Algebra 1 G9

CHAPTER Polynomials

8 and Factoring

Download videos VIDEO Chapter Preview 1 Equivalence
connecting math Essential Question  Can two algebraic
to your world.. 8-1 Adding and Subtracting Polynomials expressions that appear to be different
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Math definitions VOC ABUL ARY Vocabulary DOMAINS
in English and • Seeing Structure in Expressions
Spanish English/Spanish Vocabulary Audio Online:
• Arithmetic with Polynomials and Rational
English Spanish Expressions

binomial, p. 487 binomio

Online access degree of a monomial, p. 486 grado de un monomio
to stepped-out
problems aligned degree of a polynomial, p. 487 grado de un polinomio
to Common Core
Get and view difference of two squares, p. 525 diferencia de dos
your assignments cuadrados
online.
O NLINE factoring by grouping, p. 529 factor común por
RKME WO agrupación de términos
HO
monomial, p. 486 monomio

perfect-square trinomial, p. 523 trinomio cuadrado
perfecto

Extra practice polynomial, p. 487 polinomio
and review
online standard form of a forma normal de
polynomial, p. 487 un polinomio

trinomial, p. 487 trinomio

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PERFORMANCE TASK

Common Core Performance Task

Planning a Garden Plot

Each rectangular plot in a community garden consists of a planting area
surrounded by a border of wood chips. Kelly and Roberto each design a plot for
the community garden. They use identical square flower beds in their plots, but
they design different borders, as shown below.

Kelly’s Plan Roberto’s Plan
1 ft 1.5 ft

x ft x ft
3 ft 3 ft 3 ft 3 ft

x ft x ft

1 ft    1.5 ft

After some discussion, Kelly and Roberto decide to scrap their original plans and
combine their plots to make one larger rectangular plot. They plan to place the
two x ft-by-x ft square flower beds side by side to make a 2x ft-by-x ft flower bed,
centered inside a rectangular border. The area of the new plot will equal the sum
of the areas of the two original plots.

Task Description

Draw and label a sketch of the new flower bed and border. Include expressions for
the length and width of the new plot. Each expression must be a linear polynomial
with integer coefficients.

Connecting the Task to the Math Practices MATHEMATICAL

As you complete the task, you’ll apply several Standards for Mathematical PRACTICES
Practice.

• You’ll analyze the polynomials that describe Kelly’s and Roberto’s original
plots. (MP 6)

• You’ll write an expression for the area of each of the original plots. (MP 2, MP 3)

• You’ll write and factor an expression for the area of the new plot. (MP 7)

Chapter 8  Polynomials and Factoring 485

8-1 Adding and Subtracting CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Polynomials
MA-AAFPSR.9.A1.21. AU-AndPeRrs.1ta.n1d  Uthnadteprsotlaynndomthiatlspfoolrymnoamials
Objective To classify, add, and subtract polynomials sfoyrsmtema saynsatelomgoaunsaltoogtohuesintotetgheersin, tneagmeresl,y,ntahmeyelay,rethey
calroesceldosuendduerndtheer tohpeeroapteiorantsioonfsaodfdiatdiodnit,isounb,tsruabctrioanct,iaond,
manudltmipulicltaiptiloicna;taiodnd;,asdudb,trsaucbtt,ranctd, manudltmipulyltpipollyynomials.
MpoPlyn1o, mMiaPls2. , MP 3, MP 4, MP 6
MP 1, MP 2, MP 3, MP 4, MP 6

Try breaking The costs of admission and parking for a museum and an aquarium
this problem into are given. What is the total cost of m people in n cars visiting only
simpler parts. the museum? Only the aquarium? Both the museum and the aquarium?
How much do Explain your reasoning.
they spend on
admission? On MUSEUM AQUARIUM
parking?
ADMISSION: $15 PER PERSON ADMISSION: $20 PER PERSON
PARKING: $6 PER CAR
PARKING: $5 PER CAR

MATHEMATICAL

PRACTICES

In some cases, you can model a situation with an expression composed of monomials.
A monomial is a real number, a variable, or a product of a real number and one or more
variables with whole-number exponents. Here are some examples of monomials.

18 z - 4x2 2.5xy3 a
3
Lesson
Essential Understanding  You can use monomials to form larger expressions
Vocabulary called polynomials. Polynomials can be added and subtracted.
• monomial
• degree of a The degree of a monomial is the sum of the exponents of its variables. The degree of a
nonzero constant is 0. Zero has no degree.
monomial
• polynomial Problem 1 Finding the Degree of a Monomial
• standard form of What is the degree of each monomial?

a polynomial
• degree of a

polynomial
• binomial
• trinomial

A 5x Degree: 1 5x = 5x1. The exponent is 1.

Why is the degree of B 6x3y2 Degree: 5 The exponents are 3 and 2. Their sum is 5.

a nonzero constant 0? C 4 Degree: 0 4 = 4x0. The degree of a nonzero constant is 0.
You can write a nonzero
constant c as cx0. The
c. 11
ex ponent is 0, so the Got It? 1. What is the degree of each monomial?
de gree is 0 also. a. 8xy b. - 7y4z

486 Chapter 8  Polynomials and Factoring

You can add or subtract monomials by adding or subtracting like terms.

Problem 2 Adding and Subtracting Monomials

Will the sum of two What is the sum or difference?
monomials always be
A 3x2 + 5x2 = 8x2 Combine like terms. B 4x3y − x 3y = 3x 3y Combine like terms.
a monomial?
No. The monomials must

b e like terms. Got It? 2. What is the sum - 6x4 + 11x 4? What is the difference 2x2y4 - 7x2y4?

A polynomial is a monomial or a sum of monomials. The following polynomial is the
sum of the monomials 3x 4, 5x 2, - 7x, and 1.

3x4 + 5x2 - 7x + 1
c c c c
4 2 1 0
Degree of each monomial

The polynomial shown above is in standard form. Standard form of a polynomial
means that the degrees of its monomial terms decrease from left to right. The degree
of a polynomial in one variable is the same as the degree of the monomial with the
greatest exponent. The degree of 3x4 + 5x2 - 7x + 1 is 4.

You can name a polynomial based on its degree or the number of monomials it contains.

Polynomial Degree Name Using Number Name Using
6 0 Degree of Terms Number of Terms
1
5x ϩ 9 2 Constant 1 Monomial
4x2 ϩ 7x ϩ 3 3 Linear 2 Binomial
4
2x3 Quadratic 3 Trinomial
8x4 Ϫ 2x3 ϩ 3x
Cubic 1 Monomial

Fourth degree 3 Trinomial

Problem 3 Classifying Polynomials

Write each polynomial in standard form. What is the name of the polynomial based

Why do you need to on its degree and number of terms?

combine like terms A 3x + 4x2 B 4x − 1 + 5x3 + 7x

iT no part (B)? 4x 2 + 3x Place terms in order. 5x3 + 4x + 7x - 1 Place terms in order.
name a polynomial

c orrectly based on its This is a quadratic binomial. 5x 3 + 11x - 1 Combine like terms.
nm uumstbfeirrsotfctoemrmbisn, eyoaull
This is a cubic trinomial.

like terms. 2x - 3 + 8x 2 in standard form. What is the name of the

Got It? 3. a. Write

polynomial based on its degree and number of terms?

b. Reasoning  How does writing a polynomial in standard form help you

name the polynomial?

Lesson 8-1  Adding and Subtracting Polynomials 487

You can add polynomials by adding like terms.

Problem 4 Adding Polynomials

Travel  A researcher studied the number of overnight stays in U.S. National Park
Service campgrounds and in the backcountry of the national park system over a
5-yr period. The researcher modeled the results, in thousands, with the following
polynomials.

Campgrounds: − 7.1x2 − 180x + 5800
Backcountry: 21x2 − 140x + 1900

In each polynomial, x = 0 corresponds to the first year in the 5-yr period. What
polynomial models the total number of overnight stays in both campgrounds and
backcountry?

• Overnight stays in campgrounds: A polynomial for the total The word both implies
-7.1x2 - 180x + 5800 number of overnight addition, so add the two
stays in campgrounds and polynomials to find a
• Overnight stays in backcountry: backcountry polynomial that represents
21x2 - 140x + 1900 the total.

Method 1 Add vertically.
Line up like terms. Then add the coefficients.

- 7.1x2 - 180x + 5800
+ 21x2 - 140x + 1900
13.9x2 - 320x + 7700

Method 2 Add horizontally.
Group like terms. Then add the coefficients.

1 -7.1x2 - 180x + 58002 + 121x2 - 140x + 19002
= 1 -7.1x2 + 21x22 + 1 -180x - 140x2 + 15800 + 19002

= 13.9x2 - 320x + 7700

A polynomial that models the number of stays (in thousands) in campgrounds and
backcountry over the 5-yr period is 13.9x2 - 320x + 7700.

Got It? 4. A nutritionist studied the U.S. consumption of carrots and celery and of
broccoli over a 6-yr period. The nutritionist modeled the results, in millions
of pounds, with the following polynomials.

Carrots and celery: -12x 3 + 106x 2 - 241x + 4477
Broccoli: 14x 2 - 14x + 1545
In each polynomial, x = 0 corresponds to the first year in the 6-yr period.

What polynomial models the total number of pounds, in millions, of carrots,
celery, and broccoli consumed in the United States during the 6-yr period?

488 Chapter 8  Polynomials and Factoring

Recall that subtraction means to add the opposite. So when you subtract a polynomial,
change each of the terms to its opposite. Then add the coefficients.

How can Problem 4 Problem 5 Subtracting Polynomials
help you solve this What is a simpler form of (x 3 − 3x 2 + 5x) − (7x 3 + 5x 2 − 12)?
problem?
Change the subtraction Method 1 Subtract vertically.
to addition by adding
the opposite. Then use x 3 - 3x 2 + 5x Line up like terms.
the vertical or horizontal
method from Problem 4 - 17x 3 + 5x 2 - 122
to add the polynomials.
x 3 - 3x 2 + 5x tThheenpoaldydnothmeiaolpbpeoinsigtesuobf teraacchtetde.rm in
- 7x 3 - 5x 2 + 12

- 6x 3 - 8x 2 + 5x + 12

Method 2 Subtract horizontally.

1x3 - 3x 2 + 5x2 - 17x 3 + 5x 2 - 122

= x 3 - 3x 2 + 5x - 7x 3 - 5x 2 + 12 pWorliytneotmheiaolpbpeoinsigtesuobfteraacchtetde.rm in the

= 1x 3 - 7x 32 + 1 -3x 2 - 5x 22 + 5x + 12 Group like terms.

= - 6x 3 - 8x 2 + 5x + 12 Simplify.

Got It? 5. What is a simpler form of 1 -4m3 - m + 92 - 14m2 + m - 122?

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL
Find the degree of each monomial.
PRACTICES

1. - 7x 4 2. 8y 2z 3 Vocabulary  Name each polynomial based on its degree
and number of terms.

5. 5x 2 + 2x + 1 6. 3z - 2

Simplify each sum or difference. 7. Compare and Contrast  How are the processes of
adding monomials and adding polynomials alike?
3. 15r3 + 82 + 16r3 + 32 How are the processes different?
4. 1x2 - 22 - 13x + 52

Practice and Problem-Solving Exercises MATHEMATICAL

A Practice Find the degree of each monomial. PRACTICES See Problem 1.
11. 2b8c2
8. 3x 9. 8a3 10. 20 15. 0
14. 12w4
12. -7y3z 13. - 3

Lesson 8-1  Adding and Subtracting Polynomials 489

Simplify. 17. 2m3n3 + 9m3n3 18. 8w 2x + w 2x See problem 2.
16. 12p 2 + 8p 2 21. 30v 4w 3 - 12v 4w 3 22. 7x 2 - 2x 2 19. 3t 4 + 11t 4
20. x 3 - 9x 3 23. 5bc 4 - 13bc 4

Write each polynomial in standard form. Then name each polynomial based See problem 3.

on its degree and number of terms.

24. 5y - 2y 2 25. -2q + 7 26. x 2 + 4 - 3x

27. 6x 2 - 13x 2 - 4x + 4 28. c + 8c 3 - 3c7 29. 3z 4 - 5z - 2z 2

Simplify. 31. 6x 2 + 7 See problem 4.
30. 4w - 5 + 3x 2 + 1 32. 2k 2 - 3k + 3
+ 9w + 2
+ 5k 2 + 3k - 7
33. 15x2 + 32 + 115x2 + 22
34. 12g4 - 3g + 92 + 1 -g3 + 12g2

35. Education  The number of students at East High School and the number of
students at Central High School over a 10-year period can be modeled by the
following polynomials.

