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C H A P T E R 2Polynomial and Rational Functions
Section 2.1 Quadratic Functions . . . . . . . . . . . . . . . . . . . . . 88
Section 2.2 Polynomial Functions of Higher Degree . . . . . . . . . . 99
Section 2.3 Real Zeros of Polynomial Functions . . . . . . . . . . . . 112
Section 2.4 Complex Numbers . . . . . . . . . . . . . . . . . . . . . 126
Section 2.5 The Fundamental Theorem of Algebra . . . . . . . . . . . 132
Section 2.6 Rational Functions and Asymptotes . . . . . . . . . . . . 142
Section 2.7 Graphs of Rational Functions . . . . . . . . . . . . . . . 150
Section 2.8 Quadratic Models . . . . . . . . . . . . . . . . . . . . . . 165
Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
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C H A P T E R 2Polynomial and Rational Functions
Section 2.1 Quadratic Functions
88
1. opens upward and has vertexMatches graph (c).�2, 0�.
f �x� � �x � 2�2 2. opens downward and has vertexMatches graph (d).�0, 3�.
f �x� � 3 � x2
3. opens upward and has vertex Matches graph (b).
�0, 3�. f �x� � x2 � 3 4. opens downward and has vertexMatches graph (a).�4, 0�.
f �x� � ��x � 4�2
You should know the following facts about parabolas.
■ is a quadratic function, and its graph is a parabola.
■ If the parabola opens upward and the vertex is the minimum point. If the parabola opensdownward and the vertex is the maximum point.
■ The vertex is
■ To find the -intercepts (if any), solve
■ The standard form of the equation of a parabola is
where
(a) The vertex is
(b) The axis is the vertical line x � h.
�h, k�.a � 0.
f�x� � a�x � h�2 � k
ax2 � bx � c � 0.
x
��b�2a, f��b�2a��.
a < 0,a > 0,
f�x� � ax2 � bx � c, a � 0,
5.
(a) vertical shrink
(b) vertical shrink and vertical shift one unit downward
(c) vertical shrink and horizontalshift three units to the left
(d) horizontal shift threeunits to the left, vertical shrink, reflection in -axis, and vertical shift one unit downwardx
y � �12 �x � 3�2 � 1,
y � 12 �x � 3�2,
y � 12 x2 � 1,
y � 12 x2,
−9
−6
9
6
abc
d
6.
(a) vertical stretch
(b) vertical stretch, followed by a vertical shift upward one unit
(c) horizontal shift three units to theright, followed by a vertical stretch
(d) horizontal shift three unitsto the right, a vertical stretch, a reflection in thex-axis, and a vertical shift one unit upward
y � �32�x � 3�2 � 1,
y � 32�x � 3�2,
y � 32x2 � 1,
y � 32x2,
−9
−6
9
6
a
bc
d
Vocabulary Check
1. nonnegative integer, real 2. quadratic, parabola 3. axis
4. positive, minimum 5. negative, maximum
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Section 2.1 Quadratic Functions 89
7.
Vertex:
-intercepts:
201510−10−15−20
30
25
5
x
y
��5, 0�, �5, 0�x�0, 25�
f�x� � 25 � x2 8.
Vertex:
Intercepts:
54321−3 −1−4−5
2
1
−2−3−4
−8
x
y
�±�7, 0��0, �7�
f �x� � x2 � 7 9.
Vertex:
-intercepts:
–4 –3 –1 1 2 3 4
–5
–3
–2
1
2
3
x
y
�±2�2, 0�x�0, �4�
f �x� � 12 x2 � 4
10.
Vertex:
Intercepts:
18
12108642
−4−6
1210642−2−4−6−10x
y
�±8, 0��0, 16�
f �x� � 16 � 14 x2 11.
Vertex:
-intercepts:
5
4
3
2
1
−2−3−4
21−1−3−4−7−8x
y
��4 ± �3, 0�x��4, �3�
f �x� � �x � 4�2 � 3 12.
Vertex:
No -intercepts
6
12
18
24
30
42
36
6 12 18 24 30−6x
y
x
�6, 3�
f �x� � �x � 6�2 � 3
13.
Vertex:
-intercepts:
–4 4 8 12 16
4
8
12
16
20
x
y
�4, 0�x�4, 0�
h�x� � x2 � 8x � 16 � �x � 4�2 14.
Vertex:
Intercept:
−4 −3 −2 −1 1 2
1
2
3
4
5
6
x
y
��1, 0���1, 0�
g�x� � x2 � 2x � 1 � �x � 1�2
15.
Vertex:
-intercepts: None
−2 −1 1 2 3
1
3
4
5
x
y
x
�12, 1�f�x� � x2 � x � 54 � �x � 12�2 � 1 16.
Vertex:
Intercepts:
–5 –4 –3 –2 –1 1 2
–3
–2
1
2
3
4
x
y
��32 ± �2, 0���32, �2�
f�x� � x2 � 3x � 14 � �x � 32�2 � 2
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90 Chapter 2 Polynomial and Rational Functions
17.
Vertex:
-intercepts:
–4 2 6
–4
–2
6
x
y
�1 � �6, 0�, �1 � �6, 0�x�1, 6�
f�x� � �x2 � 2x � 5 � ��x � 1�2 � 6 18.
Vertex:
Intercepts:
–6 –5 –3 –2 –1 1 2
–3
–2
1
2
4
5
x
y
��2 ± �5, 0���2, 5�
� ��x � 2�2 � 5
� �1��x � 2�2 � 5�
f�x� � �x2 � 4x � 1 � �1�x2 � 4x � 1�
19.
Vertex:
-intercept: None
–8 –4 4 8
10
20
x
y
x
�12, 20�h�x� � 4x2 � 4x � 21 � 4�x � 12�2 � 20 20.
Vertex:
No -interceptsx
−1−2−3 1 2 3
1
3
4
5
6
x
y�14, 78� � 2�x � 14�2 � 78 � 2�x � 14�2 � 18 � 1 � 2�x2 � 12 x � � 1
f�x� � 2x2 � x � 1
21.
Vertex:
-intercepts:
−10
−6
8
6
��3, 0�, �1, 0�x
��1, 4�
f�x� � ��x2 � 2x � 3� � ��x � 1�2 � 4 22.
Vertex:
Intercepts:
� ��x � 12�2 � 1214 � ��x2 � x � 14� � 14 � 30
f �x� � ��x2 � x � 30�
�5, 0�, ��6, 0�
��12, 1214 �
−27
−3
27
33
23.
Vertex:
-intercepts: ��4 ±�5, 0�x−13
−6
5
6��4, �5�
g�x� � x2 � 8x � 11 � �x � 4�2 � 5
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Section 2.1 Quadratic Functions 91
24.
Vertex:
Intercepts:
� �x � 5�2 � 11
� �x2 � 10x � 25� � 11
f �x� � x2 � 10x � 14
��1.683, 0�, ��8.317, 0�
��5, �11�
−15
−12
6
2 25.
Vertex:
-intercept:
−5
−10
13
2
�4 ± 12�2, 0�x�4, 1�
� �2�x � 4�2 � 1
� �2�x2 � 8x � 16 � 12� � �2�x2 � 8x � 312 �
f �x� � �2x2 � 16x � 31
26.
Vertex:
No -intercepts
� �4�x � 3�2 � 5
� �4�x2 � 6x � 9� � 36 � 41
f �x� � �4x2 � 24x � 41
x
�3, �5�
−5
−50
10
5 27. is the vertex.
Since the graph passes through the point we have:
Thus, Note that ison the parabola.
��3, 0�f �x� � ��x � 1�2 � 4.
�1 � a
0 � 4a � 4
0 � a�1 � 1�2 � 4
�1, 0�,
f �x� � a�x � 1�2 � 4
��1, 4�
28. is the vertex.
Since the graph passes through we have:
Thus, y � �x � 2�2 � 1.
1 � a
4 � 4a
3 � 4a � 1
3 � a�0 � 2�2 � 1
�0, 3�,
f�x� � a�x � 2�2 � 1
��2, �1� 29. is the vertex.
Since the graph passes through the point we have:
f �x� � 1�x � 2�2 � 5 � �x � 2�2 � 5
1 � a
4 � 4a
9 � a�0 � 2�2 � 5
�0, 9�,
f�x� � a�x � 2�2 � 5
��2, 5�
30. is the vertex.
Since the graph passes through the point we have:
f�x� � �2�x � 4�2 � 1
�2 � a
�8 � 4a
�7 � 4a � 1
�7 � a�6 � 4�2 � 1
�6, �7�,
f�x� � a�x � 4�2 � 1
�4, 1� 31. is the vertex.
