that the area is a perfect-square trinomial. Reverse the rules for multiplying squares of binomials to factor the trinomial.

For example, a cube with a surface area of 24x2 1 24x 1 6 has a side measure of 2x 1 1.

6s2 5 24x2 1 24x 1 6

s2 524x2 1 24x 1 6

6 5 4x2 1 4x 1 1

s2 5 (2x 1 1)(2x 1 1)

s 5 2x 1 1

Th e surface area of a rectangular prism with two square faces is determined by the formulaSA 5 4ls 1 2s2, where l is the length and s is the measure of the side of the square face. If you are given the surface area and the area of the square face, you can determine the dimensions of the rectangular prism.

Suppose a rectangular prism has a surface area of 24x 1 30 and each square face measures 9 cm2.

24x 1 30 2 18 5 24x 1 12 Subtract the area of the square faces.

24x 1 124 5 6x 1 3 Divide by 4 to get the area of each remaining side.

6x 1 33 5 2x 1 1 Divide by the side length of the square base, or the square

root of the base’s area.

1. Th e surface area of a cube is 96x2 1 144x 1 54. What is the measure of each side?



4. Th e surface area of a rectangular prism is 100x 1 90. Th e areas of the two square faces of

the prism are 25 m2 each. What are the dimensions of the rectangular prism?

5. Th e surface area of a rectangular prism is 2x2 1 48x 1 88. Th e areas of the two square faces of the prism are x2 1 4x 1 4 each. What are the dimensions of the rectangular prism?

4x 1 3

3x 2 1

12x 1 5

5, 5, and 5x 1 2

10, x 1 2, and x 1 2

Name Class Date


69

8-7 ReteachingFactoring Special Cases

Th e area of a square is given by A 5 s2, where s is a side length. When the side length is a binomial, the area can be written as a perfect-square trinomial. If you are given the area of such a square, you can use factoring to write an expression for a side length.

Problem

A mosaic is made of small square tiles called tesserae. Suppose the area of one tessera is 9x2 1 12x 1 4. What is the length of one side of a tessera?

Because the tile is a square, you know the side lengths must be equal. Th erefore, the binomial factors of the trinomial must be equal.

9x2 1 12x 1 4 5 ( u u u )2 This is a perfect square trinomial and can be factored as the square of a binomial.

9x2 5 (3x)2 9x2 and 4 are perfect squares. Write them as squares.

4 5 22

2(3x)(2) 5 12x Check that 12x is twice the product of the fi rst and last terms. It is, so you are sure that you have a perfect-square trinomial.

9x2 1 12x 1 4 5 (3x 1 2)2 Rewrite the equation as the square of a binomial.

Multiply to check your answer.

(3x 1 2)(3x 1 2) 5 9x2 1 6x 1 6x 1 4 5 9x2 1 12x 1 43

Th e length of one side of the square is 3x 1 2.

ExercisesFactor each expression to fi nd the side length.

1. Th e area of a square oil painting is 4x2 1 28x 1 49. What is the length of one side of the painting?

2. You are installing linoleum squares in your kitchen. Th e area of each linoleum square is 16x2 2 24x 1 9. What is the length of one side of a linoleum square?

3. You are building a table with a circular top. Th e area of the tabletop is (25x2 2 40x 1 16)π. What is the radius of the tabletop?

4. A fabric designer is making a checked pattern. Each square in the pattern has an area of x2 2 16x 1 64. What is the length of one side of a check?

s

s A 5 s2

2x 1 7

4x 2 3

5x 2 4

x 2 8

Name Class Date


70



Some binomials are a diff erence of two squares. To factor these expressions, write the factors so the x-terms cancel and you are left with two perfect squares.

Problem

What is the factored form of 4x2 2 9?

4x2 2 9 5 ( u 1 u )( u 2 u ) Both 4x2 and 9 are perfect squares. You know the signs of the factors will be opposite, so the x-terms will cancel out.

"4x2 5 2x Find the square root of each term.

!9 5 3

(2x 1 3)(2x 2 3) Write each term as a binomial with opposite signs, so the x-terms will cancel out.


(2x 1 3)(2x 2 3) 5 4x2 1 6x 2 6x 2 9

5 4x2 2 93

Th e factored form of 4x2 2 9 is (2x 1 3)(2x 2 3).

ExercisesFactor each expression.

5. 9x2 2 4 6. 25x2 2 49 7. 144x2 2 1

8. 64x2 2 25 9. 49x2 2 16 10. 36x2 2 49

11. 81x2 2 16 12. 16x2 2 121 13. 25x2 2 144

14. 16x2 2 9 15. x2 2 81 16. 4x2 2 49

(3x 1 2)(3x 2 2)

(8x 1 5)(8x 2 5)

(9x 1 4)(9x 2 4)

(4x 1 3)(4x 2 3)

(5x 1 7)(5x 2 7)

(7x 1 4)(7x 2 4)

(4x 1 11)(4x 2 11)

(x 1 9)(x 2 9)

(12x 1 1)(12x 2 1)

(6x 1 7)(6x 2 7)

(5x 1 12)(5x 2 12)

(2x 1 7)(2x 2 7)


71

Name Class Date

8-8 Additional Vocabulary SupportFactoring by Grouping

Use the list to complete the diagram.

common factors factor GCF pair of binomial factors squares

Steps for Factoring a Polynomial Completely

1. Factor out the .

2. If the polynomial has two or three terms, look for a difference of two ______________ , a perfect-square trinomial, or a ____________________.

3. If the polynomial has four or more terms, group terms and __________ to find common binomial factors.

4. Make sure there are no ______________ other than 1.

GCF

squares

common factors

factor

pair of binomial factors


72

Name Class Date

8-8 Think About a PlanFactoring by Grouping

Art Th e pedestal of a sculpture is a rectangular prism with a volume of

63x3 2 28x . What expressions can represent the dimensions of the pedestal? Use factoring.

KNOW

1. Th e pedestal of the sculpture is shaped like a .

2. Th e volume of the pedestal is .

3. Th e formula you can use to fi nd the dimensions of the pedestal is .

NEED

4. To solve the problem you need to fi nd

PLAN

5. Factor out the GCF from the volume of the pedestal.

6. What type of expression is of the remaining expression?

7. Factor the expression completely.

8. What expressions represent possible dimensions of the pedestal?

rectangular prism

63x3 2 28x

V 5 lwh

3 factors

7x(9x2 2 4)

difference of two squares

7x(3x 2 2)(3x 1 2)

7x, (3x 2 2), and (3x 1 2)

Name Class Date


73

8-8 Practice Form G

Factoring by Grouping

Find the GCF of the fi rst two terms and the GCF of the last two terms for each polynomial.

