11-5 Dividing Polynomials
The quotient is
.
2
Find each quotient.
4. (t + 5t + 4) ÷ (t + 4)
2
1. (8a + 20a) ÷ 4a
SOLUTION: SOLUTION: The quotient is t + 1.
The quotient is 2a + 5.
2
5. (x + 3x − 28) ÷ (x + 7)
3
2. (4z + 1) ÷ 2z
SOLUTION: Rewrite as a rational expression and factor the
numerator. Then, divide out common factors. SOLUTION: The quotient is x − 4.
2
6. (x + x − 20) ÷ (x – 4)
The quotient is
3
.
SOLUTION: Rewrite as a rational expression and factor the
numerator. Then, divide out common factors. 2
3. (12n – 6n + 15) ÷ 6n
SOLUTION: The quotient is x + 5.
7. CHEMISTRY The number of beakers that can be
The quotient is
.
2
4. (t + 5t + 4) ÷ (t + 4)
SOLUTION: eSolutions Manual - Powered by Cognero
filled with 50 + x milliliters of a solution is given by
(400 + 3x) ÷ (50 + x). How many beakers can be
filled?
SOLUTION: Write each polynomial in decreasing order and then
use long division to find the quotient.
Page 1
11-5 Dividing Polynomials
The quotient is x + 5.
The quotient is
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?
SOLUTION: Write each polynomial in decreasing order and then
use long division to find the quotient.
3
.
2
10. (4h + 6h − 3) ÷ (2h + 3)
SOLUTION: The quotient is
.
3
11. (9n − 13n + 8) ÷ (3n − 1)
SOLUTION: 2
So, represents the number of beakers that
Don't forget to insert a n -term that has a coefficient
of 0. can be filled.
Find each quotient. Use long division.
2
8. (n + 3n + 10) ÷ (n – 1)
SOLUTION: The quotient is
The quotient is
.
.
Find each quotient.
2
12. (14x + 7x) ÷ 7x
SOLUTION: 2
9. (4y + 8y + 3) ÷ (y + 2)
SOLUTION: The quotient is
.
The quotient is 2x + 1.
3
2
10. (4h + 6h − 3) ÷ (2h + 3)
SOLUTION: eSolutions Manual - Powered by Cognero
3
2
13. (a + 4a − 18a) ÷ a
SOLUTION: Page 2
11-5The
Dividing
Polynomials
quotient
is 2x + 1.
3
The quotient is
2
.
2
16. (8k − 6) ÷ 2k
13. (a + 4a − 18a) ÷ a
SOLUTION: SOLUTION: 2
The quotient is a + 4a − 18.
The quotient is
3
14. (5q + q) ÷ q
SOLUTION: .
2
17. (9m + 5m) ÷ 6m
SOLUTION: 2
The quotient is 5q + 1.
The quotient is
.
2
15. (6n − 12n + 3) ÷ 3n
2
18. (a + a − 12) ÷ (a − 3)
SOLUTION: SOLUTION: Rewrite as a rational expression and factor the
numerator. Then, divide out common factors. The quotient is
.
2
16. (8k − 6) ÷ 2k
SOLUTION: The quotient is a + 4.
2
19. (x − 6x − 16) ÷ (x + 2)
eSolutions Manual - Powered by Cognero
SOLUTION: Rewrite as a rational expression and factor the
numerator. Then, divide out common factors. Page 3
11-5 Dividing Polynomials
The quotient is a + 4.
2
19. (x − 6x − 16) ÷ (x + 2)
SOLUTION: Rewrite as a rational expression and factor the
numerator. Then, divide out common factors. The quotient is k + 3.
2
2
22. (y − 36) ÷ (y + 6y)
SOLUTION: Rewrite as a rational expression and factor the
numerator. Then, divide out common factors. The quotient is x − 8.
2
20. (r − 12r + 11) ÷ (r − 1)
SOLUTION: Rewrite as a rational expression and factor the
numerator. Then, divide out common factors. The quotient is
3
.
2
23. (a − 4a ) ÷ (a − 4)
SOLUTION: Rewrite as a rational expression and factor the
numerator. Then, divide out common factors. The quotient is r − 11.
2
21. (k − 5k − 24) ÷ (k − 8)
SOLUTION: Rewrite as a rational expression and factor the
numerator. Then, divide out common factors. 2
The quotient is a .
3
24. (c − 9c) ÷ (c − 3)
SOLUTION: Rewrite as a rational expression and factor the
numerator. Then, divide out common factors. The quotient is k + 3.