East High School: -11x2 + 133x + 1200
Central High School: -7x2 + 95x + 1100

In each polynomial, x = 0 corresponds to the first year in the 10-year period. What
polynomial models the total number of students at both high schools?

Simplify. See problem 5.

36. 5n - 2 37. 6x 3 + 17 38. 2c 2 + 17c - 1
-(3n + 8) - 14x3 + 192 - 1c 2 - 10c + 42

39. 114h4 + 3h32 - 19h4 + 2h32 40. 1 -6w 4 + w 22 - 1 -2w 3 + 4w 2 - w2

B Apply 41. Think About a Plan  The perimeter of a triangular park is 16x + 3. 5x – 1
What is the missing length?
• What is the sum of the two given side lengths? 2x + 5
• What operation should you use to find the remaining side length?

42. Geometry  The perimeter of a trapezoid is 39a - 7. Three sides
have the following lengths: 9a, 5a + 1, and 17a - 6. What is the
length of the fourth side?

43. Error Analysis  Describe and correct the error in finding the difference of
the polynomials.

(4x2 – x + 3) - (3x2 – 5x - 6) = 4x2 – x + 3 - 3x2 – 5x - 6
= 4x2 – 3x2 - x - 5x + 3 - 6
= x2 – 6x - 3

490 Chapter 8  Polynomials and Factoring

Simplify. Write each answer in standard form. 45. 1y 3 - 4y 2 - 22 - 16y 3 + 4 - 6y 22
47. 13z3 - 4z + 7z22 + 18z2 - 6z - 52
44. 15x2 - 3x + 7x2 + 19x2 + 2x2 + 7x2
46. 1 -9r3 + 2r - 12 - 1 -5r2 + r + 82

48. a. Is the sum of two polynomials always a polynomial? Explain. yp q
b. Is the difference of two polynomials always a polynomial? Explain. 6 6x
4
C Challenge 49. a. Write the equations for line p and line q. Use slope-intercept form. 2

b. Use your equations from part (a) to write a function for the Ϫ2 O 2 4
vertical distance D(x) between points on lines p and q with
the same x-value.

c. For what value of x does D(x) equal zero?
d. Reasoning  How does the x-value in part (c) relate to the graph?

Simplify each expression.

50. 1ab2 + ba32 + 14a3b - ab2 - 5ab2

51. 19pq 6 - 11p 4q2 - 1 -5pq 6 + p 4q 42

PERFORMANCE TASK

Apply What You’ve Learned MATHEMATICAL

PRACTICES

MP 6

Look back at Kelly’s and Roberto’s original plots on page 485. Choose from the

following words, numbers, and expressions to complete the sentences below.

monomial binomial trinomial 1 2

x2 x+2 x2 + 2 x+6 2x + 6

Two polynomials that represent the length and width of Kelly’s plot are
a. ? and b. ? . Each of these polynomials is an example of a c. ? . The
polynomial that represents the area of Roberto’s flower bed is d. ? . This
polynomial is an example of a e. ? . The degree of this polynomial is f. ? .

Lesson 8-1  Adding and Subtracting Polynomials 491

8-2 Multiplying CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
and Factoring
AM-AAFPSR.9.A1.21. AU-AndPeRrs.1ta.n1d  Uthnadteprsotlaynndomthiatlspfoolrymnoamials
Objectives To multiply a monomial by a polynomial sfoyrsmtema saynsatelomgoaunsaltoogtohuesintotetgheersin, tneagmeresl,y,ntahmeyelay,rethey
To factor a monomial from a polynomial calroesceldosuendduerndtheer tohpeeroapteiorantsioonfsaodfdiatdiodnit,isounb,tsruabctrioanct,iaond,
manudltmipulicltaiptiloicna;taiodnd;,asdudb,trsaucbtt,ranctd, manudltmipulyltpipollyynomials.
MpoPlyn1o, mMiaPls2. , MP 3, MP 4
MP 1, MP 2, MP 3, MP 4

Sketch a diagram. You set aside part of a rectangular plot of 10 ft
A diagram can help land for a garden and seed the rest of the 45 ft
you understand all plot with grass, as shown. Grass seed costs
the parts of this $.03 per square foot. Write an expression x ft
problem. for the total cost of the seed. Suppose you
buy $50 worth of seed. How wide can the
section of grass be? Explain your reasoning.

MATHEMATICAL Essential Understanding  You can use the Distributive Property to multiply a

PRACTICES monomial by a polynomial. xxx 1

For example, consider the product 2x(3x + 1). x x2 x2 x2 x
2x
2x(3x + 1) = 2x(3x) + 2x(1)
= 6x 2 + 2x x x2 x2 x2 x

You can show why the multiplication makes sense using 3x ϩ 1
the area model at the right.

Problem 1 Multiplying a Monomial and a Trinomial
Multiple Choice  What is a simpler form of − x 3(9x 4 − 2x 3 + 7)?

- 9x 12 + 2x 9 - 7x 3 - 9x 7 - 2x 3 + 7
9x 7 - 2x 6 + 7x 3 - 9x 7 + 2x 6 - 7x 3
What should I
keep in mind when - x3(9x4 - 2x3 + 7) = - x3(9x4) - x3( - 2x3) - x3(7) Use the Distributive Property.
multiplying?
Remember to distribute = - 9x 3 + 4 + 2x 3 + 3 - 7x 3 Multiply coefficients and
- x 3 to all of the terms. add exponents.
Also remember to add
the exponents instead of = - 9x 7 + 2x 6 - 7x 3 Simplify.
multiplying them.
The correct answer is D.

Got It? 1. What is a simpler form of 5n(3n3 - n2 + 8)?

492 Chapter 8  Polynomials and Factoring

Essential Understanding  Factoring a polynomial reverses the multiplication
process. When factoring a monomial from a polynomial, the first step is to find the
greatest common factor (GCF) of the polynomial’s terms.

Problem 2 Finding the Greatest Common Factor
What is the GCF of the terms of 5x 3 + 25x 2 + 45x?

List the prime factors of each term. Identify the factors common to all terms.
# # # 5x3 = 5 x x x
Why use the factors # # # 25x2 = 5 5 x x Remember to list only
5 and x to form the # # # 45x = 3 3 5 x the prime factors of
GCF, but not 3? the variables.

Both 5 and x are factors #The GCF is 5 x, or 5x.

of every term of the

polynomial, but 3 is only

a factor of the last term. Got It? 2. What is the GCF of the terms of 3x 4 - 9x 2 - 12x?

Once you find the GCF of a polynomial’s terms, you can factor it out of the polynomial.

Problem 3 Factoring Out a Monomial
What is the factored form of 4x 5 − 24x 3 + 8x?

To factor the polynomial, # # # # # # 4x5 = 2 2 x x x x x
first factor each term. # # # # # # 24x3 = 2 2 2 3 x x x
# # # 8x = 2 2 2 x
Find the GCF of the # #The GCF is 2 2 x, or 4x.
three terms.
4x5 − 24x3 + 8x = 4x(x4) + 4x( − 6x2) + 4x(2)
Factor out the GCF from = 4x(x4 − 6x2 + 2)
each term. Then factor it
out of the polynomial.

The factored form of the polynomial is
4x(x 4 − 6x 2 + 2).

Got It? 3. a. What is the factored form of 9x 6 + 15x 4 + 12x 2?
b. Reasoning  What is - 6x 4 - 18x 3 - 12x 2 written as the product of a

polynomial with positive coefficients and a monomial?

Lesson 8-2  Multiplying and Factoring 493

Problem 4 Factoring a Polynomial Model

How can you find the Helipads  A helicopter landing pad, or helipad, is
shaded region’s area? sometimes marked with a circle inside a square so that
The shaded region is the it is visible from the air. What is the area of the shaded
entire square except for region of the helipad at the right? Write your answer in
the circular portion. So, factored form.
subtract the area of the
circle from the area of Step 1 Find the area of the shaded region.
the square.
A1 = s 2 Area of a square
= (2x)2 Substitute 2x for s. 2x
= 4x 2 Simplify. x
Area of a circle
A2 = pr2 Substitute x for r.
= px 2

The area of the shaded region is
A1 - A2, or 4x 2 - px 2.

Step 2 Factor the expression.

First find the GCF.

# # # 4x 2 = 2 2 x x
# # px2 = p x x
# The GCF is x x, or x 2.

Step 3 Factor out the GCF.

4x 2 - px 2 = x 2(4) + x 2( - p)

= x 2(4 - p)

The factored form of the area of the shaded region is x 2(4 - p).

Got It? 4. In Problem 4, suppose the side length of the square is 6x and the radius of the
circle is 3x. What is the factored form of the area of the shaded region?

Lesson Check Do you UNDERSTAND? MATHEMATICAL

Do you know HOW? PRACTICES
1. What is a simpler form of 6x(2x 3 + 7x)?
2. What is the GCF of the terms in 4a4 + 6a2? Match each pair of monomials with its GCF.

Factor each polynomial. 5. 14n2, 35n4 A. 1
3. 6m2 - 15m 6. 21n3, 18n2 B. 7n 2
4. 4x 3 + 8x 2 + 12x 7. 7n2, 9 C. 3n 2

8. Reasoning  Write a binomial with 9x 2 as the GCF of
its terms.

494 Chapter 8  Polynomials and Factoring

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Simplify each product. See Problem 1.
11. 3m2(10 + m)
9. 7x(x + 4) 10. (b + 11)2b 14. - 8y 3(7y 2 - 4y - 1)
13. 4x(2x 3 - 7x 2 + x)
12. - w 2(w - 15)

Find the GCF of the terms of each polynomial. See Problem 2.

15. 12x + 20 16. 8w2 - 18w 17. 45b + 27
20. 14z4 - 42z 3 + 21z 2
18. a3 + 6a2 - 11a 19. 4x 3 + 12x - 28

Factor each polynomial. 22. t 2 + 8t See Problem 3.
21. 9x - 6 25. 14x 3 - 2x 2 + 8x 23. 14n3 - 35n2 + 28
24. 5k 3 + 20k 2 - 15 26. g4 + 24g 3 + 12g 2 + 4g

27. Art  A circular mirror is surrounded by a square metal frame. The radius of the See Problem 4.
mirror is 5x. The side length of the metal frame is 15x. What is the area of the metal
frame? Write your answer in factored form. 6x
3x
28. Design  A circular table is painted yellow with a red square in the middle. The
radius of the tabletop is 6x. The side length of the red square is 3x. What is the
area of the yellow part of the tabletop? Write your answer in factored form.

B Apply Simplify. Write in standard form.

29. -2x(5x 2 - 4x + 13) 30. - 5y 2(- 3y 3 + 8y) 31. 10a( -6a2 + 2a - 7)
33. t 2(t + 1) - t(2t 2 - 1) 34. 3c(4c 2 - 5) - c(9c)
32. p(p + 2) - 3p(p - 5)

35. Think About a Plan  A rectangular wooden frame has side lengths 5x and 7x + 1.
The rectangular opening for a picture has side lengths 3x and 5x. What is the area of

the wooden part of the frame? Write your answer in factored form.
• How can drawing a diagram help you solve the problem?
• How can you express the area of the wooden part of the frame as a difference

of areas?

36. Error Analysis  Describe and correct the error made -3x(2x - 5) = -3x(2x) - 3x(5)
in multiplying. = -6x2 – 15x

Factor each polynomial.

37. 17xy 4 + 51x 2y 3 38. 9m4n5 - 27m2n3 39. 31a6b3 + 63a5

40. a. Factor n2 + n.
b. Writing  Suppose n is an integer. Is n2 + n always, sometimes, or never an even

integer? Justify your answer.

Lesson 8-2  Multiplying and Factoring 495

41. Reasoning  The GCF of two numbers p and q is 7. What is the GCF of p2 and q2?

Justify your answer.

C Challenge 42. a. Geometry  How many sides does the polygon have? How many of its
diagonals come from one vertex?

b. A polygon has n sides. How many diagonals will it have from one vertex?
n
c. The number of diagonals from all the vertices is 2 (n - 3). Write this

polynomial in standard form.

d. A polygon has 8 sides. How many diagonals does it have?

STEM 43. Manufacturing  The diagram shows a cube of metal with a cylinder 48 in. 4s
cut out of it. The formula for the volume of a cylinder is V = pr2h, s
where r is the radius and h is the height. 4s

a. Write a formula for the volume of the cube in terms of s. 4s
b. Write a formula for the volume of the cylinder in terms of s.
c. Write a formula in terms of s for the volume V of the metal left

after the cylinder has been removed.
d. Factor your formula from part (c).
e. Find V in cubic inches for s = 15 in. Use p = 3.14.

