ANSWER: 11-5 Dividing Polynomials
Find each quotient.
2
1. (8a + 20a) ÷ 4a
2
9. (4y + 8y + 3) ÷ (y + 2)
ANSWER: ANSWER: 2a + 5
3
2. (4z + 1) ÷ 2z
ANSWER: 3
2
3. (12n – 6n + 15) ÷ 6n
ANSWER: 2
4. (t + 5t + 4) ÷ (t + 4)
ANSWER: t+1
3
2
10. (4h + 6h − 3) ÷ (2h + 3)
ANSWER: 3
11. (9n − 13n + 8) ÷ (3n − 1)
ANSWER: Find each quotient.
2
12. (14x + 7x) ÷ 7x
ANSWER: 2x + 1
2
5. (x + 3x − 28) ÷ (x + 7)
ANSWER: x−4
3
2
13. (a + 4a − 18a) ÷ a
ANSWER: 2
2
6. (x + x − 20) ÷ (x – 4)
ANSWER: x +5
7. CHEMISTRY The number of beakers that can be
filled with 50 + x milliliters of a solution is given by
(400 + 3x) ÷ (50 + x). How many beakers can be
filled?
a + 4a − 18
3
14. (5q + q) ÷ q
ANSWER: 2
5q + 1
2
15. (6n − 12n + 3) ÷ 3n
ANSWER: ANSWER: 2
16. (8k − 6) ÷ 2k
Find each quotient. Use long division.
2
8. (n + 3n + 10) ÷ (n – 1)
ANSWER: ANSWER: 2
17. (9m + 5m) ÷ 6m
ANSWER: 2
9. (4y + 8y + 3) ÷ (y + 2)
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ANSWER: 2
18. (a + a − 12) ÷ (a − 3)
16. (8k − 6) ÷ 2k
2
25. (4t − 1) ÷ (2t + 1)
ANSWER: ANSWER: 2t − 1
11-5 Dividing Polynomials
3
2
17. (9m + 5m) ÷ 6m
2
26. (6x + 15x − 60x + 39) ÷ 3x
ANSWER: 2
ANSWER: 3
2
27. (2h + 8h − 3h − 12) ÷ (h + 4)
2
18. (a + a − 12) ÷ (a − 3)
ANSWER: ANSWER: a +4
2
2h − 3
3
2
28. GEOMETRY The area of a rectangle is (x − 4x )
square units, and the width is (x − 4) units. What is
the length?
2
19. (x − 6x − 16) ÷ (x + 2)
ANSWER: x−8
ANSWER: 2
2
x units
20. (r − 12r + 11) ÷ (r − 1)
ANSWER: r − 11
2
29. MANUFACTURING The expression −n + 18n +
850 represents the number of baseball caps produced
2
by n workers. Find (−n + 18n + 850) ÷ n to write an
expression for the average number of caps produced
per person.
2
21. (k − 5k − 24) ÷ (k − 8)
ANSWER: k +3
2
ANSWER: 2
22. (y − 36) ÷ (y + 6y)
ANSWER: Find each quotient. Use long division.
2
30. (b + 3b − 9) ÷ (b + 5)
ANSWER: 3
2
23. (a − 4a ) ÷ (a − 4)
b −2+
ANSWER: a
2
2
31. (a + 4a + 3) ÷ (a − 1)
3
24. (c − 9c) ÷ (c − 3)
ANSWER: ANSWER: a+5+
2
c + 3c 2
2
32. (2y − 3y + 1) ÷ (y − 2)
25. (4t − 1) ÷ (2t + 1)
ANSWER: ANSWER: 2t − 1
3
2
26. (6x + 15x − 60x + 39) ÷ 3x
2y + 1 +
2
2
ANSWER: 33. (4n − 3n + 6) ÷ (n − 2)
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3
2
27. (2h + 8h − 3h − 12) ÷ (h + 4)
4n + 5 +
Page 2
32. (2y − 3y + 1) ÷ (y − 2)
ANSWER: ANSWER: 2w + 4
11-52y
Dividing
+ 1 + Polynomials
2
Use long division to find the expression that
represents the missing length.
33. (4n − 3n + 6) ÷ (n − 2)
ANSWER: 4n + 5 +
3
39. 2
34. (p − 4p + 9) ÷ (p − 1)
ANSWER: x +3
ANSWER: 2
p − 3p − 3 +
3
35. (t − 2t − 4) ÷ (t + 4)
ANSWER: 40. 2
ANSWER: 2x + 4
t − 4t + 14 −
3
2
36. (6x + 5x + 9) ÷ (2x + 3)
3
41. Determine the quotient when x + 11x + 14 is divided
by x + 2.