2
2
22. (y − 36) ÷ (y + 6y)
SOLUTION: Rewrite as a rational expression and factor the
numerator. Then, divide out common factors. eSolutions Manual - Powered by Cognero
2
The quotient is c + 3c.
2
Page 4
11-5 Dividing Polynomials
2
The quotient is a .
The quotient is 2t − 1.
3
3
24. (c − 9c) ÷ (c − 3)
2
26. (6x + 15x − 60x + 39) ÷ 3x
SOLUTION: Rewrite as a rational expression and factor the
numerator. Then, divide out common factors. 2
SOLUTION: Rewrite as a fraction and simplify.
2
The quotient is c + 3c.
The quotient is
2
25. (4t − 1) ÷ (2t + 1)
SOLUTION: Rewrite as a rational expression and factor the
numerator. Then, divide out common factors. 3
.
2
27. (2h + 8h − 3h − 12) ÷ (h + 4)
SOLUTION: Rewrite as a rational expression and factor the
numerator. Then, divide out common factors. The quotient is 2t − 1.
3
2
26. (6x + 15x − 60x + 39) ÷ 3x
2
SOLUTION: Rewrite as a fraction and simplify.
2
The quotient is 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?
SOLUTION: eSolutions
- Powered
by Cognero
The Manual
quotient
is
3
2
.
Page 5
The average number of caps produced per person is
11-5 Dividing Polynomials
.
2
The quotient is 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?
Find each quotient. Use long division.
2
30. (b + 3b − 9) ÷ (b + 5)
SOLUTION: SOLUTION: The quotient is b − 2 +
.
2
31. (a + 4a + 3) ÷ (a − 1)
SOLUTION: 2
The length of the rectangle is x units.
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.
SOLUTION: The quotient is a + 5 +
.
2
32. (2y − 3y + 1) ÷ (y − 2)
SOLUTION: The average number of caps produced per person is
.
Find each quotient. Use long division.
2
30. (b + 3b − 9) ÷ (b + 5)
SOLUTION: The quotient is 2y + 1 +
.
2
33. (4n − 3n + 6) ÷ (n − 2)
SOLUTION: eSolutions Manual - Powered by Cognero
The quotient is b − 2 +
Page 6
.
11-5The
Dividing
Polynomials
quotient
is 2y + 1 +
2
The quotient is p − 3p − 3 +
.
2
.
3
33. (4n − 3n + 6) ÷ (n − 2)
35. (t − 2t − 4) ÷ (t + 4)
SOLUTION: SOLUTION: 2
Don't forget to insert a t -term that has a coefficient
of 0. The quotient is 4n + 5 +
3
.
2
34. (p − 4p + 9) ÷ (p − 1)
SOLUTION: Don't forget to insert a p -term that has a coefficient
of 0. 2
The quotient is t − 4t + 14 −
3
.
2
36. (6x + 5x + 9) ÷ (2x + 3)
SOLUTION: Don't forget to insert a x-term that has a coefficient
of 0. 2
The quotient is p − 3p − 3 +
.
3
35. (t − 2t − 4) ÷ (t + 4)
SOLUTION: 2
Don't forget to insert a t -term that has a coefficient
of 0. 2
The quotient is 3x − 2x + 3.
3
37. (8c + 6c − 5) ÷ (4c − 2)
SOLUTION: 2
Don't forget to insert a c -term that has a coefficient
of 0. 2
The Manual
quotient
is t − by
4t Cognero
+ 14 −
eSolutions
- Powered
3
2
36. (6x + 5x + 9) ÷ (2x + 3)
.
Page 7
11-5 Dividing Polynomials
2
The quotient is 3x − 2x + 3.
2
The quotient is 2c + c + 2 −
3
37. (8c + 6c − 5) ÷ (4c − 2)
SOLUTION: 2
Don't forget to insert a c -term that has a coefficient
of 0. .
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)
SOLUTION: 2
The quotient is 2c + c + 2 −
.
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)
Solve using long division.
SOLUTION: The measure of the base of the triangle is 2w + 4
units.
Use long division to find the expression that
represents the missing length.
39. Solve using long division.
SOLUTION: Use long division to solve.
eSolutions Manual - Powered by Cognero
Page 8
11-5The
Dividing
Polynomials
measure
of the base of the triangle is 2w + 4
units.
The width of the rectangle is x + 3 units.
Use long division to find the expression that
represents the missing length.
