5
8. 2x – 12 + 3x
8-1 Adding and Subtracting Polynomials
ANSWER: 5
2x + 3x – 12 ; 2
Determine whether each expression is a
polynomial. If it is a polynomial, find the degree
and determine whether it is a monomial,
binomial, or trinomial.
2
1. 7ab + 6b – 2a
3
2
9. 4z – 2z – 5z
4
ANSWER: 4
2
–5z – 2z + 4z; –5
3
2
10. 2a + 4a – 5a – 1
ANSWER: yes; 3; trinomial
ANSWER: 3
2
4a – 5a + 2a – 1, 4
2. 2y – 5 + 3y
2
Find each sum or difference.
3
3
11. (6x − 4) + (−2x + 9)
ANSWER: yes; 2; trinomial
3. 3x
ANSWER: 2
3
4x + 5
ANSWER: yes; 2; monomial
3
2
2
12. (g − 2g + 5g + 6) − (g + 2g)
ANSWER: 3
4. 2
g − 3g + 3g + 6
ANSWER: No; a monomial cannot have a variable in the
denominator.
2
ANSWER: 2
−13y + 11y
+ 6q
No;
3
3
variable in the denominator.
Write each polynomial in standard form. Identify
the leading coefficient.
4
2
7. –4d + 1 – d
ANSWER: ANSWER: 3
2
−8z − 3z − 2z + 13
2
2
16. (−3d − 8 + 2d) + (4d − 12 + d )
ANSWER: 2
−2d + 6d − 20
2
–4d – d + 1; –4
5
8. 2x – 12 + 3x
ANSWER: 5
2x + 3x – 12 ; 2
2
9. 4z – 2z – 5z
4
ANSWER: eSolutions
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2
2
15. (−4z − 2z + 8) − (4z + 3z − 5)
, and a monomial cannot have a
4
2
14. (8y − 4y ) + (3y − 9y )
ANSWER: 4
ANSWER: −a + 6a − 3
ANSWER: yes; 5; binomial
6. 5q
2
2
2 3
5. 5m p + 6
–4
2
13. (4 + 2a − 2a) − (3a − 8a + 7)
–5z – 2z + 4z; –5
2
17. (y + 5) + (2y + 4y – 2)
ANSWER: 2
4y + 3y + 3
3
2
2
3
18. (3n − 5n + n ) − (−8n + 3n )
ANSWER: 2
9n − 5n
Page 1
19. CCSS SENSE-MAKING The total number of
students T who traveled for spring break consists of
2
17. (y + 5) + (2y + 4y – 2)
21. ANSWER: 8-1 Adding and Subtracting Polynomials
2
4y + 3y + 3
3
2
2
3
18. (3n − 5n + n ) − (−8n + 3n )
ANSWER: yes; 0; monomial
4
ANSWER: yes; 4; trinomial
ANSWER: 2
9n − 5n
19. CCSS SENSE-MAKING The total number of
students T who traveled for spring break consists of
two groups: students who flew to their destinations F
and students who drove to their destination D. The
number (in thousands) of students who flew and the
total number of students who flew or drove can be
modeled by the following equations, where n is the
number of years since 1995.
23. d + 3d
24. a – a
b. Predict the number of students who will drive to
their destination in 2012.
2
ANSWER: yes; 2; binomial
3
3
25. 5n + nq
ANSWER: yes; 4; binomial
a. Write an equation that models the number of
students who drove to their destination for this time
period.
c
ANSWER: No; the exponent is a variable.
T = 14n + 21
F = 8n + 7
2
22. c – 2c + 1
Write each polynomial in standard form. Identify
the leading coefficient.
2
26. 5x – 2 + 3x
ANSWER: 2
5x + 3x – 2; 5
c. How many students will drive or fly to their
destination in 2015?
ANSWER: a. D(n) = 6n + 14
27. 8y + 7y
3
ANSWER: 3
7y + 8y; 7
b. 116,000 students
c. 301,000 students
28. 4 – 3c – 5c
2
ANSWER: Determine whether each expression is a
polynomial. If it is a polynomial, find the degree
and determine whether it is a monomial,
binomial, or trinomial.
2
–5c – 3c + 4; –5
3
2
29. –y + 3y – 3y + 2
ANSWER: 3
20. 2
–y – 3y + 3y + 2; –1
ANSWER: No; a monomial cannot have a variable in the
denominator.