1. 12x3 1 3x2 1 20x 1 5 2. 6v3 1 42v2 1 5v 1 35

3. 8t3 1 36t2 1 2t 1 9 4. 10s3 1 35s2 1 6s 1 21

5. 9m3 2 6m2 1 12m 2 8 6. 8w3 1 6w2 2 28w 2 21

7. 7r3 1 16r2 2 9r 2 72 8. 21x3 2 28x2 2 6x 1 8


9. 8j3 1 4j2 1 10j 1 5 10. 2m3 1 8m2 1 9m 1 36

11. 10s3 1 25s2 1 8s 1 20 12. 6x3 1 9x2 1 2x 1 3

13. 21x3 1 6x2 2 28x 2 8 14. 8w3 1 12w2 1 10w 1 15

15. 18r3 2 12r2 1 21r 2 14 16. 36n3 2 27n2 2 8n 1 6

17. 110b3 1 77b2 2 60b 2 42 18. 64d3 2 40d2 2 24d 1 15

19. 10s3 1 80s2 2 7s 2 56 20. 25j3 1 15j2 2 5j 2 3

21. 24c3 2 84c2 1 10c 2 35 22. 27f 31 9f 22 24f 2 8

3x2, 5

4t2, 1

r2, 29

(4j2 1 5)(2j 1 1)

(5s2 1 4)(2s 1 5)

(3x2 2 4)(7x 1 2)

(6r2 1 7)(3r 2 2)

(11b2 2 6)(10b 1 7)

(10s2 2 7)(s 1 8)

(12c2 1 5)(2c 2 7)

6v2, 5

5s2, 3

3m2, 4 2w2, 27

7x2, 22

(2m2 1 9)(m 1 4)

(3x2 1 1)(2x 1 3)

(4w2 1 5)(2w 1 3)

(9n2 2 2)(4n 2 3)

(8d2 2 3)(8d 2 5)

(5j2 2 1)(5j 1 3)

(9f 22 8)(3f 1 1)

Name Class Date


74



Factor completely.

23. 32x3 1 8x2 1 48x 1 12 24. 45w4 2 36w3 1 15w2 2 12w

25. 32k4 2 16k3 1 12k2 2 6k 26. 6g3 1 18g2 1 60g 1 180

27. 30b4 2 45b3 2 10b2 1 15b 28. 32m3 1 72m2 2 80m 2 180

29. 63j4 1 84j3 2 18j2 2 24j 30. 96n3 2 240n2 2 168n 1 420

31. 12e4 1 18e3 1 36e2 1 54e 32. 60a5 2 72a4 2 210a3 1 252a2

Find linear expressions for the possible dimensions of each rectangular prism.

33. 34.

35. 36.

37. A shipping box in the shape of a rectangular prism has a volume of 12x3 1 32x2 1 20x . What linear expressions can represent possible dimensions of the box?

38. Error Analysis Describe and correct the error made in factoring completely.

39. Open-Ended Write a 3-term expression for the volume of a rectangular prism that you can factor by grouping. Factor your polynomial.

16x4 + 24x3 + 64x2 + 96x = 4x(4x3 + 6x2 + 16x + 24) = 4x[2x2 (2x + 3) + 8(2x + 3)] = 4x(2x2 + 8)(2x + 3)

V = 32p3 - 224p2 + 360p

V = 18d3 + 84d2 + 48d

V = 24y3 + 54y2 -15y

V = 15x3 + 52x2 + 32x

4(2x2 1 3)(4x 1 1)

2k(8k2 1 3)(2k 2 1)

5b(3b2 2 1)(2b 2 3)

3j(7j2 2 2)(3j 1 4)

6e(e2 1 3)(2e 1 3)

x, 5x 1 4, 3x 1 8

3y, 4y 2 1, 2y 1 5

4x, 3x 1 5, x 1 1

In the fi rst step, the GCF is 8x, not 4x.

Answers may vary. Sample: x5 1 4x4 1 3x3 5 x3(x 1 3)(x 1 1)

3w(3w2 1 1)(5w 2 4)

6(g2 1 10)(g 1 3)

4(2m2 2 5)(4m 1 9)

12(4n2 2 7)(2n 2 5)

6a2(2a2 2 7)(5a 2 6)

6d, 3d 1 2, d 1 4

8p, 2p 2 5, 2p 2 9

Name Class Date


75

8-8 Practice Form K


Find the GCF of the fi rst two terms and the GCF of the last two terms for each polynomial.

1. 6n3 1 3n2 1 10n 1 5 2. 12z3 1 36z2 1 4z 1 12

3. 9k3 1 45k2 1 2k 1 10 4. 11a3 1 33a2 1 8a 1 24

5. 2f 3 1 5f 2 2 4f 2 10 6. 16d3 2 24d2 2 6d 1 9


7. 6x3 2 4x2 1 15x 2 10 8. 5q3 2 40q2 2 4q 1 32

9. 28m3 1 7m2 2 8m 2 2 10. 3p3 1 5p2 1 9p 1 15

11. 18y3 2 6y2 2 63y 1 21 12. 3t3 2 18t2 1 5t 2 30

13. 250c3 2 250c2 1 100c 2 100 14. 18g3 2 33g2 1 30g 2 55

15. 88n3 1 77n2 2 72n 2 63 16. 50h3 2 40h2 1 60h 2 48

17. 24b3 2 96b2 2 14b 1 56 18. 54r3 1 9r2 2 6r 2 1

(2x2 1 5)(3x 2 2) (5q2 2 4)(q 2 8)

(7m2 2 2)(4m 1 1) (p2 1 3)(3p 1 5)

3(2y2 2 7)(3y 2 1) (3t2 1 5)(t 2 6)

50(5c2 1 2)(c 2 1) (3g2 1 5)(6g 2 11)

(11n2 2 9)(8n 1 7) 2(5h2 1 6)(5h 2 4)

2(12b2 2 7)(b 2 4) (9r2 2 1)(6r 1 1)

3n2; 5 12z2; 4

9k2; 2 11a2; 8

f 2; 22 8d2; 23

Name Class Date


76



Factor completely.

19. 49s3 1 14s2 1 14s 1 4 20. 32h4 1 72h3 1 36h2 1 81h

21. 42z4 2 48z3 2 7z2 1 8z 22. 60p3 1 48p2 1 25p 1 20

23. 26n4 2 14n3 1 91n2 2 49n 24. 40t3 1 28t2 2 30t 2 21

25. 45k4 2 9k3 1 10k2 2 2k 26. 18b5 2 3b4 1 30b3 2 5b2


27. 28.