Since the graph passes through the point we have:
f�x� � 4�x � 1�2 � 2
4 � a
16 � 4a
14 � 4a � 2
14 � a��1 � 1�2 � 2
��1, 14�,
f�x� � a�x � 1�2 � 2
�1, �2�
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92 Chapter 2 Polynomial and Rational Functions
32. is the vertex.
Since the graph passes through the point we have:
f�x� � 54 �x � 4�2 � 1
a � 54
5 � 4a
4 � a��2 � 4�2 � 1
��2, 4�,
f�x� � a�x � 4�2 � 1
��4, �1� 33. is the vertex.
Since the graph passes through the point we have:
f�x� � �104125�x � 12�2 � 1 �104125 � a
�265 �254 a
�215 �254 a � 1
�215 � a��2 � 12�2 � 1��2, �215 �,
f�x� � a�x � 12�2 � 1�12, 1�
34. is the vertex.
Since the graph passes through the point
we have:
f�x� � ��x � 14�2 � 1 a � �1
� 116 �116a
�1716 �116a � 1
�1716 � a�0 � 14�2 � 1��1, �1716 �,
f�x� � a�x � 14�2 � 1��14, �1� 35.
-intercepts:
x � 5 or x � �1
0 � �x � 5��x � 1�
0 � x2 � 4x � 5
�5, 0�, ��1, 0�x
y � x2 � 4x � 5
39.
-intercepts: �0, 0�, �4, 0�x
x � 0 or x � 4
0 � x�x � 4�
0 � x2 � 4x
−4
−5
8
3
y � x2 � 4x 40.
-intercepts:
x � 0, x � 5
0 � x��2x � 10�
0 � �2x2 � 10x
�0, 0�, �5, 0�x
−9
−1
15
15
y � �2x2 � 10x 41.
-intercepts: ��52, 0�, �6, 0�xx � �52 or x � 6
0 � �2x � 5��x � 6�
0 � 2x2 � 7x � 30
−20
−40
20
5
y � 2x2 � 7x � 30
36.
-intercepts:
x � 12, �3
0 � �2x � 1��x � 3�
0 � 2x2 � 5x � 3
�12, 0�, ��3, 0�xy � 2x2 � 5x � 3 37.
-intercept:
x � �4
0 � �x � 4�2 0 � x2 � 8x � 16
��4, 0�x
y � x2 � 8x � 16 38.
-intercept:
x � 3
0 � �x � 3�20 � x2 � 6x � 9
�3, 0�x
y � x2 � 6x � 9
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Section 2.1 Quadratic Functions 93
42.
-intercepts:
x � �7, 34
� �x � 7��4x � 3�
0 � 4x2 � 25x � 21
��7, 0�, �0.75, 0�x
−8
−70
2
10
y � 4x2 � 25x � 21 43.
-intercepts: ��1, 0�, �7, 0�x
x � �1, 7
0 � �x � 1��x � 7�
0 � x2 � 6x � 7
0 � �12�x2 � 6x � 7�
−6
−3
12
9
y � �12�x2 � 6x � 7� 44.
-intercepts:
x � 3, �15
� �x � 3��x � 15�
0 � x2 � 12x � 45
0 � 710 �x2 � 12x � 45�
�3, 0�, ��15, 0�x
−18
−60
6
10
y � 710 �x2 � 12x � 45�
45. opens upward
opens downward
Note: has -intercepts for all real numbers a � 0.��1, 0� and �3, 0�
xf�x� � a�x � 1��x � 3�
� �x2 � 2x � 3
� ��x2 � 2x � 3�
� ��x � 1��x � 3�
g �x� � ��x � ��1���x � 3�,
� x2 � 2x � 3
� �x � 1��x � 3�
f�x� � �x � ��1���x � 3�, 46.
Many correct answers.
opens upward.
opens downward.
�x2 � 10xf �x� � �x�x � 10� �
x2 � 10xf �x� � x�x � 10� �
f �x� � a�x � 0��x � 10� � ax�x � 10�
47. opens upward
opens downward
Note: has -intercepts for all real numbers a � 0.��3, 0� and ��12, 0�
xf�x� � a�x � 3��2x � 1�
� �2x2 � 7x � 3
g �x� � ��2x2 � 7x � 3�,
� 2x2 � 7x � 3
� �x � 3��2x � 1�
� �x � 3��x � 12��2�f�x� � �x � ��3���x � ��12���2�, 48.
opens downward
Many other answers possible.
g�x� � �2x2 � x � 10
g�x� � �f�x�,
� 2x2 � x � 10, opens upward
� 2�x � 52��x � 2�f�x� � 2�x � ��52���x � 2�
49. Let the first number and the second number. Then the sum is
The product is
The maximum value of the product occurs at the vertex of and is 3025. This happens when x � y � 55.P�x�
� ��x � 55�2 � 3025
� ���x � 55�2 � 3025�
� ��x2 � 110x � 3025 � 3025�
P�x� � �x2 � 110x
P�x� � xy � x�110 � x� � 110x � x2.
x � y � 110 ⇒ y � 110 � x.y �x �
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94 Chapter 2 Polynomial and Rational Functions
51. Let be the first number and be the secondnumber. Then
The product is
Completing the square,
The maximum value of the product occurs at thevertex of the parabola and equals 72. This happenswhen and x � 24 � 2�6� � 12.y � 6
P
� �2�y � 6�2 � 72.
� �2� y2 � 12y � 36� � 72
P � �2y2 � 24y
P � xy � �24 � 2y�y � 24y � 2y2.x � 2y � 24 ⇒ x � 24 � 2y.
yx
52. Let first number and second number. Then The product is
The maximum value of the product is 147, and occurs when and y � 13 �42 � 21� � 7.x � 21
� �13 �x � 21�2 � 147.
� �13 �x2 � 42x � 441� � 147
� �13 �x2 � 42x�
P�x� � �13 x2 � 14x
14x � 13 x2.P�x� � xy � x13�42 � x� �
y � 13 �42 � x�.x � 3y � 42,y �x �
(c) Distance traveled around track in one lap:
(e)
The area is maximum when and
y �200 � 2�50�
��
100�
.
x � 50
00
100
2000
y �200 � 2x
�
�y � 200 � 2x
d � �y � 2x � 200
(d) Area of rectangular region:
The area is maximum when and
y �200 � 2�50�
��
100�
.
x � 50
� �2�
�x � 50�2 � 5000�
� �2�
�x2 � 100x � 2500 � 2500�
� �2�
�x2 � 100x�
�1�
�200x � 2x2�
A � xy � x�200 � 2x� �
53. (a)
y
x (b) Radius of semicircular ends of track:
Distance around two semicircular parts of track:
d � 2�r � 2��12y� � �y
r �12
y
50. Let first number and second number.Then, The product is
The maximum value of the product occurs at thevertex of and is This happens whenx � y � S�2.
S2�4.P�x�
� ��x � S2�2
�S2
4
� ��x2 � Sx � S2
4�
S2
4 � � �x2 � Sx
P�x� � Sx � x2P�x� � xy � x�S � x�.
x � y � S, y � S � x.y �x �
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Section 2.1 Quadratic Functions 95
54. (a)
(b)
Maximum area if
(c)
Maximum if x � 25, y � 3313
00
50
2000
A �8x�50 � x�
3
x � 25, y � 3313
4x � 3y � 200 ⇒ y � 13
�200 � 4x� ⇒ A � 2xy � 2x 13
�200 � 4x� �8x
3�50 � x�
x y Area
2
4
6
8
10
12 2xy � 121613�200 � 4�12��
2xy 106713�200 � 4�10��
2xy � 89613�200 � 4�8��
2xy � 70413�200 � 4�6��
2xy 49113�200 � 4�4��
2xy � 25613�200 � 4�2��
x y Area
20
22
24
26
28
30 2xy � 160013�200 � 4�30��
2xy 164313�200 � 4�28��
2xy � 166413�200 � 4�26��
2xy � 166413�200 � 4�24��
2xy 164313�200 � 4�22��
2xy � 160013�200 � 4�20��
(d)
The maximum area occurs at the vertex and is5000 3 square feet. This happens when feetand feet. Thedimensions are feet by 33 feet.
(e) The results are the same.
132x � 50
y � �200 � 4�25���3 � 100�3x � 25�
� �8
3�x � 25�2 �
5000
3
� �8
3��x � 25�2 � 625�
� �8
3�x2 � 50x � 625 � 625�
� �8
3�x2 � 50x�
A �8
3x�50 � x�
55. (a)
(b) When feet.
(c) The vertex occurs at
The maximum height is
feet.
(d) Using a graphing utility, the zero of occurs ator 228.6 feet from the punter.x 228.6,
y
104.0
y ��162025 �
364532 �
2
�95 �
364532 � �
32
x ��b2a
��9�5
2��16�2025� �3645
32 113.9.
y �32
x � 0,
0 2500
120 56.
