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8-3 Multiplying Polynomials
ANSWER: 2
4x + 72x + 320
Find each product.
1. (x + 5)(x + 2)
ANSWER: 2
x + 7x + 10
2. (y − 2)(y + 4)
ANSWER: 2
y + 2y − 8
Find each product.
2
8. (2a − 9)(3a + 4a − 4)
ANSWER: 3
2
6a − 19a − 44a + 36
2
2
9. (4y − 3)(4y + 7y + 2)
ANSWER: 4
3
2
16y + 28y − 4y − 21y − 6
3. (b − 7)(b + 3)
ANSWER: 2
b − 4b − 21
2
2
10. (x − 4x + 5)(5x + 3x − 4)
ANSWER: 4
3
4. (4n + 3)(n + 9)
ANSWER: 2
4n + 39n + 27
2
5x − 17x + 9x + 31x − 20
2
2
11. (2n + 3n − 6)(5n − 2n − 8)
ANSWER: 4
3
2
10n + 11n − 52n − 12n + 48
5. (8h − 1)(2h − 3)
ANSWER: 2
16h − 26h + 3
6. (2a + 9)(5a − 6)
Find each product.
12. (3c − 5)(c + 3)
ANSWER: 2
3c + 4c − 15
ANSWER: 2
10a + 33a − 54
7. FRAME Hugo is designing a frame as shown. The
frame has a width of x inches all the way around.
Write an expression that represents the total area of
the picture and frame.
13. (g + 10)(2g − 5)
ANSWER: 2
2g + 15g − 50
14. (6a + 5)(5a + 3)
ANSWER: 2
30a + 43a + 15
15. (4x + 1)(6x + 3)
ANSWER: 2
24x + 18x + 3
ANSWER: 2
4x + 72x + 320
16. (5y − 4)(3y − 1)
ANSWER: 2
15y − 17y + 4
Find each product.
2
8. (2a − 9)(3a + 4a − 4)
ANSWER: 3
17. (6d − 5)(4d − 7)
ANSWER: 2
2
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24d − 62d + 35
18. (3m + 5)(2m + 3)
2
2
9. (4y − 3)(4y + 7y + 2)
ANSWER: Page 1
2
25. (2y − 11)(y − 3y + 2)
16. (5y − 4)(3y − 1)
ANSWER: ANSWER: 8-3 Multiplying Polynomials
2
15y − 17y + 4
3
2
2y − 17y + 37y − 22
2
17. (6d − 5)(4d − 7)
26. (4a + 7)(9a + 2a − 7)
ANSWER: ANSWER: 2
3
24d − 62d + 35
2
36a + 71a − 14a − 49
18. (3m + 5)(2m + 3)
2
2
27. (m − 5m + 4)(m + 7m − 3)
ANSWER: ANSWER: 2
6m + 19m + 15
4
3
2
m + 2m − 34m + 43m − 12
19. (7n − 6)(7n − 6)
2
2
28. (x + 5x − 1)(5x − 6x + 1)
ANSWER: ANSWER: 2
49n − 84n + 36
4
3
2
5x + 19x − 34x + 11x − 1
20. (12t − 5)(12t + 5)
3
2
29. (3b − 4b − 7)(2b − b − 9)
ANSWER: ANSWER: 2
144t − 25
5
4
3
2
6b − 3b − 35b − 10b + 43b + 63
21. (5r + 7)(5r − 7)
2
3
30. (6z − 5z − 2)(3z − 2z − 4)
ANSWER: 2
ANSWER: 25r − 49
5
4
3
2
18z − 15z − 18z − 14z + 24z + 8
22. (8w + 4x)(5w − 6x)
Simplify.
ANSWER: 2
2
40w − 28wx − 24x
2
2
31. (m + 2)[(m + 3m − 6) + (m − 2m + 4)]
ANSWER: 23. (11z − 5y)(3z + 2y)
3
2
2m + 5m − 4
ANSWER: 2
33z + 7yz − 10y
2
2
2
32. [(t + 3t − 8) − (t − 2t + 6)](t − 4)
ANSWER: 24. GARDEN A walkway surrounds a rectangular
garden. The width of the garden is 8 feet, and the
length is 6 feet. The width x of the walkway around
the garden is the same on every side. Write an
expression that represents the total area of the
garden and walkway.
2
5t − 34t + 56
CCSS STRUCTURE Find an expression to represent the area of each shaded region.