29. A storage bin in the shape of a rectangular prism has a volume of

10x3 1 9x2 1 2x . What linear expressions can represent possible dimensions of the bin?

30. Writing Describe the fi rst step to look for in factoring a cubic expression containing four terms.

31. Open-Ended Write a 4-term expression that you can factor by grouping. Factor your polynomial.

V x3 x2 6x V 12a3 13a2 3a

x ; (5x 1 2) ; (2x 1 1)

Check to see if you can factor a GCF from all four terms.

Answers may vary. Sample: 4x3 1 36x2 1 7x 1 63 5 (4x2 1 7)(x 1 9)

(7s2 1 2)(7s 1 2) h(8h2 1 9)(4h 1 9)

z(6z2 2 1)(7z 2 8) (12p2 1 5)(5p 1 4)

n(2n2 1 7)(13n 2 7) (4t2 2 3)(10t 1 7)

k(9k2 1 2)(5k 2 1) b2(3b2 1 5)(6b 2 1)

a by (3a 1 1) by (4a 1 3)x by (x 1 3) by (x 2 2)


77

Name Class Date

8-8 Standardized Test PrepFactoring by Grouping

Multiple Choice


1. What is the GCF of the fi rst two terms of the polynomial 4y3 1 8y2 1 5y 1 10?

A. 4y B. 4y2 C. 4y3 D. 4

2. What is the factored form of 4x3 1 3x2 1 8x 1 6?

F. (2x2 1 3)(2x 1 3)

G. (2x2 1 2)(2x 1 3)

H. (x2 1 2)(2x 1 3)

I. (x2 1 2)(4x 1 3)

3. What is the factored form of 9x4 2 6x3 1 18x2 2 12x?

A. 3x(x2 2 2x)(x 2 4)

B. 3x(x2 2 2)(3x 1 2)

C. 3x(x2 1 2)(3x 2 2)

D. 3x(3x2 2 2x)(6x 2 4)

4. What is the factored form of 20p3 1 40p2 1 15p 1 30?

F. 5(2p2 1 3)(p 1 2)

G. 5(2p2 1 6)(p 1 4)

H. 5(4p2 1 3)(p 1 2)

I. 5(4p2 1 8p)(3p 1 6)

5. A box in the shape of a rectangular prism has a volume of 9x3 1 24x2 1 12x . Which is not one of the possible dimensions? (Its dimensions are all linear expressions with integer coeffi cients.)

A. 2x 1 3 B. 3x 1 2 C. 3x D. x 1 2

Short Response

6. Th e polynomial 3πx3 1 24πx2 1 48πx represents the volume of a cylinder. Th e formula for the volume of a cylinder with radius r and height h is V 5 πr2h.

a. Factor 3πx3 1 24πx2 1 48πx. b. Write a linear expression for a possible radius of the cylinder. Explain.

B

I

C

H

A

3πx(x 1 4)2

x 1 4 because that is the term that is squared[2] Both parts answered correctly with full explanation[1] One part answered correctly or both parts answered correctly

with incomplete explanation[0] Neither part answered correctly


78

Name Class Date

8-8 EnrichmentFactoring by Grouping

Pascal’s triangle is named after French mathematician Blaise Pascal, but this special number pattern had been studied in India, China, Persia, and Italy long before Pascal. To generate Pascal’s triangle, start with the number 1 in Row 0. Each successive row has a 1 at both ends. Add the numbers directly above-left and above-right to fi nd the new value.

You can use Pascal’s triangle to quickly expand a binomial expression. Th e exponent tells you the row number to choose. Th e numbers in the correct row are the coeffi cients to use in the expansion.

To expand (a 1 b)4, look to Row 4. Th e coeffi cients are 1, 4, 6, 4, 1. Expand the variables, raising the fi rst variable to 4 and decreasing by one for each term. Raise the second variable to 0 and increase by 1 at each new term. Multiply each term by the coeffi cients: a4 1 4a3b 1 6a2b2 1 4ab3 1 b4.

You can also use Pascal’s triangle to factor polynomials that are expansions of binomial expressions. Arrange the polynomial in standard form. Check to see if the coeffi cients correspond to a row in Pascal’s triangle. Work backwards to factor.

To factor 15xy2 2 y3 1 125x3 2 75x2y, fi rst rearrange the terms in standard form: 125x3 2 75x2y 1 15xy2 2 y3. In expansions of binomial expressions the x-exponents decrease by one in every term and y-exponents increase by one. Since the fi rst and last terms have exponents of 3, the binomial is raised to the third power. Find the cube root of 125 to fi nd the coeffi cient of x: 5. Th e fi nal term is negative and has a coeffi cient of one, so the expression is (5x 2 y)3. Expand the binomial to check your answer:

(5x 2 y)3 5 1(5x)3 1 3(5x)2(2y) 1 3(5x)(2y)2 1 1(2y)3 5 125x3 2 75x2y 1 15xy2 2 y3

Expand the binomial using Pascal’s triangle.

1. (4k 1 j)4

2. (7x 2 y)7

Factor the polynomial using Pascal’s triangle. Th en expand the binomial to check your answer.

3. 8a3 1 12a2b 1 6ab2 1 b3

4. 40x2y3 1 32x5 1 10xy4 1 80x4y 1 y5 1 80x3y2

5. 1215x4y2 1 135x2y4 1 729x6 1 18xy5 1 y6 1 1458x5y 1 540x3y3

Row 0 1

Row 1 1 1

Row 2 1 2 1

Row 3 1 3 3 1

Row 4 1 4 6 4 1

Row 5 1 5 10 10 5 1

Row 6 1 6 15 20 15 6 1

Row 7 1 7 21 35 35 21 7 1

Row 8 1 8 28 56 70 56 28 8 1

256k4 1 256k3j 1 96k2j2 1 16kj3 1 j4

(2a 1 b)3

(2x 1 y)5

823,543x7 2 823,543x6y 1 352,947x5y2 2 84,035x4y3 1 12,005x3y4 2 1029x2y5 1 49xy6 2 y7

(3x 1 y)6


79

Name Class Date

8-8 ReteachingFactoring by Grouping

You can factor some higher-degree polynomials by grouping terms and factoring out the GCF to fi nd the common binomial factor. Make sure to factor out a common GCF from all terms fi rst before grouping.