Standardized Test Prep

SAT/ACT 44. Simplify the product 4x(5x2 + 3x + 7). What is the coefficient of the x2-term?

45. What is the slope of the line that passes through CD? D y4
C2
46. What is the solution of the equation 7x - 11 = 3?
47. Simplify the product 8x3(2x2). What is the exponent? Ax
B Ϫ6 Ϫ4 Ϫ2 O
48. The expression 9x3 - 15x can be factored as ax(3x2 - 5).
What is the value of a?

Mixed Review

Simplify each sum or difference. See Lesson 8-1.
49. (5x2 + 4x - 2) + (3x2 + 7) 50. (4x4 - 3x2 - 1) + (3x4 + 6x2)
51. (3x3 - 2x) - (8x3 + 4x) 52. (7x4 + 3x3 - 5x + 1) - (x3 + 8x 2 - 5x - 3)

Solve each inequality for y. Then graph the inequality. See Lesson 6-5.
55. -3y - x 7 9
53. 4x - 5y Ú 10 54. 7x - 2y … 8

Get Ready!  To prepare for Lesson 8-3, do Exercises 56–58.

Use the Distributive Property to simplify each expression. See Lesson 1-7.

56. 8(x - 5) 57. -3(w + 4) 58. 0.25(6c + 16)

496 Chapter 8  Polynomials and Factoring

Concept Byte Using Models MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
to Multiply
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msuubltripaclitciaotnio, na;naddmd,uslutibptlircaacti,oann;dadmdu,lstiupblytrpaoctly, naonmd ials.
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MP 7

You can use algebra tiles to model the multiplication of two binomials.

Find the product (x + 4)(2x + 3).

2x ϩ 3

xϩ4 2x 2 + 3x + 8x + 12 Add coefficients of like terms.
The product is 2x 2 + 11x + 12. 2x 2 + 11x + 12

You can also model products that involve subtraction. Red tiles indicate negative
variables and negative numbers.

Find the product (x − 1)(2x + 1).

2x ϩ 1

xϪ1 2x 2 + x - 2x - 1 Add coefficients of like terms.
The product is 2x 2 - x - 1. 2x 2 - x - 1

Exercises

Use algebra tiles to find each product.

1. (x + 4)(x + 2) 2. (x + 2)(x - 3) 3. (x + 1)(3x - 2) 4. (3x + 2)(2x + 1)

Concept Byte  Using Models to Multiply 497

8-3 Multiplying Binomials Common Core State Standards

A-APR.A.1  Understand that polynomials form a
system analogous to the integers, namely, they are
closed under the operations of addition, subtraction, and
multiplication; add, subtract, and multiply polynomials.

MP 1, MP 2, MP 3, MP 4, MP 7, MP 8

Objective To multiply two binomials or a binomial by a trinomial

A park has a rectangular dog run with length 30 ft x x
x
and width 20 ft. The parks department wants to x

expand each end of each side of the dog run by 20 ft

You’ve got a the same amount x. What will be the total area of
diagram. How
can you use it to the expanded dog run? Justify your reasoning. x 30 ft x
create expressions x
for the width and x
for the length?

MATHEMATICAL Essential Understanding  There are several ways to find the product of two
PRACTICES binomials, including models, algebra, and tables.

One way to find the product of two binomials is to use an area model, as shown below.

This model shows x x1
that (2x ϩ 1)(x ϩ 2)
can be written in x x2 x2 x
standard form as 11 xx xϩ2
2x2 ϩ 5x ϩ 2.
xx 11

2x ϩ 1

You can also use the Distributive Property to find the product of two binomials.

Problem 1 Using the Distributive Property

Is the product of two What is a simpler form of (2x + 4)(3x − 7)?

polynomials always (2x + 4)(3x - 7) = 2x(3x - 7) + 4(3x - 7) Distribute the second factor, 3x - 7.

a polynomial? = 6x2 - 14x + 4(3x - 7) Distribute 2x.
Yes. The set of polynomials

is closed under = 6x2 - 14x + 12x - 28 Distribute 4.
multiplication, which = 6x2 - 2x - 28 Combine like terms.
means that multiplying

polynomials always gives

y ou another polynomial. Got It? 1. What is a simpler form of (x - 6)(4x + 3)?

498 Chapter 8  Polynomials and Factoring

When you use the Distributive Property to multiply binomials, notice that you multiply
each term of the first binomial by each term of the second binomial. A table can help
you organize your work.

Problem 2 Using a Table
What is a simpler form of (x − 3)(4x − 5)?

Binomial factors Product of binomials Use a table.
written in standard
form

Is this the only table Make a table of products. When labeling the rows
you can make? and columns, think of
4x ؊5 x Ϫ 3 as x ϩ (Ϫ3). Think
No. You can write the x 4x2 Ϫ5x of 4x Ϫ 5 as 4x ϩ (Ϫ5).
terms of x - 3 in a row ؊3 Ϫ12x 15
and the terms of 4x - 5
in a column.

The product is 4x2 - 5x - 12x + 15, or 4x2 - 17x + 15.

Got It? 2. What is a simpler form of (3x + 1)(x + 4)? Use a table.

There is a shortcut you can use to multiply two binomials. Consider the product of

2x + 2 and x + 3. The large rectangle below models this product. You can divide the
large rectangle into four smaller rectangles.

The area of the large rectangle is the sum of the areas of the four smaller rectangles.

2x ϩ 2 2x 2

xϩ3 x 2x2 2x x

3 6x 63
2x 2

(2x + 2)(x + 3) = (2x)(x) + (2x)(3) + (2)(x) + (2)(3) The area of each rectangle is the
= 2x2 + 6x + 2x + 6 product of one term of 2x ϩ 2
= 2x2 + 8x + 6 and one term of x ϩ 3.

This model illustrates another way to find the product of two binomials. You find the
sum of the products of the First terms, the Outer terms, the Inner terms, and the Last
terms of the binomials. The acronym FOIL may help you remember this method.

Lesson 8-3  Multiplying Binomials 499

Problem 3 Using FOIL

What is a simpler form of (5x − 3)(2x + 1)?

How can a diagram First Outer Inner Last

help you multiply (5x Ϫ 3)(2x ϩ 1) = (5x)(2x) + (5x)(1) + ( -3)(2x) + ( -3)(1)

two binomials? = 10x2 + 5x - 6x - 3
Draw arrows from each

term of the first binomial = 10x2 - x - 3
to each term of the
The product is 10x2 - x - 3.
second binomial. This will

help you organize the

p roducts of the terms. Got It? 3. What is a simpler form of each product? Use the FOIL method.
c. 12p2 + 3212p - 52
a. (3x - 4)(x + 2) b. (n - 6)(4n - 7)

Problem 4 Applying Multiplication of Binomials

Multiple Choice  A cylinder has the dimensions shown in the xϩ1
diagram. Which polynomial in standard form best describes the xϩ4
total surface area of the cylinder?

2px2 + 4px + 2p 4px2 + 14px + 10p
2px2 + 10px + 8p 2px2 + 2px + 10p

How can you simplify The total surface area (S.A.) of a cylinder is given by the formula
(x 1 1)2? S.A. = 2pr2 + 2prh, where r is the radius of the cylinder and h is the height.
Write the expression
as (x + 1)(x + 1) S.A. = 2pr2 + 2prh Surface area of a cylinder
and multiply the
binomials. You do = 2p(x + 1)2 + 2p(x + 1)(x + 4) Substitute x + 1 for r and x + 4 for h.
not “distribute” the
exponent to each term: = 2p(x + 1)(x + 1) + 2p(x + 1)(x + 4) Write (x + 1)2 as (x + 1)(x + 1).
(x + 1)2 ≠ x2 + 12.
= 2p1x2 + x + x + 12 + 2p1x2 + 4x + x + 42 Multiply binomials.

= 2p1x2 + 2x + 12 + 2p1x2 + 5x + 42 Combine like terms.

= 2p1x2 + 2x + 1 + x2 + 5x + 42 Factor out 2p.

= 2p12x2 + 7x + 52 Combine like terms.

= 4px2 + 14px + 10p Write in standard form.

The correct answer is C.

Got It? 4. What is the total surface area of a cylinder with radius x + 2 and height
x + 4? Write your answer as a polynomial in standard form.

You can use the FOIL method when you multiply two binomials, but it is not helpful
when multiplying a trinomial and a binomial. In this case, you can use a vertical
method to distribute each term.

500 Chapter 8  Polynomials and Factoring

Problem 5 Multiplying a Trinomial and a Binomial
What is a simpler form of (3x2 + x − 5)(2x − 7)?

How should you align Multiply by arranging the polynomials vertically as shown.
the polynomials?
Write the polynomials 3x2 + x - 5
so that like terms are
vertically aligned. 2x - 7

- 21x2 - 7x + 35 Multiply by - 7.

6x3 + 2x2 - 10x Multiply by 2x.

  6x3 - 19x2 - 17x + 35 Add like terms.

The product is 6x3 - 19x2 - 17x + 35.

Got It? 5. a. What is a simpler form of (2x2 - 3x + 1)(x - 3)?
b. Reasoning  How can you use the Distributive Property to find the

product of a trinomial and a binomial?

Lesson Check Do you UNDERSTAND? MATHEMATICAL

Do you know HOW? PRACTICES
Simplify each product.
5. Reasoning  Explain how to use the FOIL method to
1. (x + 3)(x + 6)
2. (2x - 5)(x + 3) find the product of two binomials.
3. (x + 2)(x2 + 3x - 4)
6. Compare and Contrast  Simplify (3x + 8)(x + 1)
4. A rectangle has length x + 5 and width x - 3. What using a table, the Distributive Property, and the FOIL
is the area of the rectangle? Write your answer as a
polynomial in standard form. method. Which method is most efficient? Explain.

7. Writing  How is the degree of the product of two
polynomials p(x) and q(x) related to the degrees of
p(x) and q(x)?

Practice and Problem-Solving Exercises MATHEMATICAL

A Practice Simplify each product using the Distributive Property. PRACTICES

8. (x + 7)(x + 4) 9. (y - 3)(y + 8) See Problem 1.
10. (m + 6)(m - 7)
11. (c - 10)(c - 5) 12. (2r - 3)(r + 1) 13. (2x + 7)(3x - 4)

Simplify each product using a table. See Problem 2.
16. (w - 2)(w + 6)
14. (x + 5)(x - 4) 15. (a - 1)(a - 11) 19. (3p + 4)(2p + 5)

17. (2h - 7)(h + 9) 18. (x - 8)(3x + 1)

Lesson 8-3  Multiplying Binomials 501

Simplify each product using the FOIL method. See problem 3.

20. (a + 8)(a - 2) 21. (x + 4)(4x - 5) 22. (k - 6)(k + 8)
25. (9z + 4)(5z - 3)
23. (b - 3)(b - 9) 24. (5m - 2)(m + 3) 28. (8c - 1)(6c - 7)

26. (3h + 2)(6h - 5) 27. (4w + 13)(w + 2)

29. Geometry  What is the total surface area of the cylinder? xϩ2 See problem 4.
Write your answer as a polynomial in standard form.
xϩ5
30. Design  The radius of a cylindrical gift box is (2x + 3) in.
The height of the gift box is twice the radius. What is
the surface area of the cylinder? Write your answer as a
polynomial in standard form.

Simplify each product. See problem 5.
31. (x + 5)(x2 - 3x + 1)
33. (2a2 + 4a + 5)(5a - 4) 32. (k2 - 4k + 3)(k - 2)
34. (2g + 7)(3g 2 - 5g + 2)

35. Sports  A school’s rectangular athletic fields currently have a length of 125 yd and
a width of 75 yd. The school plans to expand both the length and the width of the
fields by x yards. What polynomial in standard form represents the area of the
expanded athletic field?

B Apply Simplify each product. Write in standard form.

36. (x2 + 1)(x - 3) 37. ( -n2 - 1)(n + 3) 38. (b2 - 1)(b2 + 3)
41. (4z2 + 1)(z + 3z2)
39. (2m2 + 1)(m + 5) 40. (c2 - 4)(2c + 3)

42. Error Analysis  Describe and correct the error made in (x - 2) (3x + 4) = x(3x) + x(4) - 2(4)
finding the product. = 3x2 + 4x - 8

43. Reasoning  Is the product of two polynomials always a
polynomial? Explain.

44. Think About a Plan  You are planning a rectangular dining pavilion. Its length

is three times its width x. You want a stone walkway that is 3 ft wide around the
pavilion. You have enough stones to cover 396 ft2 and want to use them all in the

walkway. What should the dimensions of the pavilion be?
• Can you draw a diagram that represents this situation?
• How can you write a variable expression for the area of the walkway?