ANSWER: 2
4x + 28x + 48
Find each product.
2
25. (2y − 11)(y − 3y + 2)
ANSWER: 3
33. ANSWER: 2
2
4πx + 12πx + 9π − 3x − 5x − 2
2
2y − 17y + 37y − 22
2
26. (4a + 7)(9a + 2a − 7)
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ANSWER: 3
2
36a + 71a − 14a − 49
Page 2
33. 39. (x − 5y)
ANSWER: 2
2
8-3 Multiplying
Polynomials
4πx + 12πx + 9π − 3x − 5x − 2
2
ANSWER: 2
x − 10xy + 25y
40. (2r − 3t)
2
3
ANSWER: 3
2
2
3
8r − 36r t + 54rt − 27t
34. 41. (5g + 2h)
ANSWER: 3
ANSWER: 2
24x −
35. VOLLEYBALL The dimensions of a sand
volleyball court are represented by a width of 6y − 5
feet and a length of 3y + 4 feet.
42. (4y + 3z)(4y − 3z)
2
ANSWER: 3
2
2
64y − 48y z − 36yz + 27z
3
a. Write an expression that represents the area of
the court.
b. The length of a sand volleyball court is 31 feet.
Find the area of the court.
43. CONSTRUCTION A sandbox kit allows you to
build a square sandbox or a rectangular sandbox as shown.
ANSWER: a. 18y 2 + 9y − 20
2
b. 1519 ft
36. GEOMETRY Write an expression for the area of
a triangle with a base of 2x + 3 and a height of 3x −
1.
a. What are the possible values of x? Explain.
ANSWER: b. Which shape has the greater area?
c. What is the difference in areas between the two?
Find each product.
2
37. (a − 2b)
ANSWER: a. x > 4; If x = 4 the width of the rectangular
sandbox would be zero and if x < 4 the width of the
rectangular sandbox would be negative.
ANSWER: 2
a − 4ab + 4b
2
38. (3c + 4d)
b. square
2
ANSWER: 2
2
9c + 24cd + 16d
39. (x − 5y)
c. 4 ft
2
2
44. MULTIPLE REPRESENTATIONS In this
problem, you will investigate the square of a sum.
ANSWER: 2
x − 10xy + 25y
40. (2r − 3t)
2
3
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ANSWER: 3
2
2
3
8r − 36r t + 54rt − 27t
a. TABULAR Copy and complete the table for
each sum.
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square
quantities), and apply the FOIL method. For
2
2
2
example, (2x + 3)( x + 5x + 7) = (2x + 3)[ x + (5x
2
2
+ 7)] = 2x(x ) + 2x(5x + 7) + 3(x ) + 3(5x + 7). Then
use the Distributive Property and simplify.
c. 4 ft
8-3 Multiplying
Polynomials
m
44. MULTIPLE REPRESENTATIONS In this
problem, you will investigate the square of a sum.
p
46. CHALLENGE Find (x + x )(x
m−1
−x
1−p
p
+ x ).
ANSWER: a. TABULAR Copy and complete the table for
each sum.
47. OPEN ENDED Write a binomial and a trinomial
involving a single variable. Then find their product.
ANSWER: 2
2
Sample answer: x − 1, x − x − 1. (x − 1) · (x − x −
3
2
1) = x − 2x + 1
b. VERBAL Make a conjecture about the terms of
the square of a sum.
c. SYMBOLIC For a sum of the form a + b, write
an expression for the square of the sum.
ANSWER: a.
b. The first term of the square of a sum is the first
term of the sum squared. The middle term of the sum
is two times the first term of the sum multiplied by
the last term of the sum. The third term of the square
of the sum is the last term of the sum squared.
c. a 2 + 2ab + b 2
45. REASONING Determine if the following statement
is sometimes, always, or never true. Explain your
reasoning.
The FOIL method can be used to multiply a
binomial and a trinomial.
ANSWER: Always; by grouping two adjacent terms, a trinomial
can be written as a binomial (the sum of two
quantities), and apply the FOIL method. For
2
2
example, (2x + 3)( x + 5x + 7) = (2x + 3)[ x + (5x
2
2
+ 7)] = 2x(x ) + 2x(5x + 7) + 3(x ) + 3(5x + 7). Then
use the Distributive Property and simplify.
m
p
46. CHALLENGE Find (x + x )(x
m−1
−x
1−p
p
+ x ).