Problem

What is the factored form of 2b4 2 8b3 1 10b2 2 40b?

2b4 2 8b3 1 10b2 2 40b 5 2b(b3 2 4b2 1 5b 2 20) 2b is the GCF of all four terms. Factor out 2b from each term.

5 2bfb2(b 2 4) 1 5(b 2 4)g Group terms into pairs and look for the GCF of each pair. b2 is the GCF of the fi rst pair, and 5 is the GCF of the second pair.

5 2b(b2 1 5)(b 2 4) b 2 4 is the common binomial factor. Use the Distributive Property to rewrite the expression.


2b(b2 1 5)(b 2 4) 5 2b(b3 1 5b 2 4b2 2 20) Multiply b2 1 5 and b 2 4.

5 2b4 1 10b2 2 8b3 2 40b Multiply by 2b.

5 2b4 2 8b3 1 10b2 2 40b 3 Reorder the terms by degree.

Th e factored form of 2b4 2 8b3 1 10b2 2 40b is 2b(b2 1 5)(b 2 4).

Exercises

Factor completely. Show your steps.

1. 4x4 1 8x3 1 12x2 1 24x 2. 24y4 1 6y3 1 36y2 1 9y

3. 72z4 1 48z3 1 126z2 1 84z 4. 2e4 2 8e3 1 18e2 2 72e

5. 12f 32 36f 2 1 60f 2 180 6. 16g4 2 56g3 1 64g2 2 224g

7. 56m3 2 28m2 2 42m 1 21 8. 40n4 2 60n3 2 50n2 1 75n

9. 60x3 2 90x2 2 30x 1 45 10. 12p5 1 8p4 1 18p3 1 12p2

11. 6r3 1 9r2 2 60r 12. 20s6 2 50s5 2 30s4

4x(x2 1 3)(x 1 2)

6z(4z2 1 7)(3z 1 2)

12(f 21 5)(f 2 3)

7(4m2 2 3)(2m 2 1)

15(2x2 2 1)(2x 2 3)

3r(2r 2 5)(r 1 4)

3y(2y2 1 3)(4y 1 1)

2e(e2 1 9)(e 2 4)

8g(g2 1 4)(2g 2 7)

5n(4n2 2 5)(2n 2 3)

2p2(2p2 1 3)(3p 1 2)

10s4(2s 1 1)(s 2 3)


80

Name Class Date



Polynomials can be used to express the volume of a rectangular prism. Th ey can sometimes be factored into 3 expressions to represent possible dimensions of the prism. Th e three factors are the length, width, and height.

Problem

Th e plastic storage container to the right has a volume of 12x3 1 8x2 2 15x . What linear expressions could represent possible dimensions of the storage container?

12x3 1 8x2 2 15x 5 x(12x2 1 8x 2 15) Factor out x, the GCF for all three terms.

5 x(12x2 1 18x 2 10x 2 15) ac is –180 and b is 8. Break 8x into two terms that have a sum of 8x and a product of 2180x2.

5 xf6x(2x 1 3) 2 5(2x 1 3)g Group the terms into pairs and factor out the GCF from each pair. The GCF of the fi rst pair is 6x. The GCF of the second pair is 25.

5 x(6x 2 5)(2x 1 3) 2x 1 3 is the common binomial term. Use the Distributive Property to reorganize the factors.


x(6x 2 5)(2x 1 3) 5 x(12x2 1 18x 2 10x 2 15) Multiply 6x 2 5 and 2x 1 3.

5 x(12x2 1 8x 2 15) Combine like terms.

5 12x3 1 8x2 2 15x 3 Multiply by x.

Possible dimensions of the storage container are x, 6x 2 5, and 2x 1 3.

Exercises


13. 14.

15. 16.

V = 12x3 + 8x2 -15x

V = 60x3 - 68x2 -16x V = 12x3 - 15x2 -18x

V = 10x3 + 65x2 +105xV = 12x3 + 34x2 +14x

2x, 3x 1 7, 2x 1 1

4x, 5x 1 1, 3x 2 4

5x, 2x 1 7, x 1 3

3x, 4x 1 3, x 2 2

Name Class Date


81

Chapter 8 Quiz 1 Form G

Lessons 8-1 through 8-4

Do you know HOW?


1. 8x3 2. 57 3. 6p3q2 4. 81x6y3

Simplify.

5. (7t2 1 9) 1 (6t2 1 8) 6. 5x3y2 2 7x3y2

7. (3m2 1 2m 2 8) 1 (4m2 2 5m 1 6)


8. 3n(4n2 1 5n) 9. 4k2(3 2 4k) 10. 27y3(4y2 1 y 2 3)


11. 18s 2 63 12. 30b2 1 48b 2 24 13. w5 1 4w4 1 10w3 1 40w2


14. (x 1 7)(x 1 5) 15. (j 1 3)(j 2 4) 16. (3x 2 1)(x 2 6)

17. (d 1 4)(d 1 4) 18. (3a 1 7)(3a 2 7) 19. (2z 2 3)2

20. A rectangle has length x 1 9 and width 2x 2 1. What is the area of the rectangle?

21. A square has side length (5x 2 3) cm. What is the area of the square?

Do you UNDERSTAND?

22. Vocabulary What are the parts that make up a polynomial?

23. Open-Ended Write a trinomial with 3x as the GCF of its terms.

24. Writing Explain how to use the Distributive Property to fi nd the product of two binomials.

3

13t2 1 17 22x3y2

7m2 2 3m 2 2

12n3 1 15n2

9(2s 2 7)

x2 1 12x 1 35

d2 1 8d 1 16

2x2 1 17x 2 9

(25x2 2 30x 1 9) sq cm

1 or more monomials

Answers may vary. Sample 3x3 1 6x2 1 3x

Answers may vary. Sample (a 1 b)(c 1 d) 5 a(c 1 d) 1 b(c 1 d)

216k3 1 12k2

6(5b 2 2)(b 1 2)

j2 2 j 2 12

9a2 2 49

228y5 2 7y4 1 21y3

w2(w2 1 10)(w 1 4)

3x2 2 19x 1 6

4z2 2 12z 1 9

0 5 9

Name Class Date


82

Chapter 8 Quiz 2 Form G


Do you know HOW?


1. x2 1 11x 1 24 2. s2 2 7s 1 12 3. 2m2 1 27m 1 70

4. 4z2 2 16z 1 15 5. 8y2 2 22y 2 21 6. 9x2 1 48x 1 64

7. g2 2 64 8. 4s2 2 25 9. 49t2 2 9

10. 6r3 1 15r2 1 8r 1 20 11. 10c3 2 12c2 1 15c 2 18 12. 16w3 1 8w2 1 28w 1 14

13. Th e area of a rectangular fi eld is given by the trinomial t2 2 4t 2 45. Th e length of the rectangle is t 1 5. What is the expression for the width of the fi eld?