The maximum height of the dive occurs at
the vertex,
The height at is
The maximum height of the dive is 16 feet.
�4
9�3�2 �
24
9�3� � 12 � 16.
x � 3
x ��b
2a� �
24�92�� 4�9�
� 3.
y � �4
9x2 �
24
9x � 12
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96 Chapter 2 Polynomial and Rational Functions
57.
From the table, the minimum cost seems to be at
The minimum cost occurs at the vertex.
is the minimum cost.
Graphically, you could graph in the window and find the vertex�20, 700�.
�0, 40� � �0, 1000�C � 800 � 10x � 0.25x2
C�20� � 700
x ��b2a
� ���10�2�0.25� �
100.5
� 20
x � 20.
C � 800 � 10x � 0.25x2
x 10 15 20 25 30
C 725 706.25 700 706.25 725
58. (a)
(b) The parabola intersects at Thus, the maximum speed is 59.4 mph.Analytically,
Using the Quadratic Formula,
The maximum speed is the positive root,59.4 mph.
��50 ± �82,732
4 �84.4, 59.4.
s ��50 ± �502 � 4�2���10,029�
2�2�
2s2 � 50s � 10,029 � 0.
2s2 � 50s � 29 � 10,000
0.002s2 � 0.05s � 0.029 � 10
s 59.4.y � 10
0.002s2 � 0.05s � 0.029
00
100
25
59. (a)
(b) The vertex occurs at
(c)
(d) Answers will vary.
� $14,400 thousand
R�24� � �25�24�2 � 1200�24�
p ��b2a
��12002��25� � $24.
� $13,500 thousand
R�30� � �25�30�2 � 1200�30�
� $14,375 thousand
R�25� � �25�25�2 � 1200�25�
� $14,000 thousand
R�20� � �25�20�2 � 1200�20�
61.
(a)
(b) The maximum consumption per year of 4306 cigarettes per person per year occurred in 1960 Answers will vary.�t � 0�.
0 440
5000
�t � 0 corresponds to 1960.�
0 ≤ t ≤ 44C�t� � 4306 � 3.4t � 1.32t2,
60. (a)
(b) The vertex occurs at
The price is $6.25 per pet.
(c) The maximum revenue is
(d) Answers will vary.
R�254 � � �12�254 �
2
� 150�254 � � $468.75.
p ��b2a
��150
2��12� �254
� 6.25.
R�8� � �12�8�2 � 150�8� � $432
R�6� � �12�6�2 � 150�6� � $468
R�4� � �12�4�2 � 150�4� � $408
(c) For 2000,
cigarettes per smoker per year,
cigarettes per smoker per day8909365
24
�2058�209,117,00048,306,000
8909
C(40) � 2058.
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Section 2.1 Quadratic Functions 97
62.
(a) The vertex is or 1993.
(b) For 2004, and orClearly the model is not
accurate past 2003.
(c) Probably not. Answers will vary.
�$78,000,000.S �78,t � 14
�b2a
��218.1
2��28.4� 3.8,
0 ≤ t ≤ 14S � �28.40t2 � 218.1t � 2435,
65. The parabola opens downward and the vertex isMatches (c) and (d).��2, �4�.
66. The parabola opens upward and the vertex is Matches (a).�1, 3�.
67. For is a
maximum when In this case, the
maximum value is Hence,
b � ±20.
400 � b2
�100 � 300 � b2
25 � �75 �b2
4��1�
c �b2
4a.
x ��b2a
.
f�x� � a�x � b2a�2
� �c � b2
4a�a < 0, 68. For is a maximum when In this case, the maximum
value is Hence,
b � ±16.
b2 � 256
�192 � 64 � b2
48 � �16 �b2
4��1�
c �b2
4a.
x ��b2a
.
f�x� � a�x � b2a�2
� �c � b2
4a�a < 0,
69. For is a
minimum when In this case, the minimum
value is Hence,
b � ±8.
b2 � 64
40 � 104 � b2
10 � 26 �b2
4
c �b2
4a.
x ��b2a
.
f�x� � a�x � b2a�2
� �c � b2
4a�a > 0, 70. For is a minimum when In this case, the minimum
value is Hence,
b � ±10.
b2 � 100
�200 � �100 � b2
�50 � �25 �b2
4
c �b2
4a.
x ��b2a
.
f�x� � a�x � b2a�2
� �c � b2
4a�a > 0,
63. True
impossible 12x2 � �1,
�12x2 � 1 � 0
64. True. For
For
In both cases, is the axis of symmetry.x � �54
�b2a
��302�12� �
�3024
��54
.g�x�,
�b2a
� ��10
2��4� � �108
� �54
.f �x�,
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98 Chapter 2 Polynomial and Rational Functions
73.
Then and
�1.2, 6.8�
y � 8 � 65 �345 .⇒ x �
65�
23 x � y � �
23 x � �8 � x� � 6 ⇒ �53 x � �2
x � y � 8 ⇒ y � 8 � x
72.
on graph:
on graph:
From the first equation,
Thus, and hence and .y � �x2 � 5x � 4b � 5,0 � 16a � 4�4 � a� � 4 � 12a � 12 ⇒ a � �1
b � 4 � a.
0 � 16a � 4b � 4�4, 0�
0 � a � b � 4�1, 0�
y � ax2 � bx � 4
74.
The graphs intersect at �4, 2�.
x � 4
11x � 44
12x � 40 � x � 4
y � 3x � 10 � 14x � 1
75.
Thus, and are the points of intersection.�2, 5���3, 0�
x � �3, x � 2
�x � 3��x � 2� � 0
x2 � x � 6 � 0
y � x � 3 � 9 � x2
76.
The graphs intersect at �2, 11�.
x � 2
�x � 2��x2 � 2x � 8� � 0
x3 � 4x � 16 � 0
y � x3 � 2x � 1 � �2x � 15 77. Answers will vary. (Make a Decision)
71. Model (a) is preferable. means the parabola opens upward and profits are increasing for to the right of the vertex,
t ≥ �b
�2a�.
ta > 0
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Section 2.2 Polynomial Functions of Higher Degree
Section 2.2 Polynomial Functions of Higher Degree 99
■ You should know the following basic principles about polynomials.
■ is a polynomial function of degree n.
■ If f is of odd degree and
(a) then
1.
2.
(b) then
1.
2.
■ If f is of even degree and
(a) then
1.
2.
(b) then
1.
2.
■ The following are equivalent for a polynomial function.
(a) is a zero of a function.
(b) is a solution of the polynomial equation
(c) is a factor of the polynomial.
(d) is an -intercept of the graph of
■ A polynomial of degree has at most distinct zeros.
■ If is a polynomial function such that and then takes on every value between andin the interval
■ If you can find a value where a polynomial is positive and another value where it is negative, then there isat least one real zero between the values.
�a, b�.f �b�f �a�ff�a� � f�b�,a < bf
nn
f.x�a, 0��x � a�
f�x� � 0.x � ax � a
f�x� → �� as x → ��.f�x� → �� as x → �.
an < 0,
f�x� → � as x → ��.f�x� → � as x → �.
an > 0,
f�x� → � as x → ��.f�x� → �� as x → �.
an < 0,
f�x� → �� as x → ��.f�x� → � as x → �.
an > 0,
f�x� � anxn � an�1xn�1 � . . . � a2x2 � a1x � a0, an � 0,
1. is a line with -intercept Matches graph (f).
�0, 3�.y f�x� � �2x � 3 2. is a parabola with intercepts and and opens upward. Matches graph (h).�4, 0�
�0, 0� f�x� � x2 � 4x
3. is a parabola with -interceptsand opens downward. Matches
graph (c).�0, 0� and ��52, 0�
x f�x� � �2x2 � 5x 4. has intercepts and
Matches graph (a).��12 � 12�3, 0�.��12 � 12�3, 0�
�0, 1�, �1, 0�, f�x� � 2x3 � 3x � 1
5. has intercepts
Matches graph (e).�0, 0� and �±2�3 , 0�. f �x� � �14 x4 � 3x2 6. has -intercept
Matches graph (d).�0, �43�.y f�x� � �13x3 � x2 � 43
Vocabulary Check
1. continuous 2. Leading Coefficient Test 3. relative extrema
4. solution, x-intercept 5. touches, crosses 6. Intermediate Value�x � a�,
n, n � 1,
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100 Chapter 2 Polynomial and Rational Functions
9.