2
5
30. 11t + 2t – 3 + t
ANSWER: 5
2
t + 2t + 11t – 3; 1
21. ANSWER: yes; 0; monomial
4
31. 2 + r – r
ANSWER: 3
–r + r + 2; –1
2
22. c – 2c + 1
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ANSWER: yes; 4; trinomial
3
Page 2
32. 2
5
2
30. 11t + 2t – 3 + t
ANSWER: 8-1 Adding and Subtracting Polynomials
5
2
t + 2t + 11t – 3; 1
31. 2 + r – r
2
39. (4a − 5b + 3) + (6 − 2a + 3b )
3
ANSWER: 2
−2b + 2a + 9
2
2
2
40. (x y − 3x + y) + (3y − 2x y)
ANSWER: ANSWER: 3
2
–r + r + 2; –1
2
−x y − 3x + 4y
2
2
41. (−8xy + 3x − 5y) + (4x − 2y + 6xy)
32. ANSWER: ANSWER: 2
7x − 2xy − 7y
2
2
33. –9b + 10b – b
ANSWER: ANSWER: 6
2
42. (5n − 2p + 2np) − (4p + 4n)
6
2
−6p + 2np + n
2
–b – 9b + 10b; –1
2
Find each sum or difference.
2
34. (2c + 6c + 4) + (5c – 7)
2
2
2
2
44. PETS From 1999 through 2009, the number of dogs
D and the number of cats C (in hundreds) adopted
from animal shelters in the United States are
modeled by the equations D = 2n + 3 and C = n + 4,
where n is the number of years since 1999.
2
35. (2x + 3x ) − (7 − 8x )
ANSWER: 2
11x + 2x − 7
a. Write an equation that models the total number T
of dogs and cats adopted in hundreds for this time
period.
2
36. (3c − c + 11) − (c + 2c + 8)
ANSWER: 2
b. If this trend continues, how many dogs and cats
will be adopted in 2013?
3c − c − 3c + 3
2
2
ANSWER: 2
7c + 6c – 3
3
2
3x − rxt − 8r x − 6rx
ANSWER: 3
2
43. (4rxt − 8r x + x ) − (6rx + 5rxt − 2x )
2
37. (z + z) + (z − 11)
ANSWER: a. T(n) = 3n + 7
ANSWER: 2
2z + z − 11
b. 4900 dogs and cats
38. (2x − 2y + 1) − (3y + 4x)
Classify each polynomial according to its degree
and number of terms.
2
45. 4x – 3x + 5
ANSWER: −2x − 5y + 1
2
2
39. (4a − 5b + 3) + (6 − 2a + 3b )
ANSWER: quadratic trinomial
ANSWER: 2
−2b + 2a + 9
2
46. 11z
2
2
40. (x y − 3x + y) + (3y − 2x y)
ANSWER: 2
2
−x y − 3x + 4y
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2
2
41. (−8xy + 3x − 5y) + (4x − 2y + 6xy)
3
ANSWER: cubic monomial
47. 9 + y
4
ANSWER: quartic binomial
Page 3
ANSWER: 3
46. 11z
ANSWER: 8-1 Adding
and Subtracting Polynomials
cubic monomial
47. 9 + y
4
ANSWER: quartic binomial
a.
b. 3030
52. CCSS REASONING The perimeter of the figure
2
shown is represented by the expression 3x − 7x + 2.
Write a polynomial that represents the measure of
the third side.
3
48. 3x – 7
ANSWER: cubic binomial
5
2
49. –2x – x + 5x – 8 ANSWER: quintic polynomial
2
ANSWER: 4x
3
50. 10t – 4t + 6t
ANSWER: cubic trinomial
51. ENROLLMENT In a rapidly growing
school system, the numbers (in hundreds) of
total students N and K-5 students P
enrolled from 2000 to 2009 are modeled
by the equations N = 1.25t 2 – t + 7.5 and
P = 0.7t 2 – 0.95t + 3.8, where t is the
number of years since 2000.
53. GEOMETRY Consider the rectangle.
2
2
a. What does (4x + 2x – 1)(2x – x + 3) represent?
b. What does 2(4x2 + 2x – 1) + 2(2x2 – x + 3)
represent?
a. Write an equation modeling the number of 6-12
students S enrolled for this time period.
b. How many 6-12 students were enrolled in the
school system in 2007? ANSWER: a.
b. 3030
52. CCSS REASONING The perimeter of the figure
2
shown is represented by the expression 3x − 7x + 2.
Write a polynomial that represents the measure of
the third side.
ANSWER: a. the area of the rectangle b. the perimeter of the rectangle
Find each sum or difference.