14. Th e area of a rectangle is given by the trinomial 10x2 2 31x 2 14. Th e length of the rectangle is 5x 1 2. What is the expression for the width of the rectangle?

15. Th e area of a square room is 16x2 1 72x 1 81. What is the length of one side of the room?

16. A rectangular prism has a volume of 4x3 1 30x2 1 36x . What linear expressions can represent possible dimensions of the prism?

Do you UNDERSTAND?

Describe how you would factor each expression.

17. 81m2 2 25 18. 4x2 2 16x 1 16 19. 9x2 1 42x 1 49

20. Reasoning In ax2 1 bx 1 c, if ac is negative and b is positive, what do you know about the factors of ac?

21. Writing Describe how to fi nd linear expressions for the possible dimensions of a rectangular prism with a volume of 8k3 1 26k2 1 6k.

22. Open-Ended Write two trinomials that you can factor into two binomials. Factor each trinomial. Th en write one trinomial that you cannot factor and explain why.

(x 1 8)(x 1 3)

(2z 2 3)(2z 2 5)

(g 1 8)(g 2 8)

(3r2 1 4)(2r 1 5)

t 2 9

2x 2 7

4x 1 9

difference of two squares: (9m 1 5)(9m 2 5)

The factors have different signs.


Factor out the GCF and then factor the other factor. 2k(4k 1 1)(k 1 3)

factor out GCF, perfect square: 4(x 2 2)2

perfect square: (3x 1 7)2

2x, 2x 1 3, x 1 6

(s 2 3)(s 2 4)

(4y 1 3)(2y 2 7)

(2s 1 5)(2s 2 5)

(2c2 1 3)(5c 2 6)

(2m 1 7)(m 1 10)

(3x 1 8)2

(7t 1 3)(7t 2 3)

2(4w2 1 7)(2w 1 1)

Name Class Date


83

Chapter 8 Test Form G

Do you know HOW?


1. 6xy 2. 23b2c4 3. 12m7n

Simplify each sum or difference.

4. 6r3 1 7r3 5. 23u2v 2 19u2v 6. (5g 2 2g) 1 (2g2 1 6g)

7. The perimeter of a pentagon is 20t 1 7. Four sides have the following lengths: 6t, 2t, 4t 2 5, and 5t 1 1. What is the length of the fifth side?


8. 3x(x 1 6) 9. 2z2(z 2 9) 10. 2x(4x2 2 7x 1 6)


11. 12x 2 9 12. 24n3 2 40n2 1 72n 13. 14b2c3 1 21bc5

14. An artist is making a square stained glass window in which a green glass circle is surrounded by blue glass. The side length of the window is shown, and the area of the green piece is 64πx2. What is the area of the blue glass? Write your answer in factored form.

Simplify each product using the stated method.

15. (x 2 2)(3x 2 4); table

16. (3x 1 2)(x 1 7); Distributive Property 17. (4x 2 1)(2x 1 5); FOIL Method

18. What is the surface area of a cylinder with radius x 1 3 and height x 1 11?


19. (x 1 6)2 20. (2s 1 7)2 21. (3x 2 8)2

Complete.

22. k2 1 9k 1 18 5 (k 1 3)(k 1u) 23. x2 2 11x 1 28 5 (x 2 4)(x 2u)


24. (v 1 7)(v 2 7) 25. (5s 2 t)2 26. (3p2 1 10q)(3p2 2 10q)

20x

2

3t 1 11

3x2 1 18x 2z3 1 9z2

3(4x 2 3) 8n(3n2 2 5n 1 9)

16x2(25 2 4π)

3x2 2 10x 1 8

3x2 1 23x 1 14

4π(x2 1 10x 1 21)

x2 1 12x 1 36

v2 2 49 25s2 2 10st 1 t2 9p4 2 100q2

6 7

4s2 1 28s 1 49 9x2 2 48x 1 64

8x2 1 18x 2 5

7bc3(2b 1 3c2)

8x3 2 14x2 1 12x

13r3 4u2v2g2 1 9g

6 8

Name Class Date


84

Chapter 8 Test (continued) Form G

Find an expression for the area of each shaded region.

27. 28.

29. The area of a rectangular coffee table is given by the trinomial t2 1 7t 2 8. The table’s length is t 1 8. What is the table’s width?


30. r2 1 12r 1 27 31. g2 2 8g 2 48 32. m2 1 2m 2 35

33. 3d2 2 13d 1 12 34. 8y2 1 60y 1 72 35. 9w2 2 75w 2 54

Factor completely.

36. 6n3 2 24n2 1 n 2 4 37. 2p4 1 6p3 2 8p2 2 4p 38. 8h2 1 36h 1 16

39. A cereal box in the shape of a rectangular prism has a volume of 18x3 2 3x2 2 6x . What are three possible linear expressions for the dimensions of the cereal box?

40. The area of a rectangular serving tray is 3x2 1 17x 2 56. The width of the tray is x 1 8. What is the length of the tray?

Do You UNDERSTAND?

41. Writing Write a binomial with 2x3y2 as the GCF of its terms. Explain how you

found your answer.

42. Error Analysis Describe and correct the error made in simplifying the product.

43. Reasoning Let x2 1 7x 2 18 5 (x 1 p)(x 1 q) and z2 2 7z 2 18 5 (z 1 r)(z 1 s).

a. What do you know about the signs of p and q? r and s?

b. Suppose u p u . u q u and u r u . u s u . What is the value of p 2 r?

x 2 2

x 2 2

x 1 5

x 1 5 2x

2x

3x 1 4

3x 1 4

(r2 + 7s)(r2 - 7s) = r4 + 14r2s - 49s2

14x 1 21

t 2 1

(r 1 9)(r 1 3)

(3d 2 4)(d 2 3)

(g 1 4)(g 2 12)

4(2y 1 3)(y 1 6)

2p(p3 1 3p2 2 4p 2 2)

(m 1 7)(m 2 5)

3(3w 1 2)(w 2 9)

4(2h 1 1)(h 1 4)(6n2 1 1)(n 2 4)

3x, 3x 2 2, 2x 1 1

3x 2 7

18

they are opposite; they are opposite

Answers may vary. Sample: 2x4y2 1 2x3y2 ; mental math: multiplied by (x 1 1)

The student did not notice that the r2s terms should cancel. Correct answer is r4 2 49s2.