(a)
Horizontal shift twounits to the right
(c)
Reflection in the -axis and a vertical
shrinkx
–4 –3 –2 2 3 4
–4
–3
–2
1
2
3
4
x
yf�x� � �12x3
–3 –2 2 3 4 5
–4
–3
–2
1
2
3
4
x
yf�x� � �x � 2�3y � x3
(b)
Vertical shift two unitsdownward
(d)
Horizontal shift twounits to the right and a vertical shift two unitsdownward
–3 –2 1 2 4 5
–5
–4
–3
–2
1
2
3
x
yf�x� � �x � 2�3 � 2
–4 –3 –2 2 3 4
–5
–4
1
2
3
x
yf�x� � x3 � 2
10.
(a)
Horizontal shift five units to the left
(c)
Reflection in the -axis and then a vertical shift four units upward
x
–4 –3 –2 1 2 3 4
–2
1
2
3
5
6
x
–1
y
f�x� � 4 � x4
5
4
3
6
2
1
−2−3−4
2 31−1−3−5−6−7 −2−4x
y
f�x� � �x � 5�4y � x4
(b)
Vertical shift five units downward
(d)
Horizontal shift one unit to the right and a vertical shrink
–4 –3 –2 –1 1 2 3 4
–2
x
y
f�x� � 12�x � 1�4
4
3
2
1
−6
54321−2−3−4−5x
y
f�x� � x4 � 5
7. has intercepts Matches graph (g).
�0, 0� and ��2, 0�.f�x� � x4 � 2x3 8. has intercepts Matches (b).��1, 0�, �3, 0�, ��3, 0�.
�0, 0�, �1, 0�,f�x� � 15x5 � 2x3 �95x
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Section 2.2 Polynomial Functions of Higher Degree 101
11.
−12
−8
12
8
f
g
f�x� � 3x3 � 9x � 1; g�x� � 3x3 12.
−12
−8
12
8
gf
f�x� � �13 �x3 � 3x � 2�, g�x� � �13 x
3
13.
−8
−20
8
12
f
g
f�x� � ��x4 � 4x3 � 16x�; g�x� � �x4 14.
−9
−4
9
8
g
f
f�x� � 3x4 � 6x2, g�x� � 3x4
15.
Degree: 4
Leading coefficient: 2
The degree is even and the leading coefficientis positive. The graph rises to the left and right.
f�x� � 2x4 � 3x � 1 16.
Degree: 6
Leading coefficient:
The degree is even and the leading coefficient isnegative. The graph falls to the left and right.
�1
h�x� � 1 � x6
17.
Degree: 2
Leading coefficient:
The degree is even and the leading coefficientis negative. The graph falls to the left and right.
�3
g�x� � 5 � 72x � 3x2 18.
Degree: 3
Leading coefficient:
The degree is odd and the leading coefficient ispositive. The graph falls to the left and rises to the right.
13
f�x� � 13 x3 � 5x
19. Degree: 5 (odd)
Leading coefficient:
Falls to the left and rises to the right
63 � 2 > 0
20. Degree: 7 (odd)
Leading coefficient:
Falls to the left and rises to the right
34 > 0
21.
Degree: 2
Leading coefficient:
The degree is even and the leading coefficient isnegative. The graph falls to the left and right.
�23
h�t� � �23 �t2 � 5t � 3� 22.
Degree: 3
Leading coefficient:
The degree is odd and the leading coefficient is negative. The graph rises to the left and falls to the right.
�78
f�s� � �78 �s3 � 5s2 � 7s � 1�
23.
f��x � ±5
f�x� � �x � 5��x � 5�
f�x� � x2 � 25 24.
x � ±7
� �7 � x��7 � x�
f�x� � 49 � x2 25.
(multiplicity 2)h��t � 3
h�t� � �t � 3�2h�t� � t2 � 6t � 9
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102 Chapter 2 Polynomial and Rational Functions
35. (a)
(b)
(c)
t � ±1
� 12�t � 1��t � 1��t2 � 1�
g�t� � 12t4 �12
t � ±1
−6
−2
6
6 36.
(a)
(b) Zeros:
(c)
intercepts: �0, 0�, �±3, 0�x-
x � 0, ±3
0 � 14x3�x2 � 9�
0, ±3
−18
−12
18
12
y � 14x3�x2 � 9�
26.
x � �5 (multiplicity 2)
� �x � 5�2 f �x� � x2 � 10x � 25 27.
x � �2, 1
� �x � 2��x � 1�
f �x� � x2 � x � 2 28.
x � 3, 4
� 2�x � 3��x � 4�
� 2�x2 � 7x � 12�
f �x� � 2x2 � 14x � 24
29.
t � 0, 2 (multiplicity 2)
� t�t � 2�2 f �t� � t3 � 4t2 � 4t 30.
x � �4, 5, 0 (multiplicity 2)
� x2�x � 4��x � 5�
� x2�x2 � x � 20�
f �x� � x4 � x3 � 20x2
31.
� 0.5414, �5.5414
x ��5 ± �25 � 4��3�
2� �
52
±�37
2
�12
�x2 � 5x � 3�
f �x� � 12
x2 �52
x �32
32.
x �25
, �2
�13
�5x � 2��x � 2�
�13
�5x2 � 8x � 4�
f �x� � 53
x2 �83
x �43
33. (a)
(b)
(c)
x �4 ± �16 � 4
2� 2 ± �3
� 3�x2 � 4x � 1�
f �x� � 3x2 � 12x � 3
x � 3.732, 0.268
−7
−10
11
2 34.
(a)
(b) Zeros:
(c)
�1 ± �2, 0�� � �0.414, 2.414�
x �2 ± �4 � 4��1�
2� 1 ± �2
g�x� � 5�x2 � 2x � 1�
�0.414, 2.414
−8
−11
10
1
g�x� � 5x2 � 10x � 5
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Section 2.2 Polynomial Functions of Higher Degree 103
37. (a)
(b)
(c)
x � 0, ±�2
� x�x2 � 3��x2 � 2�
� x�x4 � x2 � 6� f �x� � x5 � x3 � 6x
x � 0, 1.414, �1.414
−6
−4
6
4 38.
(a)
(b) Zeros: 0,
(c)
�0, 0�, �±�3, 0� t � 0, ±�3 �� 0, ±1.732�
� t�t2 � 3�2 � t�t4 � 6t2 � 9�
g�t� � t5 � 6t3 � 9t
±1.732
−9
−6
9
6
g�t� � t5 � 6t3 � 9t
39. (a)
(b)
(c)
x � ±�5
� 2�x2 � 4��x � �5 ��x � �5 � � 2�x4 � x2 � 20�
f �x� � 2x4 � 2x2 � 40
2.236, �2.236
−10
−45
10
5 40.
(a)
(b) No real zeros
(c)
No real zeros
� 5�x2 � 1��x2 � 2� > 0
f �x� � 5�x4 � 3x2 � 2�
−6
−5
6
50
f �x� � 5x4 � 15x2 � 10
41. (a)
(b)
(c)
x � ±5, 4
� �x � 5��x � 5��x � 4�
� �x2 � 25��x � 4�
� x2�x � 4� � 25�x � 4� f �x� � x3 � 4x2 � 25x � 100
x � 4, 5, �5
−6
−10
6
130 42.
(a)
(b) Zeros:
(c)
x-intercepts: ��2, 0�, � 12, 0� x � �2, 12
� �2x � 1��2x � 1��x � 2�
� �2x � 1��2x2 � 3x � 2�
0 � 4x3 � 4x2 � 7x � 2
�2, 12
−9
−2
9
10
y � 4x3 � 4x2 � 7x � 2
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104 Chapter 2 Polynomial and Rational Functions
43. (a)
(b)
(c)
(multiplicity 2)x � 0 or x � 52
0 � x�2x � 5�20 � 4x3 � 20x2 � 25x
y � 4x3 � 20x2 � 25x
x � 0, 52
−2
−4
6
12 44.
(a)
(b) Zeros:
(c)
Zeros: 0,
�0, 0�, �±1, 0�, �±2, 0�
±1, ±2
� x�x � 2��x � 2��x � 1��x � 1�
� x�x2 � 4��x2 � 1�
� x�x4 � 5x2 � 4�
y � x5 � 5x3 � 4x
0, ±1, ±2
−9
−6
9
6
y � x5 � 5x3 � 4x
45.
Zeros:
Relative maximum:
Relative minimums:�1.225, �3.5�, ��1.225, �3.5�
�0, 1�
x � ± 0.421, ±1.680
−6
−4
6
4
f�x� � 2x4 � 6x2 � 1 46.
Real zeros:
Relative maximums:
Relative minimum: �0, 5�
��2.915, 19.688��0.915, 5.646�,
�4.142, 1.934
−6
−10
4
25
f �x� � �38x4 � x3 � 2x2 � 5
47.