ANSWER: 2
ANSWER: 3x − 2x + 3
2
3
x − 2x + 15 −
37. (8c + 6c − 5) ÷ (4c − 2)
ANSWER: 5
4
3
2
42. What is 14y + 21y − 6y − 9y + 32y + 48 divided
by 2y + 3?
2
2c + c + 2 −
ANSWER: 38. GEOMETRY The volume of a prism with a
3
2
triangular base is 10w + 23w + 5w − 2. The height
of the prism is 2w + 1, and the height of the triangle
is 5w − 1. What is the measure of the base of the
triangle? (Hint: V = Bh)
ANSWER: 2w + 4
Use long division to find the expression that
represents the missing length.
4
2
7y − 3y + 16
43. CCSS STRUCTURE Consider f (x) =
.
a. Rewrite the function as a quotient plus a
remainder. Then graph the quotient, ignoring the
remainder.
b. Graph the original function using a graphing
calculator.
c. How are the graphs of the function and quotient
related?
d. What happens to the graph near the excluded
value of x?
ANSWER: a. 39. ANSWER: x + 3Manual - Powered by Cognero
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5
4
3
2
42. What is 14y + 21y − 6y − 9y + 32y + 48 divided
by 2y + 3?
ANSWER: 11-5 Dividing Polynomials
4
2
7y − 3y + 16
43. CCSS STRUCTURE Consider f (x) =
.
a. Rewrite the function as a quotient plus a
remainder. Then graph the quotient, ignoring the
remainder.
b. Graph the original function using a graphing
calculator.
c. How are the graphs of the function and quotient
related?
d. What happens to the graph near the excluded
value of x?
ANSWER: a. c. The graph of the quotient ignoring the remainder
is an asymptote of the graph of the function.
d. As x approaches 1 from the left, y approaches
negative infinity. As x approaches 1 from the right, y
approaches positive infinity.
44. ROAD TRIP The first Ski Club van has been on
the road for 20 minutes, and the second van has been
on the road for 35 minutes.
a. Write an expression for the amount of time that
each van has spent on the road after an additional t
minutes.
b. Write a ratio for the first van’s time on the road
to the second van’s time on the road and use long
division to rewrite this ratio as an expression. Then
find the ratio of the first van’s time on the road to the
second van’s time on the road after 60 minutes, 200
minutes.
ANSWER: a. t + 20; t + 35
b. b.
about 0.84; about 0.94
45. BOILING POINT The temperature at which
water boils decreases by about 0.9°F for every 500 feet above sea level. The boiling point at sea level is
212°F.
a. Write an equation for the temperature T at which
water boils x feet above sea level.
b. Mount Whitney, the tallest point in California, is
14,494 feet above sea level. At approximately what
temperature does water boil on Mount Whitney?
ANSWER: a. b. 185.9°F
c. The graph of the quotient ignoring the remainder
is an asymptote of the graph of the function.
d. As x approaches 1 from the left, y approaches
negative infinity. As x approaches 1 from the right, y
approaches positive infinity.
44. ROAD TRIP The first Ski Club van has been on
the road for 20 minutes, and the second van has been
on the road for 35 minutes.
a. Write an expression for the amount of time that
each van has spent on the road after an additional t
minutes.
b. Write a ratio for the first van’s time on the road
to the second van’s time on the road and use long
division to rewrite this ratio as an expression. Then
find the ratio of the first van’s time on the road to the
second van’s time on the road after 60 minutes, 200
minutes.
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ANSWER: a. t + 20; t + 35
46. MULTIPLE REPRESENTATIONS In this
problem, you will use picture models to help divide
expressions.
2
a. ANALYTICAL The first figure models 6 ÷ 7. Notice that the square is divided into seven equal
parts. What are the quotient and the remainder?
What division problem does the second figure model?
Page 4
b. CONCRETE Draw figures for 32 ÷ 4 and 22 ÷ 3.
temperature does water boil on Mount Whitney?
2
c. x ÷ (x + 1) = x − 1 +
ANSWER: a. 11-5 Dividing Polynomials
b. 185.9°F
; yes
d. x − 1 +
46. MULTIPLE REPRESENTATIONS In this
problem, you will use picture models to help divide
expressions.
47. ERROR ANALYSIS Alvin and Andrea are
3
dividing c + 6c − 4 by c + 2. Is either of them
correct? Explain your reasoning.
ANSWER: Andrea; Alvin did not take into account the missing
term.
2
a. ANALYTICAL The first figure models 6 ÷ 7. Notice that the square is divided into seven equal
parts. What are the quotient and the remainder?
What division problem does the second figure model?
b. CONCRETE Draw figures for 32 ÷ 4 and 22 ÷ 3.
c. VERBAL Do you observe a pattern in the
previous exercises? Express this pattern
algebraically.
2
d. ANALYTICAL Use long division to find x ÷ (x + 1). Does this result match your expression from
part c?