45. a. Simplify each pair of products.

# i. (x + 1)(x + 1) #ii. (x + 1)(x + 2) #iii. (x + 1)(x + 3)
11 12 11 13
11 11
b. Reasoning  What are the similarities between your two answers in each pair

of products?

46. Geometry  The dimensions of a rectangular prism are n, n + 7, and n + 8. Use the
formula V = /wh to write a polynomial in standard form for the volume of the prism.

502 Chapter 8  Polynomials and Factoring

C Challenge For Exercises 47–49, each expression represents the side length of a cube. Write
a polynomial in standard form for the surface area of each cube.

47. x + 2 48. 3a + 1 49. 2c2 + 3

50. Financial Planning  Suppose you deposit $1500 for college in a savings account
that has an annual interest rate r (expressed as a decimal). At the end of 3 years, the
value of your account will be 1500(1 + r)3 dollars.

a. Rewrite the expression 1500(1 + r)3 by finding the product
1500(1 + r)(1 + r)(1 + r). Write your answer in standard form.

b. How much money is in the account after 3 yr if the interest rate is 3% per year?

PERFORMANCE TASK

Apply What You’ve Learned MATHEMATICAL

PRACTICES

MP 2, MP 3

Look back at the information on page 485 about a community garden. Kelly’s

and Roberto’s plans for their original plots are shown again below. In the Apply

What You’ve Learned in Lesson 8-1, you wrote binomials for the length and

width of Kelly’s plot.

Kelly’s Plan Roberto’s Plan
1 ft 1.5 ft

x ft x ft
3 ft 3 ft 3 ft 3 ft

x ft x ft

1 ft    1.5 ft

a. Use a product of two binomials to find a polynomial that represents the area of
Kelly’s plot.

b. Use a product of two binomials to find a polynomial that represents the area of
Roberto’s plot.

c. Explain how you can use a specific value of x to check that you found the products
in parts (a) and (b) correctly.

d. Based on the polynomials you wrote in parts (a) and (b), can you conclude that the
area of Roberto’s plot is greater than the area of Kelly’s plot for any value of x? Give
an argument to support your answer.

Lesson 8-3  Multiplying Binomials 503

8-4 Multiplying CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Special Cases
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closed under the operraattiioonnssooffaaddddiittiioonn,,ssuubbttrraaccttioionn,,aanndd
multiplication; add, ssuubbttrraacctt,,aannddmmuullttiippllyyppoolylynnoommiaialsls..

MP 1, MP 2, MP 3, MP 4, MP 7, MP 8

Objectives To find the square of a binomial and to find the product of a sum and difference

x

You are making square invitations for a party. x
You start with a square piece of paper with
6-in. sides. You reduce both its length and 6 in.
its width by x, as shown. What is the area of
How is this Solve the invitation? Justify your reasoning.
It like the one for
Lesson 8-3? How is 6 in.
it different?

MATHEMATICAL

PRACTICES

Essential Understanding  There are special rules you can use to simplify the

square of a binomial or the product of a sum and difference.

Squares of binomials have the form (a + b)2 or (a - b)2. You can algebraically
simplify the product or you can use an area model to discover the rule for simplifying
(a + b)2, as shown below.

Simplify the product. Area Model

(a + b)2 = (a + b)(a + b) Multiply the binomials. a aϩb
= a2 + ab + ba + b2 Simplify. aϩb ab
= a2 + 2ab + b2
a2 ab

b ab b2
a2 ϩ 2ab ϩ b2

Key Concept  The Square of a Binomial

Words  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.

Algebra  Examples 
(a + b)2 = a2 + 2ab + b2 (x + 4)2 = x2 + 8x + 16
(a - b)2 = a2 - 2ab + b2 (x - 3)2 = x2 - 6x + 9

504 Chapter 8  Polynomials and Factoring

Problem 1 Squaring a Binomial

What rule can you What is a simpler form of each product?
use to simplify this
A (x + 8)2 = x2 + 2x(8) + 82 Square the binomial.

product? = x 2 + 16x + 64 Simplify.
(2m - 3)2 may not look
like (a - b)2, but it has B (2m - 3)2 = (2m)2 - 2(2m)(3) + 3 Square the binomial.

the same form. Use the = 4m2 - 12m + 9 Simplify.
rule for (a - b)2 and let
a = 2m and b = 3. Got It? 1. What is a simpler form of each product?
a. (n - 7)2 b. (2x + 9)2

Problem 2 Applying Squares of Binomials

How do you find the Exterior Design  A square outdoor patio is surrounded by a 3 ft
area of the walkway? brick walkway as shown. What is the area of the walkway?
The area of the walkway x ft
is the difference of the Step 1 Find the total area of the patio and walkway. 3 ft 3 ft
total area and the area of x ft
the patio. (x + 6)2 = x 2 + 2(x)(6) + 62 Square the binomial. (x + 6) ft (x + 6) ft 3 ft

= x 2 + 12x + 36 Simplify.

#Step 2 Find the area of the patio.

The area of the patio is x x, or x2.

Step 3 Find the area of the walkway.

Area of walkway = Total area - Area of patio

= (x2 + 12x + 36) - x2 Substitute.

= (x2 - x2) + 12x + 36 Group like terms.

= 12x + 36 Simplify.

The area of the walkway is (12x + 36) ft2.

Got It? 2. In Problem 2, suppose the brick walkway is 4 ft wide. What is its area?

What number close Using mental math, you can square a binomial to find the square of a number.
to 39 can you square
mentally? Problem 3 Using Mental Math
The nearest multiple of What is 392? Use mental math.
10 to 39 is 40, which is 392 = (40 - 1)2 Write 392 as the square of a binomial.
a number you should be
able to square mentally. = 402 - 2(40)(1) + 12 Square the binomial.
= 1600 - 80 + 1 Simplify.
= 1521 Simplify. 505

Lesson 8-4  Multiplying Special Cases

Got It? 3. a. What is 852? Use mental math.
b. Reasoning  Is there more than one way to find 852 using mental math?

Explain your reasoning.

The product of the sum and difference of the same two terms also produces a pattern.

(a + b)(a - b) = a2 - ab + ba - b2 Notice that the sum
= a2 - b2 of Ϫab and ba is 0,
leaving a2 Ϫ b2.

Key Concept  The Product of a Sum and Difference

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

Algebra  Examples 
(a + b)(a - b) = a2 - b2 (x + 2)(x - 2) = x2 - 22 = x2 - 4

Problem 4 Finding the Product of a Sum and Difference
What is a simpler form of (x 3 + 8)(x 3 − 8)?

How do you choose Write the original product. (x 3 + 8)(x 3 − 8)
which rule to use? a = x 3; b = 8
The first factor in the Identify which terms
product is the sum of x3 correspond to a and b in the (x 3 + 8)(x 3 − 8) = (x 3)2 − (8)2
and 8. The second factor rule for the product of a sum = x 6 − 64
is the difference of x3 and difference.
and 8. So, use the rule for
the product of a sum and Substitute for a and b in
difference. the rule.

Simplify.

Got It? 4. What is a simpler form of each product? c. (3c - 4)(3c + 4)
a. (x + 9)(x - 9) b. (6 + m2)(6 - m2)

506 Chapter 8  Polynomials and Factoring

You can use the rule for the product of a sum and difference to calculate products using
mental math.

#How can you write Problem 5 Using Mental Math 35 84

64 56 as the #What is 64 56? 00000 0
product of a sum and # 64 56 = (60 + 4)(60 - 4) Write as a product of a sum and a difference. 11111 1
difference? 22222 2
33333 3
Find the number halfway = 602 - 42 Use (a + b)(a - b) = a2 - b2. 44444 4
between the factors. 55555 5
66666 6
60 is 4 units from each = 3600 - 16 Simplify powers. 77777 7
88888 8
factor. Write the factors = 3584 Simplify. 99999 9
in terms of 60 and 4.
# Got It? 5. What is 52 48? Use mental math.

Lesson Check Do you UNDERSTAND? MATHEMATICAL

Do you know HOW? PRACTICES
Simplify each product.
1. (c + 3)(c + 3) What rule would you use to find each product? Why?
2. (g - 4)2
3. (2r - 3)(2r + 3) 5. (3x - 1)2

4. A square has side length (2x + 3) in. What is the area 6. (4x - 9)(4x + 9)
of the square?
7. (7x + 2)(7x + 2)

8. Reasoning  How do you know whether it is
convenient to use the rule for the product of a sum
and difference to mentally multiply two numbers?

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Simplify each expression. See Problem 1.
12. (2n + 7)2
9. (w + 5)2 10. (h + 2)2 11. (3s + 9)2 16. (4x - 6)2
15. (5m - 2)2
13. (a - 8)2 14. (k - 11)2

Geometry  The figures below are squares. Find an expression for the area of See Problem 2.
each shaded region. Write your answers in standard form.

17. x ϩ 4 18. x ϩ 3

xϪ1 xϩ4 x xϩ3
xϪ1 x

Lesson 8-4  Multiplying Special Cases 507

19. Interior Design  A square green rug has a blue square in the center. The side length
of the blue square is x inches. The width of the green band that surrounds the blue
square is 6 in. What is the area of the green band?

Mental Math  Simplify each product. 22. 482 23. 4032 See Problem 3.
20. 612 21. 792 24. 3022

Simplify each product. 26. (b + 1)(b - 1) See Problem 4.
25. (v + 6)(v - 6) 29. (10 + y)(10 - y) 27. (z - 5)(z + 5)
28. (x - 3)(x + 3) 30. (t - 13)(t + 13)

Mental Math  Simplify each product. #33. 63 57 #34. 399 401 #See Problem 5.

# # 31. 42 38 32. 79 81 35. 303 297

B Apply Simplify each product.

36. (m + 3n)2 37. (2a + b)2 38. (4s - t)2 39. (g - 7h)2
43. (p4 - 9q2)2
40. (9k + 2q)2 41. (8r - 5s)2 42. (s + 6t2)2 46. (2g + 9h)(2g - 9h)
49. (3w3 - z2)(3w3 + z2)
44. (4x + 7y)(4x - 7y) 45. (a - 6b)(a + 6b)

47. (r 2 + 3s)(r 2 - 3s) 48. (2p2 + 7q)(2p2 - 7q)

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

(3a - 7)2 = 9a2 - 21a + 49

51. Think About a Plan  A company logo is a white square inside a red square. The side
length of the white square is x + 2. The side length of the red square is three times
the side length of the white square. What is the area of the red part of the logo?

Write your answer in standard form.
• How can drawing a diagram help you solve the problem?
• How can you express the area of the red part of the logo as a difference of areas?

STEM 52. Construction  A square deck has a side length of x + 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.

53. Reasoning  Use the area model at the right to write a second aϪb b(a Ϫ b)b
expression for the area of the square labeled (a - b)2. Then simplify b(a Ϫ b) b2 b
the expression to derive the rule for the square of a binomial of the a
(a Ϫ b)2 aϪb
form a - b.
a
54. Open-Ended  Give a counterexample to show that
(x + y)2 = x2 + y2 is false.

( ) 55. Reasoning  Does 3 1 2 = 941? Explain.
2

508 Chapter 8  Polynomials and Factoring

C Challenge 56. Simplify (a + b + c)2.

57. Number Theory  You can use factoring to show that the sum of two multiples of 3 is
also a multiple of 3.

If m and n are integers, then 3m and 3n are multiples of three.
3m + 3n = 3(m + n)
Since m + n is an integer, 3(m + n) is a multiple of three.
a. Show that if an integer is one more than a multiple of 3, then its square is also

one more than a multiple of 3.
b. Reasoning  If an integer is two more than a multiple of 3, is its square also two

more than a multiple of 3? Explain.

58. The formula V = 34 pr 3 gives the volume of a sphere with radius r. Find the volume
of a sphere with radius x + 3. Write your answer in standard form.

Standardized Test Prep

SAT/ACT 59. What is a simpler form of (2x + 5)(2x - 5)?