Zeros:
Relative maximum:
Relative minimum: �0.324, 5.782�
��0.324, 6.218�
x � �1.178
−9
−1
9
11
f�x� � x5 � 3x3 � x � 6 48.
Real zero:
Relative maximum:
Relative minimum: ��1, �5�
�0.111, �2.942�
�1.819
−6
−7
6
1
f �x� � �3x3 � 4x2 � x � 3
49.
Note: haszeros 0 and 4 for all nonzero real numbers a.
f �x� � a�x � 0��x � 4� � ax�x � 4�
f �x� � �x � 0��x � 4� � x2 � 4x 50. f �x� � �x � 7��x � 2� � x2 � 5x � 14
51.
Note: has zeros 0,and for all nonzero real numbers a.�3
�2,f �x� � ax�x � 2��x � 3�
f �x� � �x � 0��x � 2��x � 3� � x3 � 5x2 � 6x 52. f �x� � �x � 0��x � 2��x � 5� � x3 � 7x2 � 10x
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Section 2.2 Polynomial Functions of Higher Degree 105
53.
Note: has zeros4, 3, and 0 for all nonzero real numbers a.�3,
f�x� � a�x4 � 4x3 � 9x2 � 36x�
� x4 � 4x3 � 9x2 � 36x
� �x � 4��x2 � 9�x
f�x� � �x � 4��x � 3��x � 3��x � 0� 54.
Note:has zeros 0, 1, 2 for all nonzero real numbers a.
�2, �1, f �x� � a x�x � 2��x � 1��x � 1��x � 2�
� x5 � 5x3 � 4x
� x�x4 � 5x2 � 4�
� x�x2 � 4��x2 � 1�
� x�x � 2��x � 1��x � 1��x � 2�
f �x� � �x � ��2���x � ��1���x � 0��x � 1��x � 2�
55.
Note: has zeros and for all nonzero real numbers a.1 � �3
1 � �3f�x� � a�x2 � 2x � 2�
� x2 � 2x � 2
� x2 � 2x � 1 � 3
� �x � 1�2 � ��3 �2 � ��x � 1� � �3� ��x � 1� � �3�
f�x� � �x � �1 � �3 �� �x � �1 � �3 �� 56.
Note:
has zeros and for all nonzero realnumbers a.
6 � �36 � �3
f �x� � a�x � �6 � �3���x � �6 � �3�� � x2 � 12x � 33
� x2 � 12x � 36 � 3
� �x � 6�2 � 3
� ��x � 6� � �3���x � 6� � �3�f �x� � �x � �6 � �3���x � �6 � �3��
57.
Note: has zeros
2, and for all nonzero realnumbers a.
4 � �54 � �5,
f �x� � a�x � 2���x � 4�2 � 5�
� x3 � 10x2 � 27x � 22
� �x � 2���x � 4�2 � 5�
� �x � 2���x � 4� � �5���x � 4� � �5 � f �x� � �x � 2��x � �4 � �5 ���x � �4 � �5 �� 58.
Note: has zeros
4, for all nonzero real numbers a.2 ± �7
f�x� � a�x � 4��x2 � 4x � 3�
� x3 � 8x2 � 13x � 12
� �x � 4��x2 � 4x � 3�
� �x � 4���x � 2�2 � 7� � �x � 4���x � 2� � �7���x � 2� � �7�
f �x� � �x � 4��x � �2 � �7���x � �2 � �7��
59.
Note: has zeros and for all nonzero real numbers a.�1
�2, �2, f �x� � a�x � 2�2�x � 1�
f �x� � �x � 2�2�x � 1� � x3 � 5x2 � 8x � 4 60.
Note: has zeros 3, 2, 2, 2for all nonzero real numbers a.
f �x� � a�x � 3��x � 2�3 � x4 � 9x3 � 30x2 � 44x � 24
f �x� � �x � 3��x � 2�3
61.
Note: has zeros 3, 3 for all nonzero real numbers a.
�4,�4,f �x� � a�x � 4�2�x � 3�2 � x4 � 2x3 � 23x2 � 24x � 144
f �x� � �x � 4�2�x � 3�2 62.
Note: has zeros 0, 0 for all nonzero real numbers a.
�5,�5,�5,f �x� � a�x � 5�3x2 � x5 � 15x4 � 75x3 � 125x2
f �x� � �x � 5�3�x � 0�2
63.
Note: has zerosrises to the left, and falls to the right.�2,�1,�1,
a < 0,f �x� � a�x � 1�2�x � 2�2,
� �x3 � 4x2 � 5x � 2
f �x� � ��x � 1�2�x � 2� 64.
Note: has zeros4, 4, falls to the left and falls to the right.�1,�1,
a < 0,f �x� � a�x � 1�2�x � 4�2,
� �x4 � 6x3 � x2 � 24x � 16
f �x� � ��x � 1�2�x � 4�2
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106 Chapter 2 Polynomial and Rational Functions
65.
For example,
� �x3 � 3x � 2. f�x� � ��x � 2��x � 1�2
−3 −1
−2
−5−6−7
1 2 3 4 5 6 7
2
1
3
x
y
y = −x3 + 3x − 2
66.
For example, f�x� � �x � 2��x � 1��x � 1�2.
−4−6−8−10−12−14 2
2
4
6
8
10
x
y
y = x4 + x3 − 3x2 − x + 2
67.
−2−3 1 2 3 4 5 6 7
2
3
4
5
x
y
y = x5 − 5x2 − x + 2
68.
−6−10−14−18 2
2
4
x
y
y = −x4 + x3 + 3x2 − 6x
69. (a) The degree of is odd and the leading coefficient is 1. The graph falls to the left and rises to the right.
(b)
Zeros:
(c) and (d)
−2
2
4
−4−6−8
108642−4−6−8x
y
0, 3, �3
f �x� � x3 � 9x � x�x2 � 9� � x�x � 3��x � 3�
f 70. (a) The degree of is even and the leading coefficient is 1. The graph rises to the left and rises to the right.
(b)
Zeros:
(c) and (d)4
3
2
1
−4
431−1−4x
y
0, 2, �2: �0, 0�, �±2, 0�
� x2�x � 2��x � 2�
g�x� � x4 � 4x2 � x2�x2 � 4�
g
71. (a) The degree of is odd and the leading coefficient is 1. The graph falls to the left and rises to the right.
(b)
Zeros: 0, 3
(c) and (d)
1
2
3
4
5
542 61−2 −1−3−4x
y
f �x� � x3 � 3x2 � x2�x � 3�
f 72. (a) The degree of is odd and the leading coefficient is 3. The graph falls to the left and rises to the right.
(b)
Zeros: 0, 8
(c) and (d)
2 3 4 5 6 7 9
−175−200−225
2550
y
x
f �x� � 3x3 � 24x2 � 3x2�x � 8�
f©
Hou
ghto
n M
ifflin
Com
pany
. All
right
s re
serv
ed.
Section 2.2 Polynomial Functions of Higher Degree 107
73. (a) The degree of is even and the leading coefficient is The graph falls to the left and falls to the right.
(b)
Zeros:
(c) and (d)
−4−8−12 4
2
6 8 1210x
y
�±�5, 0��±2, 0�,±�5:±2,
f�x� � �x4 � 9x2 � 20 � ��x2 � 4��x2 � 5�
�1.f 74. (a) The degree is even and the leading coefficient
is The graph falls to the left and falls to the right.
(b)
Zeros: 2:
(c) and (d)
−2−3−4−5 1 3 4 5
12
16
20
x
y
�2, 0���1, 0�,�1,
f �x� � �x6 � 7x3 � 8 � ��x3 � 1��x3 � 8�
�1.
75. (a) The degree is odd and the leading coefficient (c) and (d)is 1. The graph falls to the left and rises to the right.
(b)
Zeros: ��3, 0��3, 0�,3, �3:
� �x � 3��x � 3�2 � �x2 � 9��x � 3�
x3 � 3x2 � 9x � 27 � x2�x � 3� � 9�x � 3�−12−20 8 12 16 20
4
8
x
y
76. (a) The degree is odd and the leading coefficient (c) and (d)is 1. The graph falls to the left and rises to the right.
(b)
Zeros: �±2, 0��2, 2:
� �x � 2��x2 � 2x � 4��x � 2��x � 2�
� �x3 � 8��x2 � 4�
x5 � 4x3 � 8x2 � 32 � x3�x2 � 4� � 8�x2 � 4� −1−3−4−5 1 3 4 5−8
−32
4
8
x
y
77.
(a) Falls to left and falls to right;
(b)
zeros
(c) and (d)
−3−4−5 1 2 3 4 5
−4−5
1
2
−6−7−8
x
y
t � �2, �2, 2, 2 ⇒ ��2, 0�, �2, 0�;
g�t� � �14�t4 � 8t2 � 16� � �14�t2 � 4�2
��14 < 0�g�t� � �14t4 � 2t2 � 4 78.
