6. (2a + 9)(5a − 6)
SOLUTION: 8-3 Multiplying Polynomials
Find each product.
1. (x + 5)(x + 2)
SOLUTION: 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.
2. (y − 2)(y + 4)
SOLUTION: SOLUTION: The total length is 2x + 20 and the width is 2x + 16.
3. (b − 7)(b + 3)
SOLUTION: Find each product.
4. (4n + 3)(n + 9)
SOLUTION: 2
8. (2a − 9)(3a + 4a − 4)
SOLUTION: 5. (8h − 1)(2h − 3)
SOLUTION: 2
2
9. (4y − 3)(4y + 7y + 2)
6. (2a + 9)(5a − 6)
SOLUTION: SOLUTION: 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.
2
2
10. (x − 4x + 5)(5x + 3x − 4)
SOLUTION: eSolutions Manual - Powered by Cognero
SOLUTION: Page 1
13. (g + 10)(2g − 5)
SOLUTION: 8-3 Multiplying Polynomials
2
2
10. (x − 4x + 5)(5x + 3x − 4)
14. (6a + 5)(5a + 3)
SOLUTION: SOLUTION: 15. (4x + 1)(6x + 3)
SOLUTION: 16. (5y − 4)(3y − 1)
SOLUTION: 2
2
11. (2n + 3n − 6)(5n − 2n − 8)
SOLUTION: 17. (6d − 5)(4d − 7)
SOLUTION: 18. (3m + 5)(2m + 3)
SOLUTION: 19. (7n − 6)(7n − 6)
Find each product.
12. (3c − 5)(c + 3)
SOLUTION: SOLUTION: 20. (12t − 5)(12t + 5)
SOLUTION: 13. (g + 10)(2g − 5)
SOLUTION: 21. (5r + 7)(5r − 7)
SOLUTION: 14. (6a + 5)(5a + 3)
SOLUTION: 22. (8w + 4x)(5w − 6x)
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15. (4x + 1)(6x + 3)
SOLUTION: Page 2
21. (5r + 7)(5r − 7)
SOLUTION: 8-3 Multiplying Polynomials
22. (8w + 4x)(5w − 6x)
SOLUTION: 2
26. (4a + 7)(9a + 2a − 7)
SOLUTION: 23. (11z − 5y)(3z + 2y)
SOLUTION: 2
2
27. (m − 5m + 4)(m + 7m − 3)
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.
SOLUTION: SOLUTION: Let 2x + 8 = the width of the garden and walkway
and let 2x + 6 = the length of the garden and
walkway.
2
Find each product.
2
25. (2y − 11)(y − 3y + 2)
2
28. (x + 5x − 1)(5x − 6x + 1)
SOLUTION: SOLUTION: 2
26. (4a + 7)(9a + 2a − 7)
2
SOLUTION: SOLUTION: eSolutions Manual - Powered by Cognero
2
3
29. (3b − 4b − 7)(2b − b − 9)
2
27. (m − 5m + 4)(m + 7m − 3)
Page 3
8-3 Multiplying Polynomials
3
2
CCSS STRUCTURE Find an expression to represent the area of each shaded region.
29. (3b − 4b − 7)(2b − b − 9)
SOLUTION: 33. SOLUTION: Find the area of the circle.
2
3
30. (6z − 5z − 2)(3z − 2z − 4)
SOLUTION: Find the area of the rectangle.
Simplify.
2
2
31. (m + 2)[(m + 3m − 6) + (m − 2m + 4)]
SOLUTION: Subtract the area of the rectangle from the area of
the circle.
2
2
32. [(t + 3t − 8) − (t − 2t + 6)](t − 4)
The area of the shaded region is represented by the
SOLUTION: 2
2
expression 4πx + 12πx + 9π − 3x − 5x − 2.
CCSS STRUCTURE Find an expression to represent the area of each shaded region.
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33. 34. SOLUTION: Find the area of the rectangle.
Page 4
The area of the shaded region is represented by the
2
2
8-3 Multiplying Polynomials
expression 4πx + 12πx + 9π − 3x − 5x − 2.
The area of the shaded region is represented by the
2
expression 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.
a. Write an expression that represents the area of
the court.
34. SOLUTION: Find the area of the rectangle.
b. The length of a sand volleyball court is 31 feet.