40. 39. SOLUTION: SOLUTION: Use long division to solve.
Use long division to solve.
The length of the rectangle is 2x + 4 units.
The width of the rectangle is x + 3 units.
3
41. Determine the quotient when x + 11x + 14 is divided
by x + 2.
SOLUTION: 40. SOLUTION: Use long division to solve.
2
The quotient is x − 2x + 15 −
5
4
3
.
2
42. What is 14y + 21y − 6y − 9y + 32y + 48 divided
by 2y + 3?
SOLUTION: The length of the rectangle is 2x + 4 units.
eSolutions Manual - Powered by Cognero
Page 9
3
41. Determine the quotient when x + 11x + 14 is divided
by x + 2.
2
11-5The
Dividing
Polynomials
quotient
is x − 2x + 15 −
.
b.
5
4
3
2
42. What is 14y + 21y − 6y − 9y + 32y + 48 divided
by 2y + 3?
SOLUTION: c. The graph of the quotient ignoring the remainder
(y = 3) is an asymptote of the graph of the function.
d. To find the excluded values, set the denominator
equal to zero.
4
2
The quotient is 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?
SOLUTION: a.
The excluded value is x = 1. 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.
The graph of the quotient is the straight line y = 3.
SOLUTION: a. After an additional t minutes, the first van has
been on the road for t + 20 minutes, and the second
van has been on the road for t + 35 minutes.
b.The ratio of the first van’s time on the road to the
second van’s time on the road is
The function rewritten as a quotient plus a remainder
.
is
.
Use long division to rewrite as an expression.
b.
eSolutions Manual - Powered by Cognero
The ratio can be rewritten as
.
Page
After 60 minutes, the ratio of the first van’s time
on 10
the road to the second van’s time on the road is 80 ÷ 95 or about 0.84.
The excluded value is x = 1. As x approaches 1 from
the left, y approaches negative infinity. As x
11-5approaches
Dividing Polynomials
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.
SOLUTION: a. After an additional t minutes, the first van has
been on the road for t + 20 minutes, and the second
van has been on the road for t + 35 minutes.
b.The ratio of the first van’s time on the road to the
second van’s time on the road is
.
Use long division to rewrite as an expression.
The ratio can be rewritten as
the road to the second van’s time on the road is 80 ÷ 95 or about 0.84.
After 200 minutes, the ratio of the first van’s time on
the road to the second van’s time on the road is 220
÷ 235 or 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?
SOLUTION: a. The boiling point of water at sea level is 212°F. The temperature decreases by 0.9°F for every 500 feet you are above sea level. Let x = the number of
feet above sea level. Then, the temperature at which
water boils x feet above sea level is
.
b. Substitute 14,494 for x in the equation
.
.
After 60 minutes, the ratio of the first van’s time on
the road to the second van’s time on the road is 80 ÷ 95 or about 0.84.
After 200 minutes, the ratio of the first van’s time on
the road to the second van’s time on the road is 220
÷ 235 or about 0.94.
The water will boil at about 185.9°F.
46. MULTIPLE REPRESENTATIONS In this
problem, you will use picture models to help divide
expressions.
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?
SOLUTION: a. The boiling point of water at sea level is 212°F. The temperature decreases by 0.9°F for every 500 feet you are above sea level. Let x = the number of
feet above sea level. Then, the temperature at which
water boils x feet above sea level is
.
eSolutions Manual - Powered by Cognero
b. Substitute 14,494 for x in the equation
a. ANALYTICAL The first figure models 62 ÷ 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?
2
2
b. CONCRETE Draw figures for 3 ÷ 4 and 2
÷ 11
Page
3.
c. VERBAL Do you observe a pattern in the
x−1+
a. ANALYTICAL The first figure models 62 ÷ 7. thatPolynomials
the square is divided into seven equal
11-5Notice
Dividing
parts. What are the quotient and the remainder?
What division problem does the second figure model?
2
2
b. CONCRETE Draw figures for 3 ÷ 4 and 2 ÷ 3.
c. VERBAL Do you observe a pattern in the
previous exercises? Express this pattern
algebraically.
d. ANALYTICAL Use long division to find x2 ÷ (x + 1). Does this result match your expression from
part c?
SOLUTION: a. In the first figure, there are 5 squares in each of
the seven equal parts, plus 1 left over. Therefore, the
quotient and remainder are
2
Yes, there is a pattern, x ÷ (x + 1) = x − 1 +
.
d.