(a) Rises to right and rises to left;
(b)
Zeros:
(c) and (d)
−3 −2−4−5 1 2 3 4 5−1
1
2
3
4
5
6
7
8
9
x
y
�3, 0���1, 0�,
g�x� � 110�x � 1�2�x � 3�2� 110 > 0�
g�x� � 110�x4 � 4x3 � 2x2 � 12x � 9�
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108 Chapter 2 Polynomial and Rational Functions
(c)
79.
(a)
The function has three zeros.They are in the intervals
and
(b) Zeros: �0.879, 1.347, 2.532
�2, 3�.��1, 0�, �1, 2�
−5
−3
7
5
f �x� � x3 � 3x2 � 3
(c)
81.
(a)
The function has two zeros.They are in the intervals
and
(b) Zeros: �1.585, 0.779
�0, 1�.��2, �1�
−6
−5
6
3
g�x� � 3x4 � 4x3 � 3
(c) x
0.2768
0.09515
�0.7356�1.54
�0.5795�1.55
�0.4184�1.56
�0.2524�1.57
�0.0812�1.58
�1.59
�1.6
y1 x
0.75
0.76
0.77
0.78 0.00866
0.79 0.14066
0.80 0.2768
0.81 0.41717
�0.1193
�0.2432
�0.3633
y1
x
0.0708
0.14514
0.21838
0.2905�0.84
�0.85
�0.86
�0.87
�0.0047�0.88
�0.0813�0.89
�0.159�0.9
y1 x
1.3 0.127
1.31 0.09979
1.32 0.07277
1.33 0.04594
1.34 0.0193
1.35
1.36 �0.0333
�0.0071
y1 x
2.5
2.51
2.52
2.53
2.54 0.03226
2.55 0.07388
2.56 0.11642
�0.0084
�0.0482
�0.087
�0.125
y1
80.
(a)
The function has three zeros.They are in the intervals
and
(b) Zeros: 0.642�0.832,�2.810,
�0, 1�.��1, 0�,��3, �2�,
−5 5
−6
4
f�x� � �2x3 � 6x2 � 3
x
0.277
0.137
�0.269�2.79
�0.136�2.80
� 0�2.81
�2.82
�2.83
y1 x
0.010
0.068�0.82
�0.83
�0.048�0.84
�0.107�0.85
�0.166�0.86
y1 x
0.62 0.217
0.63 0.119
0.64 0.018
0.65
0.66 �0.189
�0.084
y1
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Section 2.2 Polynomial Functions of Higher Degree 109
83.
No symmetry
Two -interceptsx
f �x� � x2�x � 6�
−12
−5
8
35 84.
No symmetry
Two -intercepts �0, 0�, �4, 0�x
−3
−10
7
40
h�x� � x3�x � 4�2 85.
Symmetric about the -axis
Two -interceptsx
y
g�t� � �12�t � 4�2�t � 4�2
−10
−150
10
10
86.
No symmetry.
Two -intercepts ��1, 0�, �3, 0�x
−6
−6
6
2
g�x� � 18 �x � 1�2�x � 3�3 87.
Symmetric to origin
Three -interceptsx
� x�x � 2��x � 2�
f �x� � x3 � 4x
−9
−6
9
6 88.
Symmetric with respect to -axis
Three -intercepts �±�2, 0�
�0, 0�,x
y
−3
−2
3
2
f �x� � x4 � 2x2
89.
Three -intercepts
No symmetry
x
g�x� � 15 �x � 1�2�x � 3��2x � 9�
−14
−6
16
14 90.
No symmetry; two -intercepts
−6
−3
6
24
x
h�x� � 15�x � 2�2�3x � 5�2
(c) Because the function is even, we only need to verifythe positive zeros.
82.
(a)
The function has four zeros. They are in theintervals and
(b) Notice that is even. Hence, the zeros come insymmetric pairs. Zeros: ±3.130±0.452,
h
��4, �3�.��1, 0��3, 4�,�0, 1�,
−10
−24
10
4
h�x� � x4 � 10x2 � 2
x
0.42
0.43
0.44
0.45
0.46
0.47
0.48 �0.2509
�0.1602
�0.0712
0.01601
0.10148
0.18519
0.26712
y1 x
3.09
3.10
3.11
3.12
3.13
3.14
3.15 1.231
0.61571
0.01025
�0.5855
�1.171
�1.748
�2.315
y1
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110 Chapter 2 Polynomial and Rational Functions
92. (a)
(b) Domain: 0 < x < 6
� 8x�12 � x��6 � x�
� �24 � 2x��24 � 4x�x
V�x� � length � width � height
93. The point of diminishing returns (where the graph changes from curving upward to curvingdownward) occurs when The point is which corresponds to spending$2,000,000 on advertising to obtain a revenue of $160 million.
�200, 160�x � 200.
94.
Point of Diminishing Returns:15.2 years
�15.2, 27.3�
00
35
60 95.
The model is a good fit.
4 150
250
96.
4 150
200 97. For 2010, and
thousand
thousand.
Answers will vary.
y2 � $285.0
y1 � $730.2
t � 20,
98. Answers will vary. 99. True. has only one zero, 0.f �x� � x6
91. (a) Volume
Because the box is made from a square,
Thus:
� �36 � 2x�2x Volume � �length�2 � height
length � width.
� length � width � height (b) Domain:
18 > x > 0
�36 < �2x < 0
0 < 36 � 2x < 36
(c) Height, Length and Width Volume,
1
2
3
4
5
6
7
Maximum volume 3456 for x � 6
7�36 � 2�7��2 � 338836 � 2�7�
6�36 � 2�6��2 � 345636 � 2�6�
5�36 � 2�5��2 � 338036 � 2�5�
4�36 � 2�4��2 � 313636 � 2�4�
3�36 � 2�3��2 � 270036 � 2�3�
2�36 � 2�2��2 � 204836 � 2�2�
1�36 � 2�1��2 � 115636 � 2�1�
Vx (d)
when is maximum.V�x�x � 6
5 73300
3500
(c)
Maximum occurs at x � 2.54.
1 2 3 4 5 6
120
240
360
480
600
720
x
V
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Section 2.2 Polynomial Functions of Higher Degree 111
115.
or
or
or x ≤ �12 x ≥ 1
�x ≤ �12 and x ≤ 1� �x ≥ �12 and x ≥ 1�
�2x � 1 ≤ 0 and x � 1 ≤ 0� �2x � 1 ≥ 0 and x � 1 ≥ 0�
�2x � 1��x � 1� ≥ 0
2x2 � x � 1 ≥ 0 210−2 −1
−2
x
1 2x2 � x ≥ 1
116.
and or and
and or and
impossible�26 ≤ x < 7
x > 7��x ≤ �26x < 7��x ≥ �26
x � 7 > 0��x � 26 ≤ 0x � 7 < 0��x � 26 ≥ 0
x � 26x � 7
≤ 0
5x � 2 � 4�x � 7�
x � 7≤ 0
5x � 2x � 7
� 4 ≤ 0
13 26 390−13−26−39x
7 5x � 2x � 7
≤ 4
111. � fg ���1.5� �f ��1.5�g��1.5� �
�2418
� �43
112. � f � g���1� � f �g��1�� � f�8� � 109
113. �g � f ��0� � g� f �0�� � g��3� � 8��3�2 � 72 114.
�8 < x
3x � 15 < 4x � 720− 2− 4− 6− 8− 10
x3�x � 5� < 4x � 7
100. True. The degree is odd and the leading coefficient is �1.
101. False. The graph touches at but does notcross the -axis there.x
x � 1,
102. False. The graph crosses the x-axis at and x � 0.
x � �3 103. True. The exponent of is odd �3�.�x � 2�
104. False. The graph rises to the left, and rises to theright.
105. The zeros are 0, 1, 1, and the graph rises to theright. Matches (b).
106. The zeros are 0, 0, 2, 2, and the graph falls to theright. Matches (e).
107. The zeros are 1, 1, and the graph rises tothe right. Matches (a).
�2,�2,
108.
� �59 � 128 � 69
� f � g���4� � f��4� � g��4� 109.
� 72 � 39 � 33
�g � f ��3� � g�3� � f �3� � 8�3�2 � �14�3� � 3�
110. � �140849
� �28.7347� ��11��8 � 1649 �� f � g���47� � f��
47�g��
47�
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112 Chapter 2 Polynomial and Rational Functions
Section 2.3 Real Zeros of Polynomial Functions
You should know the following basic techniques and principles of polynomial division.