(5x 1 4)(x 1 4)

Name Class Date


85

Chapter 8 Quiz 1 Form K


Do you know HOW?


1. 24t5 2. 22a4b7


3. (4n3 1 12) 1 (10n3 1 1) 4. (5y2 1 3y 2 6) 2 (2y2 2 5y 1 3)


5. 5w(2w2 1 6w) 6. 23p2(5p2 1 p 2 7)

7. (t 1 1)(t 1 6) 8. (2n 2 5)(n 2 3)


9. 16k 2 40 10. 6m4 1 12m3 1 3m2 1 21m


11. (n 1 5)2 12. (j 1 9)(j 2 9)

13. A rectangle has a length of (2x 1 1) ft and a width of (3x 1 5) ft. What is the area of the rectangle?

Do you UNDERSTAND?

14. Open-Ended Write two trinomials whose difference is 5x2 2 6x 1 9.

15. Writing Explain how to use the FOIL Method to find the product of two binomials.

14n3 1 13 3y2 2 2y 2 9

10w3 1 30w2215p4 2 3p3 1 21p2

t2 1 7t 1 6 2n2 2 11n 1 15

5 11

8(2k 2 5) 3m(2m3 1 4m2 1 m 1 7)

n2 1 10n 1 25 j2 2 81

(6x2 1 13x 1 5) ft2

Answers may vary. Sample: (7x2 1 x 1 11) 2 (2x2 1 7x 1 2).

Multiply the first terms of the binomials. Multiply the outside terms of the binomials. Multiply the inside terms of the binomials. Multiply the last terms of the binomials. Add the products and combine like terms.

Name Class Date


86

Chapter 8 Quiz 2 Form K


Do you know HOW?


1. a2 2 6a 2 7 2. m2 1 11m 1 24

3. g2 1 17g 1 72 4. v2 1 11v 1 18

5. 3h2 1 32h 1 20 6. 12y2 1 16y 2 3

7. p2 2 100 8. 81x2 2 1

9. 6k3 2 2k2 1 15k 2 5 10. 80n3 1 30n2 2 56n 2 21

11. The area of a rectangular classroom is given by the trinomial a2 2 4a 2 21. The length of the rectangle is a 1 3. What is the expression for the width of the classroom?

12. The area of a square mural is 144x2 2 72x 1 9. What is the length of one side of the mural?

Do you UNDERSTAND?

Writing Describe how you would factor each expression.

13. 9t2 2 49

14. 36n2 1 60n 1 25

15. Open-Ended Find two different values that complete the expression

24x2 1 u x 2 18 so that the trinomial can be factored into the product of two binomials. Factor your trinomials.

(a 1 1)(a 2 7) (m 1 8)(m 1 3)

(g 1 8)(g 1 9)

(3h 1 2)(h 1 10) (6y 2 1)(2y 1 3)

(p 1 10)(p 2 10) (9x 1 1)(9x 2 1)

(2k2 1 5)(3k 2 1) (10n2 2 7)(8n 1 3)

(v 1 9)(v 1 2)

a 2 7

12x 2 3

This expression is the difference of two squares. You would use a2 2 b2 5 (a 2 b)(a 1 b) which gives you (3t 1 7)(3t 2 7).

This trinomial is in the form a2 1 2ab 1 b2 where a 5 6n and b 5 5. So, the factored form of the trinomial is (6n 1 5)2.

Answers may vary. Sample: 66 and 24. 8x2 1 66x 2 18 5 (3x 1 9)(8x 2 2)and 8x2 1 24x 2 18 5 (4x 1 6)(6x 2 3).

Name Class Date


87

Chapter 8 Test Form K

Do you know HOW?


1. 28mn2 2. 10a2b3


3. t4 1 9t4 4. (3n2 2 4n 1 8) 2 (24n2 1 5n)


5. 2w(w 2 7) 6. 4n(3n2 1 6n 2 9)


7. 15r 1 6 8. 18x2y 2 27x3y2

Simplify each product using the stated method.

9. (4x 1 5)(x 1 3) ; Distributive Property 10. (2x 2 9)(5x 1 4) ; FOIL Method


11. (w 1 2)2 12. (5x 2 3)2

Write the missing value in each trinomial.

13. s2 1 14s 1 33 5 (s 1 3)(s 1u) 14. c2 2 9c 1 14 5 (c 2 2)(c 2u)


15. (b 1 3)(b 2 3) 16. (6t2 1 11)(6t2 2 11)

3 5

10t4 7n2 2 9n 1 8

3(5r 1 2) 9x2y(2 2 3xy)

2w2 2 14w 12n3 1 24n2 2 36n

4x2 1 17x 1 15

w2 1 4w 1 4

b2 2 9

10x2 2 37x 2 36

25x2 2 30x 1 9

11 7

36t4 2 121

Name Class Date


88

Chapter 8 Test (continued) Form K

17. A square has side length (7x 2 2) in. What is the area of the square?


18. b2 1 3b 1 2 19. m2 2 8b 1 12

20. x2 1 2x 2 8 21. a2 2 a 2 20

22. 7y2 1 11y 2 6 23. 10x2 2 53x 2 11

24. 4k2 2 49 25. 9z2 1 42z 1 49

26. 12n3 2 3n2 1 16n 2 4 27. 3p3 2 p2 2 6p 1 2

28. A shipping box in the shape of a rectangular prism has a volume of 18x3 1 5x2 2 2x . What are three expressions that can represent possible dimensions of the shipping box?

Do you UNDERSTAND?

29. Writing Explain how you know which terms in different polynomials can be added or subtracted. What do you add or subtract?

30. Writing Describe how you would factor the expression 25t3 2 20t2 1 4t

31. Open-Ended Find two different values that complete the expression

8n2 1 u n 1 4 so that the trinomial can be factored into the product of two binomials. Factor your trinomials.

(49x2 2 28x 1 4) in.2

(b 1 2)(b 1 1)

(x 1 4)(x 2 2)

(2k 1 7)(2k 2 7)

(m 2 2)(m 2 6)

(a 2 5)(a 1 4)

(7y 2 3)(y 1 2) (5x 1 1)(2x 2 11)

(3z 1 7)2

(3n2 1 4)(4n 2 1) (p2 2 2)(3p 2 1)

x, 9x 2 2, 2x 1 1

Answers may vary. Sample: 33 and 12. 8n2 1 33n 1 4 5 (8n 1 1)(n 1 4) and 8n2 1 12n 1 4 5 (2n 1 2)(4n 1 2).