2
polynomials is 4x − x − 7 +
. What are
the polynomials?
ANSWER: 4
3
2
Sample answer: 4x + 3x + 2x + 1 and x + x + 2
49. OPEN ENDED Write a division problem involving
polynomials that you would solve by using long
division. Explain your answer.
ANSWER: 2
Sample answer: (a + 4a − 22) ÷ (a − 3); The
ANSWER: 2
polynomial a + 4a − 22 is prime, so the problem can
be solved by using long division.
2
a. 48. CCSS REGULARITY The quotient of two
; 7 ÷ 8
b.
50. WRITING IN MATH Describe the steps to find
2
(w − 2w − 30) ÷ (w + 7).
2
c. x ÷ (x + 1) = x − 1 +
ANSWER: Sample answer: Divide the first term of the dividend,
2
d. x − 1 +
; yes
47. ERROR ANALYSIS Alvin and Andrea are
3
dividing c + 6c − 4 by c + 2. Is either of them
correct? Explain your reasoning.
w , by the first term of the divisor, w. Write the
answer, w, above the division bar and multiply w and
w + 7. Subtract and bring down the −30 to get −9w −
30. Divide the first term of the partial dividend, −9w,
by the first term of the divisor, w. Write the answer,
−9, above the division bar and multiply −9 and w + 7.
Subtract. The answer is w − 9 +
51. Simplify
.
.
A 3x2 − 5x
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ANSWER: Andrea; Alvin did not take into account the missing
2
B 4x − 6x
C 3x − 5
D 5x − 3
Page 5
30. Divide the first term of the partial dividend, −9w,
by the first term of the divisor, w. Write the answer,
−9, above the division bar and multiply −9 and w + 7.
11-5Subtract.
DividingThe
Polynomials
answer is w − 9 +
C 60
D 72
ANSWER: D
.
Find each product.
51. Simplify
.
55. A 3x2 − 5x
2
ANSWER: 6x
B 4x − 6x
C 3x − 5
D 5x − 3
ANSWER: A
56. ANSWER: 52. EXTENDED RESPONSE The box shown is
designed to hold rice.
a. What is the volume of the box?
b. What is the area of the label on the box, if the
label covers all surfaces?
57. ANSWER: ANSWER: a. 360 cm3
b. 314 cm
2
53. Simplify
58. .
F x + 4
G ANSWER: 5(r + 2)
Find the zeros of each function.
59. H x + 2
J ANSWER: G
54. Susana bought cards at 6 for $10. She decorated
them and sold them at 4 for $10. She made $60 in
profit. How many cards did she buy and sell if she
had none left?
A 25
B 53
C 60
D 72
ANSWER: D
Find each product.
55. eSolutions
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ANSWER: ANSWER: −2
60. ANSWER: −1
61. ANSWER: no zero
62. SHADOWS A flagpole casts a shadow that is 10
feet long when the Sun is at an elevation of 68°. How tall is the flagpole?
ANSWER: about 24.75 ft
Solve each equation. Check your solution.
Page 6
61. 2
70. 2x + 98 = 28x
11-5ANSWER: Dividing Polynomials
no zero
62. SHADOWS A flagpole casts a shadow that is 10
feet long when the Sun is at an elevation of 68°. How tall is the flagpole?
ANSWER: about 24.75 ft
Solve each equation. Check your solution.
63. ANSWER: 81
64. ANSWER: 7
2
71. 2n − 7n − 3 = 0
ANSWER: −0.4, 3.9
2
72. 2w = − (7w + 3)
ANSWER: −3, −0.5
73. THEATER A backdrop for a play uses a series of
thin metal arches attached to the stage floor. For
each arch the height y, in feet, is modeled by the
2
equation y = −x + 6x, where x is the distance, in
feet, across the bottom of the arch.
ANSWER: no solution
ANSWER: 9
ANSWER: a.
65. a. Graph the related function and determine the
width of the arch at the floor.
b. What is the height at the top of the arch?
66. ANSWER: 29
Solve each equation by using the Quadratic
Formula. Round to the nearest tenth if
necessary.
2
67. v + 12v + 20 = 0
ANSWER: −10, −2
2
68. 3t − 7t − 20 = 0
ANSWER: 6 ft
b. 9 ft
Find each sum.
2
2
74. (3a + 2a − 12) + (8a + 7 − 2a )
ANSWER: 2
a + 10a − 5
2
69. 5y − y − 4 = 0
ANSWER: −0.8, 1
3
2
3
2
75. (2c + 3cd − d ) + (−5cd − 2c + 2d )
ANSWER: −2cd + d
2
2
70. 2x + 98 = 28x
ANSWER: 7
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71. 2n − 7n − 3 = 0
ANSWER: Page 7