Find the area of the court.
SOLUTION: a.
Find the area of the triangle.
The area of the court is represented by the
2
expression 18y + 9y − 20.
b.
Subtract the area of the triangle from the area of the
rectangle.
Substitute 9 for y in the expression for area to find
the area of the sand volleyball court when the length
is 31 feet.
The area of the shaded region is represented by the
2
expression 24x −
2
The area of the sand volleyball court is 1519 ft .
.
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.
36. GEOMETRY Write an expression for the area of
a triangle with a base of 2x + 3 and a height of 3x −
1.
SOLUTION: 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.
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SOLUTION: a.
Page 5
8-3 Multiplying
Polynomials
2
The area of the sand volleyball court is 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.
40. (2r − 3t)
3
SOLUTION: SOLUTION: The area of the triangle is represented by the
expression
.
Find each product.
2
37. (a − 2b)
41. (5g + 2h)
3
SOLUTION: SOLUTION: 38. (3c + 4d)
2
SOLUTION: 42. (4y + 3z)(4y − 3z)
2
SOLUTION: 39. (x − 5y)
2
SOLUTION: 40. (2r − 3t)
3
SOLUTION: eSolutions Manual - Powered by Cognero
Page 6
43. CONSTRUCTION A sandbox kit allows you to
build a square sandbox or a rectangular sandbox as 8-3 Multiplying Polynomials
42. (4y + 3z)(4y − 3z)
2
SOLUTION: 43. CONSTRUCTION A sandbox kit allows you to
build a square sandbox or a rectangular sandbox as shown.
a. What are the possible values of x? Explain.
43. CONSTRUCTION A sandbox kit allows you to
build a square sandbox or a rectangular sandbox as shown.
b. Which shape has the greater area?
c. What is the difference in areas between the two?
SOLUTION: a. The value of x must be greater than 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.
b.
a. What are the possible values of x? Explain.
b. Which shape has the greater area?
c. What is the difference in areas between the two?
SOLUTION: a. The value of x must be greater than 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.
b.
The square has the greatest area.
c. Subtract the area of the rectangle from the area of
the square.
2
The difference in the areas is 4 ft .
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44. MULTIPLE REPRESENTATIONS In this probl
Page 7
a. TABULAR Copy and complete the table for ea
8-3 Multiplying
Polynomials
2
The difference in the areas is 4 ft .
Then,
44. MULTIPLE REPRESENTATIONS In this probl
a. TABULAR Copy and complete the table for ea
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.
SOLUTION: 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
b. VERBAL Make a conjecture about the terms of
c. SYMBOLIC For a sum of the form a + b, write
SOLUTION: a.
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 ).
SOLUTION: 47. OPEN ENDED Write a binomial and a trinomial
involving a single variable. Then find their product.
SOLUTION: 2
Sample answer: x − 1, x − x − 1.
b. The first term of the square of a sum is the first t
times the first term of the sum multiplied by the last t
last term of the sum squared.
c. 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.
Then,
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.
SOLUTION: Always; by grouping two adjacent terms, a trinomial
can be written as a binomial (the sum of two
eSolutions Manual - Powered by Cognero
quantities), and apply the FOIL method. For
2
2
example, (2x + 3)( x + 5x + 7) = (2x + 3)[ x + (5x
SOLUTION: 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.
Consider the following examples.
Page 8
8-3 Multiplying Polynomials
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.
SOLUTION: 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.
Consider the following examples.
49. WRITING IN MATH Summarize the methods tha
multiply polynomials.
SOLUTION: The Distributive Property can be used with a vertical
distributing, multiplying, and combining like terms.
Horizontal: The FOIL method is used with a horizontal format. Y
outer, inner, and last terms of the binomials and then
49. WRITING IN MATH Summarize the methods tha
multiply polynomials.
SOLUTION: The Distributive Property can be used with a vertical
distributing, multiplying, and combining like terms.
A rectangular method can also be used by writing the
polynomials along the top and left side of a rectangle
terms and combining like terms.
Horizontal: 50. What is the product of 2x − 5 and 3x + 4?
A 5x − 1
2
B 6x − 7x − 20
C 6x2 − 20
2
D 6x + 7x − 20
The FOIL method is used with a horizontal format. Y
outer, inner, and last terms of the binomials and then
SOLUTION: Choice B is the correct answer.