Like terms in polynomials can be added or subtracted. Like terms are added or subtracted by adding or subtracting the coefficients only of the like terms.

First, factor out the GCF, t, giving you t(25t2 2 20t 1 4). What is left is inside the parentheses is in the form a2 1 2ab 1 b2 where a 5 5t and b 5 22. So, the factored form of the trinomial is t(5t 2 2)2.

Name Class Date


89

Chapter 8 Performance Task

Give complete answers. Show all your work.

TASK 1

How would you describe a polynomial to some new algebra students? What advice would you give to help them learn how to add and subtract polynomials?

TASK 2

In your own words, explain what is accomplished by factoring. As part of your explanation, write a sample problem in which you factor a monomial from a polynomial. Th en write and factor another problem that is a quadratic expression. Check your solutions by multiplying after you factor.


[4] Complete explanation of polynomials and polynomial addition and subtraction given

[3] Explanation includes minor errors or omits small details[2] Explanation covers many important points and is generally correct[1] Some parts of the explanation are correct[0] No correct information given


[4] Factoring correctly explained and two correct examples given and checked correctly

[3] Answer contains minor errors or omits one part of the task[2] At least two of the four parts of the task completed correctly[1] At least one of the four parts of the task completed correctly[0] No part of the task completed correctly

Name Class Date


90

Chapter 8 Performance Task (continued)

TASK 3

A classmate is having trouble factoring special-case polynomials. Point out the errors she has made, and write a brief suggestion to help her with similar problems.

a. 4a2 2 100

5 (2a 2 10)(2a 2 10)

5 (2a 2 10)2

b. 81m4 1 72m2n 1 16n2

5 (9m2 1 8n)(9m2 1 8n)

5 (9m2 1 8n)2

c. a10b4 2 16

5 (a5b2 2 16)

5 (a5b2 2 4)(a5b2 1 4)

d. 4d2 1 36bd 1 81

5 (2d 1 9)(2d 2 9)

TASK 4

Explain how to factor a polynomial with four terms by grouping.

Use grouping to factor 20ay 2 10ax 1 42by 2 21bx. Explain each step.

The fi rst step should be to factor the GCD, 4. In Step 2, signs should be different. Remember that a2 2 b2 5 (a 1 b)(a 2 b).

The square root of 16 is 4, not 8. Learn the common squares and square roots.

In Step 2, the parentheses are around the square of a5b2 and the exponent 2 should be used. No parentheses are needed after 16.

The expression is a perfect square, so the answer should be (2d 1 9)2. Watch the signs when you work.

[4] Correct explanation of grouping and demonstration with each step fully explained

[3] Correct explanation of grouping and demonstration with some steps inadequately explained

[2] Inadequate explanation of grouping or minor errors in demonstration

[1] Some steps in grouping explained or demonstrated correctly

[0] No part of task explained or demonstrated correctly

[4] All four errors correctly identifi ed with helpful suggestions

[3] Three errors correctly identifi ed with helpful suggestions

[2] Two errors correctly identifi ed with helpful suggestions

[1] One error correctly identifi ed with a helpful suggestion

[0] No errors correctly identifi ed

First look for the GCF of all four terms. Look for the GCF of two terms at a time.

10a(2y 2 x) 1 21b(2y 2 x) Found GCFs.

(10a 1 21b)(2y 2 x) Simplifi ed with Distributive Property

Name Class Date


91

Cumulative Review Chapters 1–8

Multiple Choice

For Exercises 1–11 choose the correct letter.

1. What are the next three terms in the sequence 6, 12, 24, 48, c? A. 72, 96, 120 B. 86, 162, 240 C. 96, 192, 384 D. 50, 52, 54

2. Solve 8y 5 2100.

F. 2800 G. 212.5 H. 800 I. 12.5

3. Find the equation of the line passing through (2, 21) and parallel to

y 5 23x 2 1.

A. y 5 23x 1 5 B. y 5 2

3x

22 1 C. y 5

x

31 5 D. y 5 3x 1 1

4. Solve 3x 1 7y 5 224x 2 3y 5 22

.

F. (24, 22) G. (24, 2) H. (4, 2) I. (4, 22)

5. Simplify 10x5y3

2x6y.

A. 5xy2 B. 5y2

x C.

5x

y2 D. x

5y2

6. Simplify (3x 2 1)(x 1 4).

F. 3x2 2 4 G. 3x2 2 11x 2 4 H. 3x2 1 11x 2 4 I. 3x2 1 13x 2 4

7. A scuba diver at a depth of 80 ft begins her ascent to the ocean surface. Her rate of change in depth is 2ft/s. Which expression represents her depth in feet t seconds after she begins her ascent?

A. 2t 2 80 B. 80 2 2t C. 280 2 2t D. 80 1 2t

8. Factor 4x2 2 x 2 14. F. (4x 1 7)(x 2 2) G. (2x 2 7)(2x 1 2) H. (4x 2 7)(x 1 2) I. (2x 1 7)(2x 2 2)

9. What is the GCF of the terms of 3x3 1 6x2 2 9x?

A. x B. 3 C. 3x D. 3x2

10. Which number is not a solution of the compound inequality 7 2 4x # 3 and 2x 2 5 . 210?

F. 5 G. 4 H. 2 I. 1

11. Which of the following is a cubic binomial?

A. w3 2 6w2 1 9 B. 7a3 1 4a22 C. 2y3 1 3y5 D. x2 2 2x3

C

G

A

I

B

H

B

F

C

F

D

Name Class Date


92

12. A city is growing at a rate of 8 percent per year. What multiplier is used to fi nd the new population each year?

13. Simplify 62 4 4 1 2(7 2 3) ? 4.

14. What is the slope of a line that passes through the origin and the point (6, 3)?

15. Evaluate x2 1 3y for x 5 4 and y 5 0.5.

16. A weight of 6 lb stretches a spring a distance of 12 in. Find the constant k for the spring.

17. Solve 18x5

2114

.

18. What is the x-intercept of the line with equation 5x 1 4y 5 30?

19. How many positive solutions are there to the equation Z 2x 2 5 Z 5 4?

20. Write an equation in standard form passing through the points (22, 0) and (23, 21).

21. Th e product of two negative integers is 36. Th e second integer is 5 more than the fi rst. Find the integers.

22. Th e length of a rectangular pizza is 4 in. less than twice its width. Th e area of

the pizza is 160 in.2. Find the dimensions of the pizza.