4x 2 - 20x - 25 4x 2 + 20x + 25 4x 2 - 25 2x 2 - 5

60. Sara and Nick sold tickets to a play. Sara sold 20 student tickets and 3 adult Ticket Sales

Short tickets for more than $60. Nick sold 15 student tickets and 5 adult tickets for 25 y
Response less than $75. This information can be represented by 20x + 3y 7 60 and 20
15x + 5y 6 75, where x is the price of a student ticket and y is the price of 15
an adult ticket. The inequalities are graphed at the right. Which could be the 10
5
price of a student ticket? 00 x
6
$1 $5.50 1 2 3 4 5

$2.75 $6

61. Graph the solutions of the system. 5x + 4y Ú 20
5x + 4y … 20

Mixed Review

Simplify each product. 63. (4m - 1)(6m - 7) See Lesson 8-3.
62. (3x + 2)(2x - 5) 64. (x + 9)(5x + 8)

Find each percent change. Describe the percent change as an increase or See Lesson 2-10.
decrease. If necessary, round to the nearest tenth. 68. $40 to $35

65. $4 to $3 66. 4 ft to 5 ft 67. 12 lb to 15 lb

Get Ready!  To prepare for Lesson 8-5, do Exercises 69–71.

Factor each polynomial. See Lesson 8-2.
71. 35x 3 + 7x 2 + 63x
69. 12x 4 + 30x 3 + 42x 70. 72x 3 + 54x 2 + 27

Lesson 8-4  Multiplying Special Cases 509

8 Mid-Chapter Quiz athXM

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Do you know HOW? Simplify each product.
Find the degree of each monomial. 18. (r + 3)2
1. - 5a8 19. (k - 3)(k + 3)
2. 4x2y3 20. (3d + 10)2
21. (g + 10)(g - 10)
Write each polynomial in standard form. Then name each 22. (2m - 7)2
polynomial based on its degree and number of terms. 23. (7h - 2)(7h + 2)
3. 4x + 3x2
4. 7p2 - 3p + 2p3 24. Woodworking  A birdhouse has a square base with
side length 3x - 4. What polynomial in standard
Simplify each sum or difference. form represents the area of the base?

5. 1x2 + 6x + 112 + 13x2 + 7x + 42 Do you UNDERSTAND?
6. 15w3 + 3w2 + 8w + 22 + 17w2 + 3w + 12 25. Writing  Can the degree of a monomial ever be
7. 14q2 + 10q + 72 - 12q2 + 7q + 52
8. 19t4 + 5t + 82 - 13t2 - 6t - 42 negative? Explain.

Simplify each product. 26. Geometry  The figures below are rectangles. What
polynomial in standard form represents the area of
9. 6x214x2 + 32 the shaded region?
10. -8c313c2 + 2c - 92

Factor each polynomial.
11. 16b4 + 8b2 + 20b xϪ4 xϩ3
12. 77x3 + 22x2 - 33x - 88 xϪ2

Simplify each product. xϩ6
13. (x + 2)(x + 9)
14. (4b - 1)(b - 8) 27. Open-Ended  Write a trinomial that has 9x 2 as the
GCF of its terms.
15. (h + 2)13h2 + h - 72
16. (z - 1)1z2 - 4z + 92 28. Open-Ended  Write a trinomial of degree 4 such that
the GCF of its terms is 1.
17. Design  You are designing a rectangular rubber
stamp. The length of the stamp is 2r + 3. The width 29. Reasoning  Suppose n represents an even number.
of the stamp is r - 4. What polynomial in standard Write a simplified expression that represents the
form represents the area of the stamp? product of the next two even numbers.

30. W riting  Describe how to simplify

18k2 + k - 12 - 1k3 - 4k2 - 7k + 152. Write your

answer as a polynomial in standard form.

510 Chapter 8  Mid-Chapter Quiz

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MP 7

You can sometimes write a trinomial as the product of two binomial factors. You can
use algebra tiles to find the factors by arranging all of the tiles to form a rectangle. The
lengths of the sides of the rectangle are the factors of the trinomial.

Write x2 + 7x + 12 as the product of two binomial factors.

Model of polynomial

x2 ϩ 7x ϩ 12

Use the tiles to form a rectangle.

First try: Second try: Third try:

xϩ6 xϩ5 xϩ4

xϩ1 xϩ2

xϩ3

There are six tiles left over.

There is one tile too few. Correct! There is the exact number
of tiles needed.
x2 + 7x + 12 = (x + 3) (x + 4)

Exercises

Use algebra tiles to find binomial factors of each trinomial.

1. x2 + 4x + 4 2. x2 + 5x + 6 3. x2 + 10x + 9
4. x2 + 7x + 10 5. x2 + 9x + 14 6. x2 + 8x + 16

7. Reasoning  Explain why you cannot use algebra tiles to represent the trinomial
x2 + 2x + 3 as a rectangle.

Concept Byte  Using Models to Factor 511

8-5 Factoring x2 + bx + c CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

MA-ASFSSE.9A1.12a.A  -InStSeErp.1re.1t ap aIrntsteorfparent epxaprrtessosfioann, such as
etexrpmress, sfaiocnto, rssu,cahnadsctoeerfmfisc,iefanctst.ors, and coefficients.

MP 1, MP 3, MP 4, MP 7, MP 8

Objective To factor trinomials of the form x2 + bx + c

Here’s a hint. The area of the rectangular solar panel is given
What two numbers by the trinomial x2 + 7x + 12. The height of the
have a sum of 7 and solar panel is x + 3. What is an expression for
a product of 12?
the length of the panel? Explain your reasoning.

MATHEMATICAL Essential Understanding  You can write some trinomials of the form x2 + bx + c

PRACTICES

as the product of two binomials.

#To understand how, consider the product of binomials below.

(x + 3)(x + 7) = x2 + (7 + 3)x + 3 7 = x2 + 10x + 21

The coefficient of the trinomial’s x2–term is 1. The coefficient of the trinomial’s x-term,
10, is the sum of the numbers 3 and 7 in the binomials. The trinomial’s constant term,
21, is the product of the same numbers, 3 and 7. To factor a trinomial of the form
x2 + bx + c as the product of binomials, you must find two numbers that have a sum of
b and a product of c.

What is an easy way Problem 1 Factoring x2 + bx + c  Where b + 0, c + 0
to organize your What is the factored form of x 2 + 8x + 15?
factoring?
Use a table to list the List the pairs of factors of 15. Identify the Factors of 15 Sum of Factors
pairs of factors of the pair that has a sum of 8. 1 and 15 16
constant term c and the 3 and 5 8
sums of those pairs of
factors. x2 + 8x + 15 = (x + 3)(x + 5)

Check (x + 3)(x + 5) = x2 + 5x + 3x + 15

= x2 + 8x + 15 ✔

Got It? 1. What is the factored form of r 2 + 11r + 24?

512 Chapter 8  Polynomials and Factoring

Some factorable trinomials have a negative coefficient of x and a positive constant
term. In this case, you need to inspect the negative factors of c to find the factors of
the trinomial.

Why look at pairs Problem 2 Factoring x2 + bx + c  Where b * 0, c + 0
of negative factors What is the factored form of x 2 − 11x + 24?
of 24? List the pairs of negative factors of 24. Identify the pair that has a sum of -11.

You want the factors of Factors of 24 Sum of Factors
24 with a sum of  - 11.
Only two negative Ϫ1 and Ϫ24 Ϫ25

numbers have a positive Ϫ2 and Ϫ12 Ϫ14

product and a negative Ϫ3 and Ϫ8 Ϫ11

sum.

Ϫ4 and Ϫ6 Ϫ10

x2 - 11x + 24 = (x - 3)(x - 8)

Check (x - 3)(x - 8) = x2 - 8x - 3x + 24

= x2 - 11x + 24 ✔

Got It? 2. a. What is the factored form of y2 - 6y + 8?
b. Reasoning  Can you factor x2 - x + 2? Explain.

When you factor trinomials with a negative constant term, you need to inspect pairs of
positive and negative factors of c.

What’s another way Problem 3 Factoring x2 + bx + c  Where c * 0
to do this problem? What is the factored form of x2 + 2x − 15?
Find two positive factors
of 15 that differ by 2. The Identify the pair of factors of -15 that has a sum of 2.
factors are 3 and 5. Then
attach a negative sign to
one of the factors so that Factors of ؊15 Sum of Factors
their sum is positive. You
get -3 and 5. 1 and Ϫ15 Ϫ14

Ϫ1 and 15 14

3 and Ϫ5 Ϫ2

Ϫ3 and 5 2

x2 + 2x - 15 = (x - 3)(x + 5)
Got It? 3. What is the factored form of each polynomial?
a. n2 + 9n - 36 b. c2 - 4c - 21

Lesson 8-5  Factoring x2 + bx + c 513

Problem 4 Applying Factoring Trinomials

Geometry  The area of a rectangle is given by the trinomial x2 − 2x − 35. What are
the possible dimensions of the rectangle? Use factoring.

The area of the rectangle Possible dimensions Area = length * width, so factor the
of the rectangle trinomial for area as the product of
binomials that represent the length
and width.

To factor x2 - 2x - 35, identify the pair of Factors of ؊35 Sum of Factors
factors of -35 that has a sum of -2. 1 and Ϫ35 Ϫ34
34
Ϫ1 and 35 Ϫ2
2
5 and Ϫ7
Column Heads
Ϫ5 and 7

x2 - 2x - 35 = (x + 5)(x - 7)

So the possible dimensions of the rectangle are x + 5 and x - 7.

Got It? 4. A rectangle’s area is x2 - x - 72. What are possible dimensions of the
rectangle? Use factoring.

You can also factor some trinomials that have more than one variable. Consider the
product (p + 9q)(p + 7q).
(p + 9q)(p + 7q) = p2 + 7pq + 9pq + 9q(7q)
= p2 + 16pq + 63q2

This suggests that a trinomial with two variables may be factorable if the first term
includes the square of one variable, the middle term includes both variables, and the
last term includes the square of the other variable.

Problem 5 Factoring a Trinomial With Two Variables
What is the factored form of x2 + 6xy − 55y2?
Is this problem
similar to one you’ve List the pairs of factors of -55. Identify the pair Factors of ؊55 Sum of Factors
seen before? that has a sum of 6. 1 and Ϫ55 Ϫ54
Yes. This problem is Ϫ1 and 55 54
similar to factoring a x2 + 6xy - 55y2 = (x - 5y)(x + 11y) 5 and Ϫ11 Ϫ6
trinomial in one Ϫ5 and 11 6
variable of the form
x2 + bx + c, where
c 6 0.

Got It? 5. What is the factored form of m2 + 6mn - 27n2?

514 Chapter 8  Polynomials and Factoring

Lesson Check Do you UNDERSTAND? MATHEMATICAL

Do you know HOW? PRACTICES
Factor each expression. Check your answer.
1. x2 + 7x + 12 Tell whether the sum of the factors of the constant
2. r 2 - 13r + 42 term should be positive or negative when you factor
3. p2 + 3p - 40 the trinomial.
4. a2 + 12ab + 32b2
6. s2 + s - 30
5. The area of a rectangle is given by the trinomial
n2 - 3n - 28. What are the possible dimensions of 7. w2 + 11w + 18
the rectangle? Use factoring.
8. x2 - x - 20

9. Reasoning  Under what circumstances should you
look at pairs of negative factors of the constant term
when factoring a trinomial of the form x2 + bx + c?

Practice and Problem-Solving Exercises MATHEMATICAL

A Practice Complete. PRACTICES

10. k2 + 5k + 6 = (k + 2)(k +■) See Problems 1 and 2.
12. t 2 - 10t + 24 = (t - 4)(t -■) 11. x2 - 7x + 10 = (x - 5)(x -■)
13. v2 + 12v + 20 = (v + 10)(v +■)

Factor each expression. Check your answer.

14. y2 + 6y + 5 15. t 2 + 10t + 16 16. x2 + 15x + 56
19. q2 - 8q + 12
17. n2 - 15n + 56 18. r 2 - 11r + 24

Complete. See Problem 3.
20. q2 + 3q - 54 = (q - 6)(q +■)
22. n2 - 5n - 50 = (n + 5)(n -■) 21. z2 - 2z - 48 = (z - 8)(z + ■)
23. y2 + 8y - 9 = (y + 9)(y -■)

Factor each expression. Check your answer.

24. r2 + 6r - 27 25. w2 - 7w - 8 26. z2 + 2z - 8
29. n2 - 3n - 10
27. x2 + 5x - 6 28. v2 + 5v - 36

STEM 30. Carpentry  The area of a rectangular desk is given by the trinomial d2 - 7d - 18. See Problem 4.
What are the possible dimensions of the desk? Use factoring.

31. Design  The area of a rectangular rug is given by the trinomial r 2 - 3r - 4. What are
the possible dimensions of the rug? Use factoring.