54. (4x + 2y − 6z) + (5y − 2z + 7x) + (−9z − 2x − 3y)
ANSWER: 9x + 4y − 17z
2
2
2
55. (5a − 4) + (a − 2a + 12) + (4a − 6a + 8)
ANSWER: 2
10a − 8a + 16
2
2
56. (3c − 7) + (4c + 7) − (c + 5c − 8)
ANSWER: 2
2c − c + 8
ANSWER: eSolutions
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4x
3
2
3
2
+ 4) 4
57. (3n + 3n − 10) − (4n − 5n) + (4n − 3n − 9n Page
ANSWER: 3
2
2
d. $336
2
56. (3c − 7) + (4c + 7) − (c + 5c − 8)
60. MULTIPLE REPRESENTATIONS In this
problem, you will explore perimeter and area.
ANSWER: 8-1 Adding and Subtracting Polynomials
2
2c − c + 8
3
2
3
2
57. (3n + 3n − 10) − (4n − 5n) + (4n − 3n − 9n + 4)
ANSWER: 3
2
7n − 7n − n − 6
58. FOOTBALL The National Football League is
divided into two conferences, the American A and
the National N. From 2002 through 2009, the total
attendance T (in thousands) for both conferences and
for the American Conference games are modeled by
the following equations, where x is the number of
years since 2002.
3
2
T = –0.69x + 55.83x + 643.31x + 10,538
3
2
A = –3.78x + 58.96x + 265.96x + 5257
Determine how many people attended National
Conference football games in 2009.
ANSWER: 8,829,000 people
a. Geometric Draw three rectangles that each have
a perimeter of 400 feet.
b. Tabular Record the width and length of each
rectangle in a table like the one shown below. Find
the area of each rectangle.
c. Graphical On a coordinate system, graph the area
of rectangle 4 in terms of the length, x. Use the
graph to determine the largest area possible.
d. Analytical Determine the length and width that
produce the largest area.
ANSWER: a.
59. CAR RENTAL The cost to rent a car for a day is
$15 plus $0.15 for each mile driven.
a. Write a polynomial that represents the cost of
renting a car for m miles.
b. If a car is driven 145 miles, how much would it
cost to rent?
b.
c. If a car is driven 105 miles each day for four
days, how much would it cost to rent a car?
d. If a car is driven 220 miles each day for seven
days, how much would it cost to rent a car?
c.
ANSWER: a. 15 + 0.15m
b. $36.75
c. $123
d. $336
60. MULTIPLE REPRESENTATIONS In this
problem, you will explore perimeter and area.
a. Geometric Draw three rectangles that each have
a perimeter of 400 feet.
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b. Tabular Record the width and length of each
rectangle in a table like the one shown below. Find
d. The length and width of the rectangle must be 100
feet each to have the largest area.
Page 5
61. CCSS CRITIQUE Cheyenne and Sebastian are
2
2
finding (2x − x) − (3x + 3x − 2). Is either of them
the sum of an odd integer 2n + 1 and the next two
consecutive odd integers.
8-1 Adding and Subtracting Polynomials
d. The length and width of the rectangle must be 100
feet each to have the largest area.
61. CCSS CRITIQUE Cheyenne and Sebastian are
2
ANSWER: 6n + 9
64. WRITING IN MATH Why would you add or
subtract equations that represent real-world
situations? Explain.
2
finding (2x − x) − (3x + 3x − 2). Is either of them
correct? Explain your reasoning.
ANSWER: Sample answer: When you add or subtract two or
more polynomial equations, like terms are combined,
which reduces the number of terms in the resulting
equation. This could help minimize the number of
operations performed when using the equations.
65. WRITING IN MATH Describe how to add and
subtract polynomials using both the vertical and
horizontal formats. ANSWER: Neither; neither of them found the additive inverse
correctly. All terms should be multiplied by −1.
62. REASONING Determine whether each of the
following statements is true or false . Explain your
reasoning.
a. A binomial can have a degree of zero.
b. The order in which polynomials are subtracted
does not matter.
ANSWER: a. False; sample answer: a binomial must have at
least one monomial term with degree greater than
zero.
b. False; sample answer: (2x – 3) – (4x – 3) = –2x,
but (4x – 3) – (2x – 3) = 2x
63. CHALLENGE Write a polynomial that represents
the sum of an odd integer 2n + 1 and the next two
consecutive odd integers.
ANSWER: 6n + 9
64. WRITING IN MATH Why would you add or
subtract equations that represent real-world
situations? Explain.
ANSWER: Sample answer: When you add or subtract two or
more polynomial equations, like terms are combined,
which reduces the number of terms in the resulting
equation. This could help minimize the number of
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operations performed when using the equations.