A rectangular method can also be used by writing the
polynomials along the top and left side of a rectangle
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terms and combining like terms.
51. Which statement is correct about the symmetry of
this design?
Page 9
SOLUTION: 8-3 Multiplying Polynomials
Choice B is the correct answer.
51. Which statement is correct about the symmetry of
this design?
axis, you can eliminate this choice.
J Since the figure is symmetrical about the y -axis,
you can eliminate this choice. Thus, Choice F is the correct answer.
52. Which point on the number line represents a number
that, when cubed, will result in a number greater than
itself?
A P
B Q
C R
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.
J The design has no symmetry.
SOLUTION: Consider each choice.
D T
SOLUTION: T is the only number greater than 1, so it is the only
number when cubed that will be greater than itself. Choice D is the correct answer.
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 For the design to be symmetrical only about the yaxis, you can fold it along the y-axis. The part to the
right and left of the y-axis should be identical. In this
case they are. So the figure is symmetrical about the
y-axis.
G For the design to be symmetrical about the xaxis, you can fold it on the x-axis. The part above
and below the x-axis, should be identical. In this case
they are not. So it is not symmetrical about the xaxis.
H Since the figure is not symmetrical about the x-
She drew a line of best fit on the graph. What is the
slope of the line that she drew?
axis, you can eliminate this choice.
J Since the figure is symmetrical about the y -axis,
you can eliminate this choice. Thus, Choice F is the correct answer.
SOLUTION: The line passes through the points (1, 1) and (5, 7).
52. Which point on the number line represents a number
that, when cubed, will result in a number greater than
itself?
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Page 10
SOLUTION: T is the only number greater than 1, so it is the only
number when cubed that will be greater than itself. 8-3 Multiplying
Polynomials
Choice D is the correct answer.
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.
She drew a line of best fit on the graph. What is the
slope of the line that she drew?
So, the slope of the line is
.
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.
SOLUTION: Let x = the amount placed into the 2% interest
savings account
Let 6000-x = the amount placed into the 4%
certificate of deposit
To calculate the amount of money that will be in the
account at the end of the year, use principle (1 +
rate) time. (The 1 + the rate will add back in the original money deposited.)
Savings account:
SOLUTION: The line passes through the points (1, 1) and (5, 7).
Certificate of deposit:
Therefore, T = 1.02x + 1.04(6000 − x)
So, the slope of the line is
.
Find each sum or difference.
2
2
55. (7a − 5) + (−3a + 10)
SOLUTION: 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.
SOLUTION: Let x = the amount placed into the 2% interest
savings account
Let 6000-x = the amount placed into the 4%
certificate of deposit
To calculate the amount of money that will be in the
account at the end of the year, use principle (1 +
rate) time. (The 1 + the rate will add back in the eSolutions
Manual
- Powered
by Cognero
original
money
deposited.)
Savings account:
2
2
56. (8n − 2n ) + (4n − 6n )
SOLUTION: 3
2
3
2
57. (4 + n + 3n ) + (2n − 9n + 6)
SOLUTION: Page 11
2
2
58. (−4u − 9 + 2u) + (6u + 14 + 2u )
2
2
56. (8n − 2n ) + (4n − 6n )
SOLUTION: 8-3 Multiplying Polynomials
3
2
3
2
57. (4 + n + 3n ) + (2n − 9n + 6)
SOLUTION: Simplify.
4 3
3 4
63. (−2t ) − 3(−2t )
SOLUTION: 2
2
58. (−4u − 9 + 2u) + (6u + 14 + 2u )
SOLUTION: 2 3
3 2
64. (−3h ) − 2(−h )
SOLUTION: 59. (b + 4) + (c + 3b − 2)
SOLUTION: 3 2
3
SOLUTION: SOLUTION: 3
3 3
65. 2(−5y ) + (−3y )
3
60. (3a − 6a) − (3a + 5a)
3
2
61. (−4m − m + 10) − (3m + 3m − 7)
SOLUTION: 4 2
2 2
66. 3(−6n ) + (−2n )
SOLUTION: 62. (3a + 4ab + 3b) − (2b + 5a + 8ab)
SOLUTION: Simplify.
4 3
3 4
63. (−2t ) − 3(−2t )
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Page 12