The result from long division is x − 1 +
which
matches the expression from part c.
.
47. ERROR ANALYSIS Alvin and Andrea are
3
dividing c + 6c − 4 by c + 2. Is either of them
correct? Explain your reasoning.
The second figure is made of sides of 7, and is
divided into eight equal parts. So, the figure is
2
modeled by the division problem 7 ÷ 8.
b.
c.
List each division along with the quotient. Notice that
the divisor is always one larger than the value that is
squared. Now, look at the quotient. Notice that the
integer value of the quotient is always one less than
the value that is squared. The remainder is always
one divided by the divisor. Finally, let x represent the
value that is squared and represent the other values
in terms of x.
SOLUTION: 2
2 ÷ 3
1
2
3 ÷ 4
2
Andrea is correct. Alvin did not take into account the
missing term.
2
6 ÷ 7
48. CCSS REGULARITY The quotient of two
2
2
7 ÷ 8
polynomials is 4x − x − 7 +
6
the polynomials?
2
x ÷ (x + 1)
. What are
x−1+
2
Yes, there is a pattern, x ÷ (x + 1) = x − 1 +
d.
eSolutions Manual - Powered by Cognero
SOLUTION: The divisor of the polynomials is the denominator of
.
2
the remainder, or x + x + 2.
Multiply the denominator of the remainder by the
quotient and add the numerator of the remainder to
get the dividend.
Page 12
SOLUTION: 2
11-5Andrea
Dividing
Polynomials
is correct.
Alvin did not take into account the
missing term.
48. CCSS REGULARITY The quotient of two
2
Consider (a + 4a − 22) ÷ (a − 3). The polynomial a
+ 4a − 22 is prime, so the problem must be solved by
using long division.
50. WRITING IN MATH Describe the steps to find
2
2
polynomials is 4x − x − 7 +
(w − 2w − 30) ÷ (w + 7).
. What are
SOLUTION: the polynomials?
SOLUTION: The divisor of the polynomials is the denominator of
2
the remainder, or x + x + 2.
Multiply the denominator of the remainder by the
quotient and add the numerator of the remainder to
get the dividend.
2
Divide the first term of the dividend, 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.
4
3
The dividend is 4x + 3x + 2x + 1. The division
4
3
2
expression is (4x + 3x + 2x + 1) ÷ ( x + x + 2).
49. OPEN ENDED Write a division problem involving
polynomials that you would solve by using long
division. Explain your answer.
The answer is w − 9 +
.
SOLUTION: 2
2
Consider (a + 4a − 22) ÷ (a − 3). The polynomial a
+ 4a − 22 is prime, so the problem must be solved by
using long division.
50. WRITING IN MATH Describe the steps to find
2
(w − 2w − 30) ÷ (w + 7).
SOLUTION: 51. Simplify
.
2
A 3x − 5x
B 4x2 − 6x
C 3x − 5
D 5x − 3
SOLUTION: The correct choice is A.
2
Divide the first term of the dividend, 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.
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The answer is w − 9 +
.
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
Page 13
label covers all surfaces?
SOLUTION: 11-5 Dividing Polynomials
The correct choice is A.
52. EXTENDED RESPONSE The box shown is
designed to hold rice.
The area of the label on the box is 314 square
centimeters.
53. Simplify
.
F x + 4
G 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?
H x + 2
J SOLUTION: a. Find the volume of the box.
SOLUTION: The box has a volume of 360 cubic centimeters.
b. Find the total surface area of the box.
The correct choice is G.
The area of the label on the box is 314 square
centimeters.
53. Simplify
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
SOLUTION: Let x = the number of cards she bought and sold.
Her profit is her revenue minus her cost. Her
revenue is
. Her cost is
.
.
F x + 4
G H x + 2
J SOLUTION: To make a profit of $60 Susana bought and sold 72
cards, so the correct choice is D.
Find each product.
The correct choice is G.
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54. Susana bought cards at 6 for $10. She decorated
them and sold them at 4 for $10. She made $60 in
55. SOLUTION: Page 14
make a Polynomials
profit of $60 Susana bought and sold 72
11-5To
Dividing
cards, so the correct choice is D.
Find each product.
55. SOLUTION: Find the zeros of each function.
59. SOLUTION: 56. SOLUTION: The function is undefined when
is undefined when x = 2 or x = 4. It
When x = –2, the numerator becomes 0, which
makes f (x) = 0. Therefore, the root of the function is
–2.