ANSWER: eSolutions Manual - Powered by Cognero
47. OPEN ENDED Write a binomial and a trinomial
involving a single variable. Then find their product.
48. CCSS REGULARITY Compare and contrast the
procedure used to multiply a trinomial by a binomial
using the vertical method with the procedure used to
multiply a three-digit number by a two-digit number.
ANSWER: The three monomials that make up the trinomial are
similar to the three digits that make up the 3-digit
number. The single monomial is similar to a 1-digit
number. With each procedure you perform 3
multiplications. The difference is that polynomial
multiplication involves variables and the resulting
product is often the sum of two or more monomials,
while numerical multiplication results in a single
number.
49. WRITING IN MATH Summarize the methods
that can be used to multiply polynomials.
ANSWER: The Distributive Property can be used with a vertical
or horizontal format by distributing, multiplying, and
combining like terms. The FOIL method is used with
a horizontal format. You multiply the first, outer,
inner, and last terms of the binomials and then
combine like terms. A rectangular method can also
be used by writing the terms of the polynomials along
the top and left side of a rectangle and then
multiplying the terms and combining like terms.
50. What is the product of 2x − 5 and 3x + 4?
A 5x − 1
B 6x2 − 7x − 20
2
C 6x − 20
D 6x2 + 7x − 20
ANSWER: B
51. Which statement is correct about the symmetry of
this design?
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2
C 6x − 20
D 6x2 + 7x − 20
D T
ANSWER: Polynomials
8-3 Multiplying
B
ANSWER: D
51. Which statement is correct about the symmetry of
this design?
53. SHORT RESPONSE For a science project, Jodi
selected three bean plants of equal height. Then, for
five days, she measured their heights in centimeters
and plotted the values on the graph below.
F The design is symmetrical only about the y-axis.
G The design is symmetrical only about the x-axis.
H The design is symmetrical about both the y- and
the x-axes.
She drew a line of best fit on the graph. What is the
slope of the line that she drew?
ANSWER: J The design has no symmetry.
ANSWER: F
52. Which point on the number line represents a number
that, when cubed, will result in a number greater than
itself?
54. SAVINGS Carrie has $6000 to invest. She puts x
dollars of this money into a savings account that
earns 2% interest per year. She uses the rest of the
money to purchase a certificate of deposit that earns
4% interest. Write an equation for the amount of
money that Carrie will have in one year.
ANSWER: T = 1.02x + 1.04(6000 − x)
A P
Find each sum or difference.
2
2
55. (7a − 5) + (−3a + 10)
B Q
ANSWER: 4a + 5
C R
2
D T
ANSWER: D
53. SHORT RESPONSE For a science project, Jodi
selected three bean plants of equal height. Then, for
five days, she measured their heights in centimeters
and plotted the values on the graph below.
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2
2
56. (8n − 2n ) + (4n − 6n )
ANSWER: 12n − 8n
2
3
2
3
2
57. (4 + n + 3n ) + (2n − 9n + 6)
ANSWER: 3
2
3n − 6n + 10
2
2
58. (−4u − 9 + 2u) + (6u + 14 + 2u )
ANSWER: 2
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3
2
3
2
57. (4 + n + 3n ) + (2n − 9n + 6)
ANSWER: 8-3 Multiplying Polynomials
3
2
3n − 6n + 10
2
2
58. (−4u − 9 + 2u) + (6u + 14 + 2u )
ANSWER: 2
−2u + 8u + 5
59. (b + 4) + (c + 3b − 2)
ANSWER: 4b + c + 2
3
3
60. (3a − 6a) − (3a + 5a)
ANSWER: −11a
3
3
2
61. (−4m − m + 10) − (3m + 3m − 7)
ANSWER: 3
2
−7m − 3m − m + 17
62. (3a + 4ab + 3b) − (2b + 5a + 8ab)
ANSWER: −2a − 4ab + b
Simplify.
4 3
3 4
63. (−2t ) − 3(−2t )
ANSWER: 12
−56t
2 3
3 2
64. (−3h ) − 2(−h )
ANSWER: −29h
6
3 2
3 3
65. 2(−5y ) + (−3y )
ANSWER: 6
50y − 27y
9
4 2
2 2
66. 3(−6n ) + (−2n )
ANSWER: 8
108n + 4n
4
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