65. WRITING IN MATH Describe how to add and
ANSWER: Sample answer: To add polynomials in a horizontal
format, you combine like terms. For the vertical
format, you write the polynomials in standard form,
align like terms in columns, and combine like terms.
To subtract polynomials in a horizontal format you
find the additive inverse of the polynomial you are
subtracting, and then combine like terms. For the
vertical format, you write the polynomials in standard
form, align like terms in columns, and subtract by
adding the additive inverse.
66. Three consecutive integers can be represented by x,
x + 1, and x + 2. What is the sum of these three
integers?
A x(x + 1)(x + 2)
B x3 + 3
C 3x + 3
D x + 3
ANSWER: C
67. SHORT RESPONSE What is the perimeter of a
square with sides that measure 2x + 3 units?
ANSWER: 8x + 12 units
68. Jim cuts a board in the shape of a regular hexagon
and pounds in a nail at each vertex, as shown. How
many rubber bands will he need to stretch a rubber
band across every possible pair of nails?
Page 6
67. SHORT RESPONSE What is the perimeter of a
square with sides that measure 2x + 3 units?
D (0, 5)
ANSWER: 8-1 Adding
and Subtracting Polynomials
8x + 12 units
68. Jim cuts a board in the shape of a regular hexagon
and pounds in a nail at each vertex, as shown. How
many rubber bands will he need to stretch a rubber
band across every possible pair of nails?
ANSWER: C
70. COMPUTERS A computer technician charges by
the hour to fix and repair computer equipment. The
total cost of the technician for one hour is $75, for
two hours is $125, for three hours is $175, for four
hours is $225, and so on. Write a recursive formula
for the sequence.
ANSWER: F 15
Determine whether each sequence is
arithmetic, geometric, or neither. Explain.
71. 8, –32, 128, –512, ...
ANSWER: Geometric; the common ratio is –4.
G 14
H 12
72. 25, 8, –9, –26, ...
ANSWER: Arithmetic; the common difference is –17.
J 9
ANSWER: F
69. Which ordered pair is in the solution set of the
system of inequalities shown in the graph?
73. ANSWER: Neither; there is no common ratio or difference.
74. 43, 52, 61, 70, ...
ANSWER: Arithmetic; the common difference is 9.
75. –27, –16, –5, 6, ...
A (−3, 0)
B (0, −3)
ANSWER: Arithmetic; the common difference is 11.
76. 200, 100, 50, 25, …
ANSWER: Geometric; the common ratio is .
C (5, 0)
D (0, 5)
ANSWER: C
77. JOBS Kimi received an offer for a new job. She
wants to compare the offer with her current job.
What is total amount of sales that Kimi must get
each month to make the same income at either job?
70. COMPUTERS A computer technician charges by
the hour to fix and repair computer equipment. The
total cost of the technician for one hour is $75, for
two hours is $125, for three hours is $175, for four
hours is $225, and so on. Write a recursive formula
for the sequence.
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ANSWER: Page 7
yes; 4
76. 200, 100, 50, 25, …
ANSWER: 8-1 Adding and Subtracting Polynomials
Geometric; the common ratio is .
77. JOBS Kimi received an offer for a new job. She
wants to compare the offer with her current job.
What is total amount of sales that Kimi must get
each month to make the same income at either job?
83. −0.3, 0.2, 0.7, 1.2, …
ANSWER: yes; 0.5
Simplify.
5
7
84. t(t )(t )
ANSWER: 3
2
3
85. n (n )(−2n )
ANSWER: −2n
8
5 2
3 4
86. (5t v )(10t v )
ANSWER: ANSWER: $80,000
8 6
50t v
4 5
Determine whether each sequence is an
arithmetic sequence. If it is, state the common
difference.
78. 24, 16, 8, 0, …
ANSWER: yes; −8
79. 4
87. (−8u z )(5uz )
ANSWER: 5 9
−40u z
2 3
88. [(3) ]
ANSWER: 729
, 13, 26, …
3 2
89. [(2) ]
ANSWER: no
80. 7, 6, 5, 4, …
ANSWER: yes; −1
81. 10, 12, 15, 18, …
ANSWER: no
82. −15, −11, −7, −3, …
ANSWER: 64
4 3 2
2 3
90. (2m k ) (−3mk )
ANSWER: 11 12
−108m k
2 2
2 2 2 3
91. (6xy ) (2x y z )
ANSWER: 8 10 6
288x y z
ANSWER: yes; 4
83. −0.3, 0.2, 0.7, 1.2, …
ANSWER: yes; 0.5
Simplify.
5
7
84. t(t )(t )
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ANSWER: Page 8