■ The Division Algorithm (Long Division of Polynomials)
■ Synthetic Division
■ is equal to the remainder of divided by
■ if and only if is a factor of
■ The Rational Zero Test
■ The Upper and Lower Bound Rule
f�x�.�x � k�f�k� � 0�x � k�.f�x�f�k�
1.
2x2 � 10x � 12
x � 3� 2x � 4, x � �3
0
4x � 12
4x � 12
2x2 � 6x
x � 3 ) 2x2 � 10x � 12 2x � 4 2.
5x2 � 17x � 12
x � 4� 5x � 3, x � 4
0
3x � 12
3x � 12
5x2 � 20x
x � 4 ) 5x2 � 17 x � 12 5x � 3
3.
x4 � 5x3 � 6x2 � x � 2x � 2
� x3 � 3x2 � 1, x � �2
0
�x � 2
�x � 2
3x3 � 6x2
3x3 � 6x2
x4 � 2x3
) x4 � 5x3 � 6x2 � x � 2x � 2 x3 � 3x2 � 1 4.
x3 � 4x2 � 17x � 6x � 3
� x2 � x � 20 �54
x � 3
�54
�20x � 60
�20x � 6
�x2 � 3x
�x2 � 17x
x3 � 3x2 x � 3 ) x3 � 4x2 � 17x � 6
x2 � x � 20
117.
or
or x ≤ �24 x ≥ 8
x � 8 ≤ �16 x � 8 ≥ 16
�x � 8� ≥ 161680−8−16−24−32
x �x � 8� � 1 ≥ 15
Vocabulary Check
1. is the dividend, is the divisor, is the quotient, and is the remainder.
2. improper, proper 3. synthetic division 4. Rational Zero
5. Descartes’s Rule, Signs 6. Remainder Theorem 7. upper bound, lower bound
r�x�q(x)d�x�f�x�
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Section 2.3 Real Zeros of Polynomial Functions 113
7.
7x3 � 3x � 2
� 7x2 � 14x � 28 �53
x � 2
�53
28x � 56
28x � 3
�14x2 � 28x
�14x2
7x3 � 14x2
x � 3 ) 7x3 � 0x2 � 0x � 3 7x2 � 14x � 28 8.
8x4 � 52x � 1
� 4x3 � 2x2 � x �12
�9�2
2x � 1
�92
�x � 12
�x � 5
2x2 � x
2x2
�4x3 � 2x2
�4x3
8x4 � 4x3
2x � 1 ) 8x4 � 0x3 � 0x2 � 0x � 5 4x3 � 2x2 � x � 12
9.
6x3 � 10x2 � x � 8
2x2 � 1� 3x � 5 �
2x � 3
2x2 � 1
�2x � 3
� �10x2 � 0x � 5� 10x2 � 2x � 8
� �6x3 � 0x2 � 3x� 2x2 � 0x � 1 ) 6x3 � 10x2 � x � 8
3x � 5 10.
x4 � 3x2 � 1x2 � 2x � 3
� x2 � 2x � 4 �2x � 11
x2 � 2x � 3
2x � 11
4x2 � 8x � 12
4x2 � 6x � 1
2x3 � 4x2 � 6x
2x3 � 0x
x4 � 2x3 � 3x2
x2 � 2x � 3 ) x4 � 0x3 � 3x2 � 0x � 1
x2 � 2x � 4
11.
x3 � 9x2 � 1
� x �x � 9x2 � 1
�x � 9
x3 � x
x2 � 1 ) x3 � 0x2 � 0x � 9 x 12.
x5 � 7
x3 � 1� x2 �
x2 � 7
x3 � 1
x2 � 7
x5 � x2 x3 � 1� x5 � 0x4 � 0x3 � 0x2 � 0x � 7
x2
5.
4x3 � 7x2 � 11x � 54x � 5
� x2 � 3x � 1, x � �54
0
� �4x � 5� 4x � 5
� ��12x2 � 15x� �12x2 � 11x
� �4x3 � 5x2� 4x � 5 ) 4x3 � 7x2 � 11x � 5
x2 � 3x � 1 6.
x �32
2x3 � 3x2 � 50x � 752x � 3
� x2 � 25,
0
�50x � 75
�50x � 75
2x3 � 3x2
2x � 3 ) 2x3 � 3x2 � 50x � 75 x2 � 25
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114 Chapter 2 Polynomial and Rational Functions
15.
x � 53x3 � 17x2 � 15x � 25
x � 5� 3x2 � 2x � 5,
5 3
3
�1715
�2
15�10
5
�2525
0
17.
6x3 � 7x2 � x � 26x � 3
� 6x2 � 25x � 74 �248
x � 3
3 6
6
718
25
�175
74
26222
248
19.
9x3 � 18x2 � 16x � 32x � 2
� 9x2 � 16, x � 2
2 9
9
�1818
0
�160
�16
32�32
0
21.
x3 � 512
x � 8� x2 � 8x � 64, x � �8
�8 1
1
0�8
�8
064
64
512�512
0
23.
x � �12
4x3 � 16x2 � 23x � 15
x � 12� 4x2 � 14x � 30,
�12 4
4
16�2
14
�23�7
�30
�1515
0
16.
5x3 � 18x2 � 7x � 6x � 3
� 5x2 � 3x � 2, x � �3
�3 5
5
18�15
3
7�9
�2
�66
0
18.
2x3 � 14x2 � 20x � 7x � 6
� 2x2 � 2x � 32 �199
x � 6
�6 2
2
14�12
2
�20�12
�32
7192
199
24.
3x3 � 4x2 � 5
x � 32� 3x2 �
1
2 x �
3
4�
49
8x � 12
32 3
3
�492
12
034
34
598
498
20.
5x3 � 6x � 8
x � 2� 5x2 � 10x � 26 �
44
x � 2
�2 5
5
0
�10
�10
6
20
26
8
�52
�44
22.
x3 � 729x � 9
� x2 � 9x � 81, x � 9
9 1
1
0
9
9
0
81
81
�729
729
0
13.
2x3 � 4x2 � 15x � 5
�x � 1�2� 2x �
17x � 5
�x � 1�2
�17x � 5
2x3 � 4x2 � 2x
x2 � 2x � 1 ) 2x3 � 4x2 � 15x � 5 2x 14.
x4
�x � 1�3� x � 3 �
6x2 � 8x � 3
�x � 1�3
6x2 � 8x � 3
3x3 � 9x2 � 9x � 3
3x3 � 3x2 � x
x4 � 3x3 � 3x2 � x
x3 � 3x2 � 3x � 1 ) x4 x � 3
�x � 1�3 � x3 � 3x2 � 3x � 1
©H
ough
ton
Miff
lin C
ompa
ny. A
ll rig
hts
rese
rved
.
Section 2.3 Real Zeros of Polynomial Functions 115
27.
� y1−9
−6
9
6
�x4 � 3x2 � 1
x2 � 5
�x4 � 8x2 � 5x2 � 40 � 39
x2 � 5
��x2 � 8��x2 � 5� � 39
x2 � 5
y2 � x2 � 8 �
39
x2 � 528.
� y1 −6 6
−2
6
�x4 � x2 � 1
x2 � 1
�x2�x2 � 1� � 1
x2 � 1
y2 � x2 �
1
x2 � 1
30.
f ��23� � 343 f �x� � �x � 23��15x3 � 6x � 4� � 343
�23 15
15
10
�10
0
�6
0
�6
0
4
4
14
�83
343
f �x� � 15x4 � 10x3 � 6x2 � 14, k � �23
32.
f ���5� � 6f �x� � �x � �5��x2 � �2 � �5�x � 2�5� � 6
��5 1
1
2
� �5
2 � �5
�5
5 � 2�5
� 2�5
�4
10
6
29.
f�4� � �0��26� � 3 � 3
f�x� � �x � 4��x2 � 3x � 2� � 3
4 1
1
�14
3
�1412
�2
11�8
3
f �x� � x3 � x2 � 14x � 11, k � 4
31.
f ��2 � � 0�4 � 6�2 � � 8 � �8f �x� � �x � �2 ��x2 � �3 � �2 �x � 3�2 � � 8
�2 1
1
3
�2
3 � �2
�2
2 � 3�2
3�2
�14
6
�8
25.
−15
−10
15
10
� y1
�x2
x � 2
�x2 � 4 � 4
x � 2
��x � 2��x � 2� � 4
x � 2
y2 � x � 2 �4
x � 226.
−18 12
−12
8
� y1
�x2 � 2x � 1
x � 3
�x2 � 2x � 3 � 2
x � 3
��x � 1��x � 3� � 2
x � 3
y2 � x � 1 �2
x � 3
©H
ough
ton
Miff
lin C
ompa
ny. A
ll rig
hts
rese
rved
.