Lesson 8-5  Factoring x2 + bx + c 515

Choose the correct factored form for each expression. See Problem 5.
B. (k - 7n)(k + 12n)
32. k2 + 5kn - 84n2 A. (k - 7n)(k - 12n) B. (p - 3q)(p + 11q)
B. (x - 4y)(x - 12y)
33. p2 - 8pq - 33q2 A. (p + 3q)(p - 11q)

34. x2 - 16xy + 48y2 A. (x - 4y)(x + 12y)

Factor each expression. 36. g 2 - 12gh + 35h2 37. m2 - 3mn - 28n2
35. r 2 + 19rs + 90s2 39. w 2 - 14wz + 40z2 40. p2 + 11pq + 24q2
38. x2 + 3xy - 18y2

B Apply 41. Writing  Suppose you can factor x2 + bx + c as (x + p)(x + q).

a. Explain what you know about p and q when c 7 0.
b. Explain what you know about p and q when c 6 0.

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

x2 - 10x - 24 = (x - 6)(x - 4)

43. Think About a Plan  The area of a parallelogram is given by the trinomial
x2 - 14x + 24. The base of the parallelogram is x - 2. What is an expression for the
height of the parallelogram?

• What is the formula for the area of a parallelogram?
• How can you tell whether the binomial that represents the height has a positive

or negative constant term?

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

Write the standard form of each polynomial modeled below. Then factor
each expression.

45. 46.

6x2 4x

4x2 10x

9x 6

2x 5

47. Reasoning  Let x2 - 13x - 30 = (x + p)(x + q).
a. What do you know about the signs of p and q?
b. Suppose ͉ p ͉ 7 ͉ q ͉ . Which number, p or q, is a negative integer? Explain.

48. Reasoning  Let x2 + 13x - 30 = (x + p)(x + q).
a. What do you know about the signs of p and q?
b. Suppose ͉ p ͉ 7 ͉ q ͉ . Which number, p or q, is a negative integer? Explain.

516 Chapter 8  Polynomials and Factoring

Factor each expression. 50. g2 - 18g + 45 51. k2 - 18k - 63
49. x2 + 27x + 50 53. s2 - 10st - 75t2 54. h2 + 9hj - 90j2
52. d2 + 30d - 64
57. r6 - 21r 3 + 80
C Challenge Factor each trinomial. 60. p6 + 14p3 - 72

Sample  n6 + n3 - 42 = (n3)2 + n3 - 42
= (n3 - 6)(n3 + 7)

55. x12 + 12x6 + 35 56. t 8 + 5t 4 - 24

58. m10 + 18m5 + 17 59. x12 - 19x6 - 120

Standardized Test Prep

SAT/ACT 61. What is the factored form of x2 + x - 42?

(x - 7)(x - 6) (x - 7)(x + 6) (x + 7)(x - 6) (x + 7)(x + 6)

62. What is the solution of the equation 6x + 7 = 25? 531 8
2 3

Short 63. A museum charges an admission price of $12 per person when you buy tickets
online. There is also a $5 charge per order. You spend $65 purchasing p tickets
Response online. Which equation best represents this situation?

12p + 5 = 65 5p + 12 = 65 12p - 5 = 65 65p + 12 = 5

64. You and your friend bike to school at the rates shown. Who
is faster? Show your work.

You: 7 mi/h Your friend: 11 ft/s

Mixed Review

Simplify each product. 66. (2v - 9)2 See Lesson 8-4.
65. (c + 4)2 67. (3w + 7)(3w - 7)

Solve each equation for x. See Lesson 2-5.
68. ab = dx
69. 8(x - d) = x 70. m = (c + x)
n

Get Ready!  To prepare for Lesson 8-6, do Exercises 71–73.

Find the GCF of the terms of each polynomial. See Lesson 8-2.
73. 6x 3 + 45x 2 + 15
71. 14x 2 + 7x 72. 24x 2 - 30x + 12

Lesson 8-5  Factoring x2 + bx + c 517

8-6 Factoring ax2 + bx + c MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

AM-ASFSSE.9A1.12a.A  -InStSeErp.1re.1t ap aIrntsteorfparent epxaprrtessosfioann, such as
etexrpmres,ssfaiocnto, rssu,cahnadsctoeerfmfisc,iefanctst.ors, and coefficients.

MP 1, MP 2, MP 3, MP 4

Objective To factor trinomials of the form ax2 + bx + c

You did this for An array of three rectangular solar
one panel in panels has area 3x2 + 21x + 36.
Lesson 8-5­—now The height of the array is x + 3.
there are more.
What is the length of the array?

Explain your reasoning.

MATHEMATICAL

PRACTICES Essential Understanding  You can write some trinomials of the form
ax 2 + bx + c as the product of two binomials.

Consider the trinomial 6x 2 + 23x + 7. To factor it, think of 23x as 2x + 21x.

6x2 + 23x + 7 = 6x 2 + 2x + 21x + 7 Rewrite 23x as 2x + 21x.

= 2x(3x + 1) + 7(3x + 1) Factor out the GCF of each pair of terms.

= (2x + 7)(3x + 1) Distributive Property

How do you know to rewrite 23x as 2x + 21x? Notice that multiplying 2 and 21 gives
42, which is the product of the x2-coefficient 6 and the constant term 7. This example
suggests that, to factor a trinomial of the form ax2 + bx + c, you should look for factors
of the product ac that have a sum of b.

Problem 1 Factoring When ac Is Positive
What is the factored form of 5x 2 + 11x + 2?

Will the process still Step 1 Find factors of ac that have sum b. Factors of 10 1, 10 2, 5
work if you write Since ac = 10 and b = 11, find positive
5x 2 + 10x + x + 2? Sum of Factors 11 7
Yes. You can rewrite this factors of 10 that have sum 11.

alternate expression as Step 2 To factor the trinomial, use the factors you
5x(x + 2) + (x + 2), found to rewrite bx.
which equals
(5x + 1)(x + 2). 5x 2 + 11x + 2 = 5x 2 + 1x + 10x + 2 Rewrite bx : 11x = 1x + 10x.

= x(5x + 1) + 2(5x + 1) Factor out the GCF of each pair of terms.

= (x + 2)(5x + 1) Distributive Property

518 Chapter 8  Polynomials and Factoring

Got It? 1. a. What is the factored form of 6x 2 + 13x + 5?
b. Reasoning In ax2 + bx + c, suppose ac is positive and b is negative. What

do you know about the factors of ac? Explain.

Can you apply the Problem 2 Factoring When ac Is Negative
steps for Problem 1 What is the factored form of 3x 2 + 4x − 15?
to this problem? Step 1 Find factors of ac that have sum b. Since ac = -45 and b = 4, find factors

Yes. Your goal is still to of -45 that have sum 4.

find factors of ac that Factors of ؊45 1, −45 −1, 45 3, −15 −3, 15 5, −9 −5, 9
Sum of Factors −44 44 −12 12 −4 4
have sum b. Because
ac 6 0,the factors must
have different signs.

Step 2 To factor the trinomial, use the factors you found to rewrite bx.

3x2 + 4x - 15 = 3x2 - 5x + 9x - 15 Rewrite bx: 4x = -5x + 9x.

= x(3x - 5) + 3(3x - 5) Factor out the GCF of each pair of terms.

= (3x - 5)(x + 3) Distributive Property

Got It? 2. What is the factored form of 10x 2 + 31x - 14?

How can you find the Problem 3 Applying Trinomial Factoring
dimensions of the Geometry  The area of a rectangle is 2x 2 − 13x − 7. What are the possible
rectangle? dimensions of the rectangle? Use factoring.
Factor the rectangle’s
area as the product Step 1 Find factors of ac that have sum b. Since ac = -14 and b = -13, find factors of
of two binomials, one -14 that have sum -13.
of which is the width.
The other must be Factors of ؊14 1, −14 −1, 14 2, −7 −2, 7
the length since
area = length ∙ width. Sum of Factors −13 13 −5 5

Step 2 To factor the trinomial, use the factors you found to rewrite bx.

2x2 - 13x - 7 = 2x2 + x - 14x - 7 Rewrite bx: -13x = x - 14x.

= x(2x + 1) - 7(2x + 1) Factor out the GCF of each pair of terms.

= (2x + 1)(x - 7) Distributive Property

The possible dimensions of the rectangle are 2x + 1 and x - 7.

Got It? 3. The area of a rectangle is 8x 2 + 22x + 15. What are the possible dimensions
of the rectangle? Use factoring.

Lesson 8-6  Factoring ax2 + bx + c 519

To factor a polynomial completely, first factor out the GCF of the polynomial’s terms.
Then factor the remaining polynomial until it is written as the product of polynomials
that cannot be factored further.

Problem 4 Factoring Out a Monomial First
What is the factored form of 18x 2 − 33x + 12?

How can you simplify Factor out the GCF. 18x2 − 33x + 12 = 3(6x2 − 11x + 4)
this problem?
Factor out the GCF of Factor 6x2 -11x + 4. Since Factors of 24 1, 24 2, 12 3, 8 4, 6
the trinomial’s terms. The ac = 24 and b = -11, find Sum of Factors 25 14 11 10
trinomial that remains negative factors of 24 that
is similar to those in have sum -11.
Problems 1–3.
Rewrite the term bx. Then
use the Distributive Property 3(6x2 − 3x − 8x + 4)
to finish factoring. 3[3x(2x − 1) − 4(2x − 1)]
3(3x − 4)(2x − 1)

Got It? 4. What is the factored form of 8x2 - 36x - 20?

Lesson Check Do you UNDERSTAND? MATHEMATICAL

Do you know HOW? PRACTICES
Factor each expression.
5. Reasoning  Explain why you cannot factor the
1. 3x 2 + 16x + 5 trinomial 2x 2 + 7x + 10.
2. 10q 2 + 9q + 2
3. 4w 2 + 4w - 3 6. Reasoning  To factor 8x 2 + bx + 3, a
4. The area of a rectangle is 6x2 - 11x - 72. What student correctly rewrites the trinomial as
8x 2 + px + qx + 3. What is the value of pq?
are the possible dimensions of the rectangle?
Use factoring. 7. Compare and Contrast  How is factoring a trinomial
ax 2 + bx + c when a ≠ 1 different from factoring a
trinomial when a = 1? How is it similar?

Practice and Problem-Solving Exercises MATHEMATICAL

A Practice Factor each expression. 9. 3d 2 + 23d + 14 PRACTICES
12. 6r 2 - 23r + 20
8. 2x 2 + 13x + 6 See Problem 1.
11. 4p 2 + 7p + 3 10. 4n 2 - 8n + 3
13. 8g 2 - 14g + 3

520 Chapter 8  Polynomials and Factoring

Factor each expression. 15. 2k 2 - 13k - 24 See Problem 2.
14. 5z 2 + 19z - 4 18. 4w 2 - 5w - 6 16. 6t 2 + 7t - 5
17. 3x 2 + 23x - 36 19. 4d 2 - 4d - 35

20. Interior Design  The area of a rectangular kitchen tile is 8x 2 + 30x + 7. See Problem 3.
What are the possible dimensions of the tile? Use factoring.

21. Crafts  The area of a rectangular knitted blanket is 15x 2 - 14x - 8. What are the
possible dimensions of the blanket? Use factoring.

Factor each expression completely. See Problem 4.
24. 6s 2 + 57s + 72
22. 12p 2 + 20p - 8 23. 8v 2 + 34v - 30 27. 9r 2 + 3r - 30

25. 20w 2 - 45w + 10 26. 12x 2 - 46x - 8

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

28. 4s 2 + ■s + 10 29. 15v 2 + ■v - 24 30. 35m 2 + ■m - 16

31. 9g 2 + ■g + 4 32. 6n 2 + ■n + 28 33. 8r 2 + ■r - 42

34. Error Analysis  Describe and correct the error 3x 2 - 16x - 12 = 3x2 + 4x - 20x - 12
made in factoring the expression at the right.

35. Think About a Plan  A triangle has area = x(3x + 4) - 4(5x + 3)

9x 2 - 9x - 10. The base of the triangle is 3x - 5. = (x - 4)(3x + 4)(5x + 3)

What is the height of the triangle?

• What is the formula for the area of a triangle?

• How does factoring the given trinomial help you solve the problem?

STEM 36. Carpentry  The top of a rectangular table has an area of 18x2 + 69x + 60. The width
of the table is 3x + 4. What is the length of the table?