23. Write a polynomial that is a diff erence of two squares using the variable m. Write the polynomial in factored and standard forms.

24. Solve the following system of inequalities by graphing:

2x 2 4y # 4

23x 2 6y . 6

Cumulative Review (continued) Chapters 1–8

6

2

1.08

41

17.5

12

y 2 x 5 2

29, 24

10 in. by 16 in.

Sample :4m2 2 9; (2m 1 3)(2m 2 3)

graph of y L 12x 2 1 and y R 2

12x 2 1

12

12

xO

y4

2

2

4

2

4 42

T E A C H E R I N S T R U C T I O N S


93

About the Project

In this project students will learn about the uses of trees. Th ey will use formulas to analyze data and predict the production of wood and fruit.

Introducing the Project• Ask students to think of something fl at that is made of wood, such as a table

top or door.

• Instruct them to estimate the number of pieces of wood, each 1 ft2, that make up their objects.

• Ask students to compare results with partners.

• Direct student attention to Activity 1. Explain that they will research types of wood and the tools carpenters use to work with wood.

Activity 1: ResearchingStudents research lumber and tool requirements for the construction of a house.

Activity 2: CalculatingStudents evaluate the given expression to calculate the useable board feet of a log.

Activity 3: CalculatingStudents use the given expression to determine the diameter of a tree.

Activity 4: GraphingStudents use the given function to calculate and graph the number of bushels of walnuts produced on an acre of land.

Finishing the Project

You may wish to plan a project day on which students share their completed projects. Encourage groups to explain their processes as well as their results. Have students review the project work and update their folders.

• Have students review the research, equations, graphs, and explanations needed for the project.

• Ask groups to share their insights that resulted from completing the project, such as any shortcuts they found for doing research, solving equations, or making graphs.

Chapter 8 Project Teacher Notes: Trees Are Us

Name Class Date


94

Beginning the Chapter Project

Many schools celebrate Arbor Day by planting young trees to replenish our ecosystem. Trees use carbon dioxide that humans and animals exhale to make oxygen. Trees anchor the soil and prevent erosion. Th ey also produce fruit. Wood from trees is used for the construction of everything from pencils to houses.

As you work through the activities, you will learn more about the uses of trees. You will use formulas to analyze data and predict the production of wood and fruit. Th en you will decide how to organize and display your results.

List of Materials• Calculator

• Graph paper

Activities

Activity 1: Researching

A board foot is a cubic measure of lumber equal to a square foot of wood 1 in. thick.

• What can you make from 10 board feet? 100 board feet? 1000 board feet? How is the size of a house related to the amount of wood used to build it?

• What diff erent types of wood are needed for cabinets, fl oors, and roofs? What tools do carpenters use to make these items?

Activity 2: Calculating

You can use the expression 0.0655l(1 2 p)(d 2 s)2 to fi nd the number of useable board feet in a log.

• Estimate the useable board feet in a 35-ft log if its diameter is 20 in. Assume the log loses 10% of its volume from the saw cuts and a total of 2 in. is trimmed off the log.

• Th e diameter of a log is 25 in. A total of 2 in. will be trimmed off the log. Th e estimated volume loss due to saw cuts is 10%. How long must the log be to yield 600 board feet of lumber?

Chapter 8 Project: Trees Are Us

i length in feet

P percent lossdue to cut of saw

d diameter in inches

s inches trimmed off the log to make boards

Name Class Date


95

Activity 3: Calculating

With aerial photography, you can study a forest of ponderosa pines without ever walking through it. To fi nd the diameter in inches of trees in the forest, use the expression:

3.76 1 (1.35 3 1022)hv 2 (2.45 3 1026)hv2 1 (2.44 3 10210)hv3.Th e variable h is the height of the tree in feet, and v is the crown diameter visible in feet (from a photograph). Determine the diameter of a 100-ft tree that has a visible crown diameter of 20 ft.

Activity 4: Graphing

You can use the function b 5 20.01t2 1 0.8t to fi nd the number of bushels b of walnuts produced on an acre of land. Th e variable t represents the number of walnut trees per acre.

• Use your graphing calculator to graph this function. Include an accurate graph in your notebook. You may wish to investigate the TABLE feature on your calculator. Use the maximum feature under the CALC menu to determine the number of trees per acre that gives the greatest yield.

• How many walnut trees would you advise a farmer to plant on 5 acres of land to produce the most walnuts possible? Explain your reasoning.

Finishing the Project

Th e answers to the four activities should help you complete your project. Assemble all the parts of your project in a folder. Add a summary telling what you have learned about the uses of trees.

Refl ect and ReviseAsk a classmate to review your project folder with you. Together, check that your graph is clearly labeled and accurate. Check that you have used formulas correctly and that your calculations are accurate. Make any revisions necessary to improve your work.

Extending the ProjectTrees have many uses that you can investigate. You can begin your research by contacting the United States Department of Agriculture Forest Service or a local, state, or national park. You can also get more information by using the Internet.

Chapter 8 Project: Trees Are Us (continued)

20 ft

Aerial View of Ponderosa Pine

Name Class Date


96

Getting StartedRead the project. As you work on the project, you will need a calculator, materials on which you will record your calculations, and materials to make accurate and attractive graphs. Keep all of your work for the project in a folder.

Checklist Suggestions

☐ Activity 1: researching lumber ☐ Measure wooden objects to help you estimate.

☐ Activity 2: calculating ☐ Use the given formula.

☐ Activity 3: calculating ☐ Have someone check your solution.

☐ Activity 4: graphing the function

☐ Select an appropriate viewing window.

☐ project report ☐ What have you learned about trees and lumber while working on this project? To whom might the formulas in the activities be most useful, and why?

Scoring Rubric3 Th orough research techniques are demonstrated with many diff erent sources

of information accessed. Calculations are correct. Th e graph is neat, accurate, and has an appropriate scale. Explanations are complete and well thought-out.

2 Good research techniques are evident. Calculations are mostly correct, but have minor errors. Th e graph is neat and mostly accurate with minor errors in scale.

1 Needed information is located with some help. Calculations contain both minor and major errors. Th e graph could be more accurate.

0 Major elements of the project are incomplete or missing.

Your Evaluation of Project Evaluate your work, based on the Scoring Rubric.

Teacher’s Evaluation of Project

Chapter 8 Project Manager: Trees Are Us

chapter 8 answer key

Documents

teaching resourcescopyright

pearson education

number of sandwiches

th ree sides

quadratic trinomial3x4

quadratic trinomial12t2

lengthsthe length

b quadratic binomial14z