60. SOLUTION: 57. SOLUTION: 58. The function is undefined when
is undefined when x = 4 or x = –3. SOLUTION: It
When x = –1, the numerator becomes 0, which
makes f (x) = 0. Therefore, the root of the function is
–1.
61. Find the zeros of each function.
59. SOLUTION: Find the values of x whenf (x) = 0.
SOLUTION: eSolutions Manual - Powered by Cognero
Page 15
is undefined when x = 4 or x = –3. the function is undefined when
x = 3 or x = –3. When x = –1, the numerator becomes 0, which
f (x)Polynomials
= 0. Therefore, the root of the function is
11-5makes
Dividing
–1.
61. SOLUTION: or when
Therefore, since –3 is excluded from the domain of f
(x), there is 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?
SOLUTION: Sketch a drawing.
Find the values of x whenf (x) = 0.
When x = –3, the numerator becomes 0. However,
the function is undefined when
or when
x = 3 or x = –3. Therefore, since –3 is excluded from the domain of f
(x), there is no zero.
Solve using trigonometry.
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?
SOLUTION: Sketch a drawing.
Solve each equation. Check your solution.
63. SOLUTION: Check:
Solve using trigonometry.
64. SOLUTION: eSolutions Manual - Powered by Cognero
Because the square root of a number cannot bePage 16
negative, there is no solution.
11-5 Dividing Polynomials
64. SOLUTION: Solve each equation by using the Quadratic
Formula. Round to the nearest tenth if
necessary.
2
67. v + 12v + 20 = 0
Because the square root of a number cannot be
negative, there is no solution.
SOLUTION: For this equation, a = 1, b = 12, and c = 20.
65. SOLUTION: Check:
The solutions are –10 and –2.
2
68. 3t − 7t − 20 = 0
SOLUTION: For this equation, a = 3, b = –7, and c = –20.
66. SOLUTION: Check:
Solve each equation by using the Quadratic
Formula. Round to the nearest tenth if
necessary.
The solutions are 4 and
.
2
69. 5y − y − 4 = 0
2
67. v + 12v + 20 = 0
eSolutions Manual - Powered by Cognero
SOLUTION: For this equation, a = 1, b = 12, and c = 20.
SOLUTION: For this equation, a = 5, b = –1, and c = –4.
Page 17
11-5The
Dividing
Polynomials
solutions
are 4 and
.
2
69. 5y − y − 4 = 0
The solution is 7.
2
71. 2n − 7n − 3 = 0
SOLUTION: For this equation, a = 5, b = –1, and c = –4.
SOLUTION: For this equation, a = 2, b = –7, and c = –3.
The solutions are –0.8 and 1.
The solutions are about –0.4 and 3.9.
2
70. 2x + 98 = 28x
SOLUTION: Rewrite the equation in standard form.
2
72. 2w = − (7w + 3)
SOLUTION: Rewrite the equation in standard form.
For this equation, a = 2, b = –28, and c = 98.
For this equation, a = 2, b = 7, and c = 3.
The solution is 7.
2
71. 2n − 7n − 3 = 0
SOLUTION: For this equation, a = 2, b = –7, and c = –3.
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The solutions are –3 and –0.5.
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73. THEATER A backdrop for a play uses a series
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thin metal arches attached to the stage floor. For
each arch the height y, in feet, is modeled by the
2
11-5 Dividing Polynomials
The solutions are –3 and –0.5.
The vertex is at (1, 9).
Plot the points and connect with a smooth curve.
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.
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?
SOLUTION: a. First determine the x-intercepts of the related
2
function f (x) = –x + 6x.
The x-intercepts are at 0 and 6. Therefore, the
width of the arch at the floor is 6 – 0 = 6 feet.
b. The height at the top of the arch is the maximum
y-value of the function, or the y-value of the vertex.
So, the height of the arch is 9 feet.
Find each sum.
2
2
74. (3a + 2a − 12) + (8a + 7 − 2a )
SOLUTION: The x-intercepts are at (0, 0) and (6, 0).
Next, find the equation of the axis of symmetry.
3
Substitute x = 3 into the equation to find the y-value
of the vertex.
2
3
2
75. (2c + 3cd − d ) + (−5cd − 2c + 2d )
SOLUTION: The vertex is at (1, 9).
Plot the points and connect with a smooth curve.
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The x-intercepts are at 0 and 6. Therefore, the
width of the arch at the floor is 6 – 0 = 6 feet.
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