37. a. Write each area as a product of two binomials.

x ϩ1

2x ϩ 2 2x

x ϩ
ϩ 4
2

b. Are the products equal?
c. Writing  Explain how the two products you found in part (a) can equal the

same trinomial.

Lesson 8-6  Factoring ax2 + bx + c 521

Factor each expression. 39. 66k2 + 57k + 12 40. 14z 2 - 53z + 14
38. 54x 2 + 87x + 28 42. 21y 2 + 72y - 48 43. 55n2 - 52n + 12
41. 28h2 + 28h - 56 45. 63g 2 - 89g + 30 46. 99v 2 - 92v + 9
44. 36p2 + 114p - 20

47. Reasoning  If a and c in ax2 + bx + c are prime numbers and the trinomial is
factorable, how many positive values are possible for b? Explain your reasoning.

C Challenge Factor each expression. 49. 49p 2 + 63pq - 36q2 50. 108g 2h - 162gh + 54h

48. 56x 3 + 43x 2 + 5x

51. The graph of the function y = x2 + 5x + 6 is shown at the right. y
a. What are the x-intercepts? 2
b. Factor x 2 + 5x + 6.
c. Reasoning  Describe the relationship between the binomial factors

you found in part (b) and the x-intercepts.

Ϫ5 Ϫ3 O x
2

PERFORMANCE TASK

Apply What You’ve Learned MATHEMATICAL

PRACTICES

MP 7

Look back at the information on page 485 about Kelly’s and Roberto’s plan to

combine their plots in a community garden. Kelly’s and Roberto’s plans for

their original plots are shown again below. In the Apply What You’ve Learned

in Lesson 8-3, you wrote trinomials for the area of Kelly’s original plot and the

area of Roberto’s original plot.

Kelly’s Plan Roberto’s Plan
1 ft 1.5 ft

x ft x ft
3 ft 3 ft 3 ft 3 ft

x ft x ft

1 ft    1.5 ft

a. Write a trinomial that represents the total area of the two original plots.

b. Factor the trinomial you wrote in part (a).

c. What do the factors in your answer to part (b) represent in relation to the new plot?
Explain how you know.

522 Chapter 8  Polynomials and Factoring

8-7 Factoring CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Special Cases
MA-ASFSSE.9A1.12a.A  -InStSeErp.1re.1t ap aIrntsteorfparent epxaprrtessosfioann, such
eaxsptreersmsiso, nfa,cstuocrhs,aasntdercmoesf,ffiacicetnotrs., AanlsdocoAe-fSfiSciEe.nAt.s1. bA,lso
MA-ASFSSE.9A1.2.A-SSE.1.1b, MAFS.912.A-SSE.1.2

MP 1, MP 2, MP 3, MP 4

Objective To factor perfect-square trinomials and the differences of two squares

Start with a plan. The diagram shows two adjacent Area Area
What do you need squares and their areas. In terms x2 ϩ 14x ϩ 49 25
to find out about of x, how much taller is the left
each square first? square than the right square?
Explain your reasoning.

MATHEMATICAL

PRACTICES

Essential Understanding  You can factor some trinomials by “reversing” the
rules for multiplying special case binomials that you learned in Lesson 8-4.

For example, recall the rules for finding squares of binomials.

Lesson (a + b)2 = (a + b)(a + b) = a 2 + 2ab + b 2

Vocabulary (a - b)2 = (a - b)(a - b) = a 2 - 2ab + b 2
• perfect-square
Any trinomial of the form a 2 + 2ab + b 2 or a 2 - 2ab + b 2 is a perfect-square trinomial
trinomial because it is the result of squaring a binomial. Reading the equations above from right
• difference of to left gives you rules for factoring perfect-square trinomials.

two squares

Key Concept  Factoring Perfect-Square Trinomials

Algebra  For every real number a and b:

a 2 + 2ab + b 2 = (a + b)(a + b) = (a + b)2
a 2 - 2ab + b 2 = (a - b)(a - b) = (a - b)2
Examples x 2 + 8x + 16 = (x + 4)(x + 4) = (x + 4)2
4n 2 - 12n + 9 = (2n - 3)(2n - 3) = (2n - 3)2

Here is how to recognize a perfect-square trinomial:

• The first and the last terms are perfect squares.
• The middle term is twice the product of one factor from the first term and one

factor from the last term.

Lesson 8-7  Factoring Special Cases 523

Problem 1 Factoring a Perfect-Square Trinomial

Will the answer have What is the factored form of x 2 − 12x + 36?
the form (a + b)2 or
(a − b)2? x 2 - 12x + 36 = x 2 - 12x + 62 Write the last term as a square.
The middle term - 12x
has a negative = x 2 - 2(x)(6) + 62 Does middle term equal - 2ab? - 12x = - 2(x)(6)  ✔

= (x - 6)2 Write as the square of a binomial.

coefficient, so the

fh aacvteortehdeefoxprmres(saio-n wb)il2l. Got It? 1. Wa. hxa2t +is 6thxe+fa9c tored form of each expressbio.n x?2 - 14x + 49

Problem 2 Factoring to Find a Length STEM

Computers  Digital images are composed of thousands of tiny pixels rendered as
squares, as shown below. Suppose the area of a pixel is 4x 2 + 20x + 25. What is the
length of one side of the pixel?

How can you find the
side length?
Since the pixel’s area is
its side length squared,
factor the expression for
area as the square of a
binomial. The binomial is
the side length.

4x 2 + 20x + 25 = (2x)2 + 20x + 52 Write first and last terms as squares.

= (2x)2 + 2(2x)(5) + 52 Does middle term equal 2ab? 20x = 2(2x)(5) ✔

= (2x + 5)2 Write as the square of a binomial.

The length of one side of the pixel is 2x + 5.

Got It? 2. You are building a square patio. The area of the patio is 16m2 - 72m + 81.
What is the length of one side of the patio?

524 Chapter 8  Polynomials and Factoring

Recall from Lesson 8-4 that (a + b)(a - b) = a 2 - b 2. So you can factor a difference of
two squares, a 2 - b 2, as (a + b)(a - b).

Key Concept  Factoring a Difference of Two Squares
Algebra  For all real numbers a and b:

a 2 - b 2 = (a + b)(a - b)
Examples x 2 - 64 = (x + 8)(x - 8)

25x 2 - 36 = (5x + 6)(5x - 6)

Problem 3 Factoring a Difference of Two Squares
What is the factored form of z 2 − 9?

Can you use the rule Rewrite 9 as a square. z2 − 9 = z2 − 32
for the difference of
two squares? Factor using the rule for a = (z + 3)(z − 3)
Yes. The binomial is a difference of two squares.
difference and both (z + 3)(z − 3) = z 2 − 3z + 3z − 9
its terms are perfect Check your answer by = z2 − 9 ✔
squares. multiplying the factored
form.

Got It? 3. What is the factored form of each expression?
a. v 2 - 100 b. s 2 - 16

Problem 4 Factoring a Difference of Two Squares

When is a term of the What is the factored form of 16x 2 − 81?
form ax 2 a perfect
16x 2 - 81 = (4x)2 - 92 Write each term as a square.

square? = (4x + 9)(4x - 9) Use the rule for the difference of squares.
ax 2 is a perfect square

when a is a perfect is the factored form of 25d 2 - 64?

s quare. For example, Got It? 4. a. What
1 6x2 is a perfect square b. Reasoning  The expression 25d 2 + 64 contains two perfect squares. Can
but 17x2 is not.
you use the method in Problem 4 to factor it? Explain your reasoning.

Lesson 8-7  Factoring Special Cases 525

When you factor out the GCF of a polynomial, sometimes the expression that remains
is a perfect-square trinomial or the difference of two squares. You can then factor this
expression further using the rules from this lesson.

Problem 5 Factoring Out a Common Factor

Is 24g2 − 6 a What is the factored form of 24g 2 − 6?

difference of two 24g 2 - 6 = 6(4g 2 - 1) Factor out the GCF, 6.

squares? = 6[(2g)2 - 12] Write the difference as a2 - b2.
No. 24g 2 and 6 are not
= 6(2g + 1)(2g - 1) Use the rule for the difference of squares.
perfect squares. To get

a difference of squares,

yt hoeuGmCuFs.t first factor out Got It? 5. Wa. h1a2tti2s t-he48fa ctored form of each expressbio.n 1?2x 2 + 12x + 3

Lesson Check Do you UNDERSTAND? MATHEMATICAL

Do you know HOW? PRACTICES
Factor each expression.
1. y 2 - 16y + 64 Identify the rule you would use to factor each expression.
2. 9q 2 + 12q + 4
3. p 2 - 36 5. 81r 2 - 90r + 25

4. The area of a square is 36w 2 + 60w + 25.What is the 6. k 2 + 12k + 36
side length of the square?
7. 9h 2 - 64

8. Reasoning  Explain how to determine whether a
binomial is a difference of two squares.

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Factor each expression. 10. v 2 - 10v + 25 See Problems 1 and 2.
13. q 2 + 2q + 1
9. h 2 + 8h + 16 16. 4r 2 + 36r + 81 11. d 2 - 20d + 100
12. m 2 + 18m + 81 19. 25z 2 + 40z + 16 14. p 2 - 4p + 4
15. 64x 2 + 112x + 49 17. 9n 2 - 42n + 49
18. 36s 2 - 60s + 25 20. 49g 2 - 84g + 36

The given expression represents the area. Find the side length of the square.

21. 22. 23.

100r2 Ϫ 220r ϩ 121 64r2 Ϫ 144r ϩ 81 25r2 ϩ 30r ϩ 9
526 Chapter 8  Polynomials and Factoring

Factor each expression. 25. a 2 - 49 See Problems 3–5.
28. k 2 - 64 26. y 2 - 121
24. w 2 - 144 31. 81r 2 - 1 29. m 2 - 225
27. t 2 - 25 34. 16x 2 - 121 32. 36v 2 - 25
30. 4p 2 - 49 37. 27w 2 - 12 35. 9n 2 - 400
33. 64q 2 - 81 38. 80g 2 - 45
36. 2h 2 - 2

B Apply 39. Rewrite the expression x4 - y4 so that it is a difference of squares. Then factor the expression

completely.

40. Error Analysis  Describe and correct the error made
in factoring.

9x2 - 49 = (9x + 7)(9x - 7)

41. Writing  Summarize the procedure for factoring a 25x2 + 40x + 16 x2 – 18x + 81
difference of two squares. Give at least two examples.

42. Think About a Plan  Two square windows and their areas are
shown at the right. What is an expression that represents the
difference of the areas of the windows? Show two different
ways to find the solution.

• How can you solve the problem without factoring?
• How can you use the factored forms of the areas to find the

difference of the areas of the windows?

43. Interior Design  A square rug has an area of
49x 2 - 56x + 16. A second square rug has an area of
16x 2 + 24x + 9. What is an expression that represents the
difference of the areas of the rugs? Show two different ways to
find the solution.

Mental Math  For Exercises 44–48, find a pair of factors for each number by
using the difference of two squares.

Sample  117 = 121 - 4 Write 117 as the difference of two squares.
= 112 - 22 Write each term as a square.
= (11 + 2)(11 - 2) Use the rule for the difference of squares.
   = (13)(9) Simplify.

44. 143 45. 99 46. 224 47. 84 48. 91

49. a. Open-Ended  Write an expression that is a perfect-square trinomial.
b. Explain how you know your trinomial is a perfect-square trinomial.

Lesson 8-7  Factoring Special Cases 527

50. a. Factor 4x 2 - 100 by removing the common monomial factor and then using the
difference-of-squares rule to factor the remaining expression.

b. Factor 4x 2 - 100 by using the difference-of-squares rule and removing the
common monomial factors.

c. Reasoning  Why can you factor 4x 2 - 100 in two different ways?
d. Can you factor 3x2 - 75 in the two ways you factored 4x 2 - 100 in parts (a)

and (b)? Explain your answer.

C Challenge Factor each expression. 52. p6 + 40p 3q + 400q 2 53. 36m 4 + 84m 2 + 49
55. x 20 - 4x 10y 5 + 4y 10 56. 256g 4 - 100h 6
51. 64r 6 - 144r 3 + 81

54. 108n6 - 147

57. The binomial 16 - 81n4 can be factored twice using the difference-of-squares rule.
a. Factor 16 - 81n4 completely.
b. Reasoning  What characteristics do 16 and 81n4 share that make this possible?
c. Open-Ended  Write another binomial that can be factored twice using the

difference of squares rule.

Standardized Test Prep

SAT/ACT 58. What is the factored form of 4x 2 - 20x + 25?

(2x + 5)(2x - 5) (2x - 5)(2x - 5) (4x - 5)(4x - 5) (4x + 5)(4x - 5)

59. Which equation has -2 as its solution?

x + 3 = 2x + 1 x - 5 = 2x - 7 2x + 5 = 5x + 11 3x + 1 = x - 5

60. Which equation illustrates the Commutative Property of Multiplication?

ab = ba a(bc) = (ab)c ab = ab a(b + c) = ab + ac

Short 61. A film club sponsors a film fest at a local movie theater. Renting the theater costs
Response $190. The admission is $2 per person.

a. Write an equation that relates the film club’s total cost c and the number of

people p who attend the film fest.
b. Graph the equation you wrote in part (a).

Mixed Review 63. 8x 2 + 18x + 9 See Lesson 8-6.
64. 12x 2 - 41x + 35
Factor each expression.
62. 18x 2 + 9x - 14

G et Ready!  To prepare for Lesson 8-8, do Exercises 65–67.

Find the GCF of the terms of each polynomial. See Lesson 8-2.
67. 16h 4 - 12h 3 - 36h 2
65. 6t 2 + 12t - 4 66. 9m 3 + 15m 2 - 21m

528 Chapter 8  Polynomials and Factoring

8-8 Factoring by Grouping CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

AM-ASFSSE.9A1.12a.A  -InStSeErp.1re.1t ap aIrntsteorfparent epxaprrtessosfioann, such
eaxsptreersmsiso, nfa,cstuocrhs,aasntdercmoesf,ffiacicetnotrs., AanlsdocoAe-fSfiSciEe.nAt.s1. bA,lso
MA-ASFSSE.9A1.2.A-SSE.1.1b, MAFS.912.A-SSE.1.2

MP 1, MP 2, MP 3, MP 4, MP 7

Objective To factor higher-degree polynomials by grouping

A packaging company sells two kinds of boxes,

Box A and Box B. The company is designing

a new box, Box C, that will have the same C

What information volume as Boxes A and B A
do you need combined. Suppose one
before you can
find the other two dimension of Box C is x.
side lengths?
What could be the other ?
MATHEMATICAL B
two dimensions? Explain 3x + 7
PRACTICES your reasoning.

x+3

xx x x+3 x?

Lesson Essential Understanding  Some polynomials of a degree greater than 2 can
be factored.
Vocabulary
• factoring by In Lesson 8-6, you factored trinomials of the form ax 2 + bx + c by rewriting bx as a
sum of two monomials. You then grouped the terms in pairs, factored the GCF from
grouping each pair, and looked for a common binomial factor. This process is called factoring by
grouping. You can extend this technique to higher-degree polynomials.

Problem 1 Factoring a Cubic Polynomial

How should you What is the factored form of 3n3 − 12n2 + 2n − 8?
group the terms of
the polynomial? 3n3 - 12n2 + 2n - 8 = 3n2(n - 4) + 2(n - 4) Fgaroctuopr out the GCF of each
of two terms.

First group the two terms = (3n2 + 2)(n - 4) Factor out the common factor n - 4.
with the highest degrees.
If that doesn’t work, try Check  (3n2 + 2)(n - 4) = 3n3 - 12n2 + 2n - 8 ✔
another grouping. Your

gbionaolmisiatol ffainctdoar. common Got It? 1. ba.. RWehaastoinsitnhge  fHacotworiesdthfoerfmacotof r8itn3g+m1e4tht 2od+ 20t + 35? 1 like
used in Problem

the method used in Lesson 8-6? How is it different?

Lesson 8-8  Factoring by Grouping 529 RM
hs

Before factoring by grouping, you may need to factor out the GCF of all the terms.

Problem 2 Factoring a Polynomial Completely

Do the terms share What is the factored form of 4q4 − 8q3 + 12q2 − 24q? Factor completely.
any numerical or
4q4 - 8q3 + 12q2 - 24q = 4q1q3 - 2q2 + 3q - 62 Factor out the GCF.
variable factors?
Yes. The terms have a = 4q3q2(q - 2) + 3(q - 2)4 Factor by grouping.

common numerical factor = 4q1q2 + 32(q - 2) Factor again.

of 4 and a common

vT harei aGbCleFfiasc4toqr. of q. Got It? 2. What is the factored form of 6h4 + 9h3 + 12h2 + 18h? Factor completely.

You can sometimes factor to find possible expressions for the length, width, and height
of a rectangular prism.

Problem 3 Finding the Dimensions of a Rectangular Prism
Entertainment  The toy shown below is made of several bars that can fold together
to form a rectangular prism or unfold to form a “ladder.” What expressions can
represent the dimensions of the toy when it is folded up? Use factoring.

V = 6x3 + 19x2 + 15x

How can you find the Step 1 Factor out the GCF.
prism’s dimensions? 6x3 + 19x2 + 15x = x(6x 2 + 19x + 15)
Factor the cubic
expression for the Step 2 To factor the trinomial, find factors of ac that have sum b.
volume of the prism Since ac = 90 and b = 19, find factors of 90 that have sum 19.
as the product of three
linear expressions. Each Factors of 90 1, 90 2, 45 3, 30 5, 18 6, 15 9, 10
linear expression is a Sum of Factors 91 47 33 23 21 19
dimension.
Step 3 To factor the trinomial, use the factors you found to rewrite bx.

x(6x 2 + 19x + 15) = x(6x 2 + 9x + 10x + 15) Rewrite bx: 19x = 9x + 10x.

= x[3x(2x + 3) + 5(2x + 3)] Factor by grouping.

= x(3x + 5)(2x + 3) Distributive Property

The possible dimensions are x, 3x + 5, and 2x + 3.

530 Chapter 8  Polynomials and Factoring

Got It? 3. Geometry  A rectangular prism has volume 60x 3 + 34x 2 + 4x. What
expressions can represent the dimensions of the prism? Use factoring.

Here is a summary of what to remember as you factor polynomials.

Summary  Factoring Polynomials
1. Factor out the greatest common factor (GCF).
2. If the polynomial has two terms or three terms, look for a difference of two squares,

a perfect-square trinomial, or a pair of binomial factors.
3. If the polynomial has four or more terms, group terms and factor to find common

binomial factors.
4. As a final check, make sure there are no common factors other than 1.

Lesson Check Do you UNDERSTAND? MATHEMATICAL

Do you know HOW? PRACTICES
Factor each expression.
1. 20r 3 + 8r 2 + 15r + 6 Vocabulary  Tell whether you would factor the
2. 6d 3 + 3d 2 - 10d - 5 polynomial by grouping. Explain your answer.
3. 24x 3 + 60x 2 + 36x + 90
5. x 2 - 6x + 9
4. A rectangular prism has a volume of
36x 3 + 36x 2 + 8x. What expressions can represent 6. 4w 2 + 23w + 15
the dimensions of the prism? Use factoring.
7. 24t 3 - 42t 2 - 28t + 49

8. Reasoning  Can you factor the polynomial
6q 3 + 2q 2 + 12q - 3 by grouping? Explain.

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Find the GCF of the first two terms and the GCF of the last two terms for See Problem 1.
each polynomial.

9. 2z 3 + 6z 2 + 3z + 9 10. 10g 3 - 25g 2 + 4g - 10

11. 2r3 + 12r 2 - 5r - 30 12. 6p 3 + 3p 2 + 2p + 1

Factor each expression. 14. 14y 3 + 8y 2 + 7y + 4 15. 14z 3 - 35z 2 + 16z - 40
13. 15q 3 + 40q 2 + 3q + 8 17. 8m 3 + 12m 2 - 2m - 3 18. 12k 3 - 27k 2 - 40k + 90
16. 11w 3 - 9w 2 + 11w - 9 20. 18h 3 + 45h 2 - 8h - 20 21. 12y 3 + 4y 2 - 9y - 3
19. 20v 3 + 24v 2 - 25v - 30

Lesson 8-8  Factoring by Grouping 531

Factor completely. 23. 3w 4 - 2w 3 + 18w 2 - 12w See Problem 2.
22. 8p 3 - 32p 2 + 28p - 112 26. 36v 3 - 126v 2 + 48v - 168 24. 5g 4 - 5g 3 + 20g 2 - 20g
25. 6q 4 + 3q 3 - 24q 2 - 12q 27. 4d 3 - 6d 2 + 16d - 24

Find expressions for the possible dimensions of each rectangular prism. See Problem 3.

28. 29.

V ϭ 3y3 ϩ 14y2 ϩ 8y V ϭ 4c3 ϩ 52c2 ϩ 160c

STEM 30. Carpentry  A trunk in the shape of a rectangular prism has a volume of
6x 3 + 38x 2 - 28x. What expressions can represent the dimensions of the trunk?

B Apply Factor completely.

31. 9t 3 - 90t 2 + 144t 32. 60y 4 - 300y 3 - 42y 2 + 210y
34. 10p 2 - 5pq - 180q 2
33. 8m 3 + 32m 2 + 40m + 160

35. Error Analysis  Describe and correct the error made in factoring completely.

4x4 + 12x3 + 8x2 + 24x = 4(x4 + 3x3 + 2x2 + 6x)
= 4[x3(x + 3) + 2x(x + 3)]
= 4(x3 + 2x)(x + 3)

36. a. Factor (20x 3 - 5x 2) + (44x - 11).
b. Factor (20x 3 + 44x) + ( - 5x 2 - 11).
c. Reasoning  Why can you factor the same polynomial using different pairs

of terms?

37. Writing  Describe how to factor the expression 6x 5 + 4x 4 + 12x 3 + 8x 2 + 9x + 6.

38. Think About a Plan  Bat houses, such as the one at the right, are large
wooden structures that people mount on buildings to attract bats.
What expressions can represent the dimensions of the bat house?

• Into how many factors should you factor the expression for
the volume?

• What is the first step in factoring this expression?

39. Open-Ended  Write a four-term polynomial that you can factor by
grouping. Factor your polynomial.

40. Art  The pedestal of a sculpture is a rectangular prism with a volume V = 4x3 + 22x2 + 24x
of 63x 3 - 28x. What expressions can represent the dimensions of the
pedestal? Use factoring.

532 Chapter 8  Polynomials and Factoring

C Challenge Factor by grouping. 42. p 2m + p 2n 5 + qm + qn 5 43. 30g 5 + 24g 3h - 35g 2h 2 - 28h 3
41. y 3 + 11y 2 - 4y - 44


44. Geometry  The polynomial 2px 3 + 12px 2 + 18px represents the volume of a
cylinder. The formula for the volume V of a cylinder with radius r and height h
is V = pr 2h.

a. Factor 2px 3 + 12px 2 + 18px.
b. Based on your answer to part (a), write an expression for a possible radius of

the cylinder.

You can write the number 63 as 25 + 24 + 23 + 22 + 21 + 20. For Exercises
45 and 46, factor each expression by grouping. Then simplify the powers of 2 to
write 63 as the product of two numbers.

45. 125 + 24 + 232 + 122 + 21 + 202 46. 125 + 242 + 123 + 222 + 121 + 202

Standardized Test Prep

SAT/ACT 47. What is 30z 3 - 12z 2 + 120z - 48 factored completely?

2(15z 3 - 6z 2 + 60z - 24) 6(5z 3 - 2z 2 + 20z - 8)

(6z 2 + 24)(5z - 2) 6(z 2 + 4)(5z - 2)

#48. What is the simplified form of 2x3 x8? 2x 24 8x 24
2x 11 8x 11

49. Which equation represents the line with slope -3 that passes through (2, 5)?

y = -3x + 17 y = -3x + 11 y = 4x - 3 y = x - 3
Short
Response 50. What is the solution of the inequality 7 6 -2x + 5?

x 7 -1 x 6 -1 x 7 1 x 6 1

51. Factor 10r 4 + 30r 3 + 5r 2 + 15r completely. Show your work.

Mixed Review

Factor each expression. See Lesson 8-7.
52. m 2 + 12m + 36
53. 64x 2 - 144x + 81 54. 49p 2 - 4

Use a mapping diagram to determine whether each relation is a function. See Lesson 4-6.

55. 5(4, 3), (3, 4), (4, 7), (7, 4)6 56. 5( -1, 8), (1, 8), (3, 8), (5, 8)6 57. 5(2, 7), (4, -7), (6, 7), (8, -7)6

Get Ready!  To prepare for Lesson 9-1, do Exercises 58–61.

Use the slope and y-intercept to graph each equation. See Lesson 5-3.
61. y = - 35x + 2
58. y = 12x + 3 59. y = -4x - 1 60. y = 2x - 3

Lesson 8-8  Factoring by Grouping 533