116 Chapter 2 Polynomial and Rational Functions
35.(a) 1 2
2
02
2
�72
�5
3�5
�2 � f �1�
f �x� � 2x3 � 7x � 3
(c) 12 2
2
0
1
1
�712
�132
3
�134
�14 � f �12�
(b) �2 2
2
0�4
�4
�78
1
3�2
1 � f ��2�
(d) 2 2
2
04
4
�78
1
32
5 � f �2�
36.
(a)
(b)
(c)
(d) �1 2
2
0�2
�2
32
5
0�5
�5
�15
4
0�4
�4
34
7 � g��1�
3 2
2
06
6
318
21
063
63
�1189
188
0564
564
31692
1695 � g�3�
1 2
2
02
2
32
5
05
5
�15
4
04
4
34
7 � g�1�
2 2
2
04
4
38
11
022
22
�144
43
086
86
3172
175 � g�2�
g�x� � 2x6 � 3x4 � x2 � 3
37.
(a) 3 1
1
�53
�2
�7�6
�13
4�39
�35 � h�3�
h�x� � x3 � 5x2 � 7x � 4
(c) �2 1
1
�5�2
�7
�714
7
4�14
�10 � h��2�
(b) 2 1
1
�52
�3
�7�6
�13
4�26
�22 � h�2�
(d) �5 1
1
�5�5
�10
�750
43
4�215
�211 � h��5�
33.
f �1 � �3 � � 0f �x� � �x � 1 � �3 ��4x2 � �2 � 4�3 �x � �2 � 2�3 ��
1 � �3 4
4
�64 � 4�3
�2 � 4�3
�1210 � 2�3
�2 � 2�3
�44
0
34.
f�2 � �2� � 0f �x� � �x � �2 � �2����3x2 � �2 � 3�2�x � 8 � 4�2�
2 � �2 �3
�3
8�6 � 3�2
2 � 3�2
10�2 � 4�2
8 � 4�2
�88
0
©H
ough
ton
Miff
lin C
ompa
ny. A
ll rig
hts
rese
rved
.
Section 2.3 Real Zeros of Polynomial Functions 117
39.
Zeros: 2, �3, 1
� �x � 2��x � 3��x � 1�
x3 � 7x � 6 � �x � 2��x2 � 2x � 3�
2 1
1
02
2
�74
�3
6�6
0
41.
Zeros: 12, 2, 5
2x3 � �2x � 1��x � 2��x � 5�
2x3 � �x � 12 ��2x2 � 14x � 20� 2x3 � 15x2 � 27x � 10
12 2
2
�151
�14
27�7
20
�1010
0
40.
Zeros: �4, �2, 6
� �x � 4��x � 6��x � 2�
x3 � 28x � 48 � �x � 4��x2 � 4x � 12�
�4 1
1
0
�4
�4
�28
16
�12
�48
48
0
42.
Zeros: 23, 34,
14
� �3x � 2��4x � 3��4x � 1�
48x3 � 80x2 � 41x � 6 � �x � 23��48x2 � 48x � 9�
23 48
48
�8032
�48
41�32
9
�66
0
43. (a)
(b)
Remaining factors:
(c)
(d) Real zeros: 1
(e)
−8
−3
7
7
12,�2,
f (x) � (x � 2��2x � 1��x � 1�
�x � 1��2x � 1�,
2x2 � 3x � 1 � �2x � 1��x � 1�
�2 2
2
1�4
�3
�56
1
2�2
0
44. (a)
(b)
Remaining factors:
(c)
(d) Real zeros: 2
(e)
−6
−20
6
30
13,�3,
f (x) � �x � 3��3x � 1��x � 2�
�x � 2��3x � 1�,
3x2 � 7x � 2 � �3x � 1��x � 2�
�3 3
3
2�9
�7
�1921
2
6�6
0
38.
(a) (b)
(c) (d) �10 4
4
�16�40
�56
7560
567
0�5670
�5670
2056,700
56,720 � f ��10�
5 4
4
�1620
4
720
27
0135
135
20675
695 � f �5�
�2 4
4
�16�8
�24
748
55
0�110
�110
20220
240 � f ��2�
1 4
4
�164
�12
7�12
�5
0�5
�5
20�5
15 � f �1�
f�x� � 4x4 � 16x3 � 7x2 � 20©
Hou
ghto
n M
ifflin
Com
pany
. All
right
s re
serv
ed.
118 Chapter 2 Polynomial and Rational Functions
46. (a)
4 8
8
�3032
2
�118
�3
12�12
0
�2 8
8
�14�16
�30
�7160
�11
�1022
12
24�24
0
47. (a)
(b)
Remaining factors:
(c)
(d) Real zeros:
(e)
−9
−20
3
320
�723,�12,
f �x� � �2x � 1��3x � 2��x � 7�
�x � 7��3x � 2�,
6x2 � 38x � 28 � �3x � 2��2x � 14�
�12 6
6
41
�3
38
�9
�19
�28
�14
14
0
48. (a)
(b)
Remaining factors:
(c)
(d) Real zeros:
(e)
−6
−10
6
15
± �512,
f �x� � �2x � 1��x � �5��x � �5��x � �5��x � �5�,
2x2 � 10 � 2�x � �5��x � �5�
12 2
2
�1
1
0
�10
0
�10
5
�5
0
(b)
Remaining factors:
(c)
(d) Real zeros:
(e)−6
−400
6
50
�2, 4, �34, 12
f �x� � �x � 2��x � 4��4x � 3��2x � 1�
�4x � 3�, �2x � 1�
8x2 � 2x � 3 � �4x � 3��2x � 1�
49.
factor of
factor of 1
Possible rational zeros:
Rational zeros: �3±1,
f �x� � x2�x � 3� � �x � 3� � �x � 3��x2 � 1�
±3±1,
q �
�3p �
f �x� � x3 � 3x2 � x � 3 50.
factor of 16
factor of 1
Possible rational zeros:
Rational zeros: 4, ±2
f �x� � x2�x � 4� � 4�x � 4� � �x � 4��x2 � 4�
±16±8,±4,±2,±1,
q �
p �
f �x� � x3 � 4x2 � 4x � 16
45. (a)
�4 1
1
1�4
�3
�1012
2
8�8
0
5 1
1
�45
1
�155
�10
58�50
8
�4040
0
(b)
Remaining factors:
(c)
(d) Real zeros:
(e)−7
−200
7
20
5, �4, 2, 1
f �x� � �x � 5��x � 4��x � 2��x � 1�
�x � 2�, �x � 1�
x2 � 3x � 2 � �x � 2��x � 1�
©H
ough
ton
Miff
lin C
ompa
ny. A
ll rig
hts
rese
rved
.
Section 2.3 Real Zeros of Polynomial Functions 119
55.
Using a graphing utility and synthetic division,and are rational zeros. Hence,
�y � 6��y � 1�2�2y � 1� � 0 ⇒ y � �6, 1, 12.
�61�2, 1,
2y4 � 7y3 � 26y2 � 23y � 6 � 054.
Using a graphing utility and synthetic division,and are rational zeros. Hence,
�x � 6��x � 5��x2 � 1� � 0 ⇒ x � �5, 6.x � �5x � 6
x4 � x3 � 29x2 � x � 30 � 0
56.
The real zeros are �2, 0, 1.
x�x � 1��x � 2��x � 1��x � 1� � 0
x�x � 1��x � 2��x2 � 2x � 1� � 0
�2 1
1
0
�2
�2
�3
4
1
2
�2
0
1 1
1
�1
1
0
�3
0
�3
5
�3
2
�2
2
0
x�x4 � x3 � 3x2 � 5x � 2� � 0
x5 � x4 � 3x3 � 5x2 � 2x � 0
57.
Using a graphing utility and synthetic division,4, are rational zeros. Hence,
�32.
12,�3,x � 4,
�x � 4��x � 3��2x � 1��2x � 3� � 0 ⇒�
32
12,�3,
4x4 � 55x2 � 45x � 36 � 0 58.
Using a graphing utility and synthetic division,
and 3 are rational zeros. Hence,
3.32,�2,x � �52,
�2x � 5��x � 2��2x � 3��x � 3� � 0 ⇒
32,�2,
�52,
4x4 � 43x2 � 9x � 90 � 0
53.
Possible rational zeros:
The only real zeros are and 2. You can verify this by graphing the function f�z� � z4 � z3 � 2z � 4.�1
z4 � z3 � 2z � 4 � �z � 1��z � 2��z2 � 2� � 0
2 1
1
�22
0
20
2
�44
0
�1 1
1
�1�1
�2
02
2
�2�2
�4
�44
0
±1, ±2, ±4
z4 � z3 � 2z � 4 � 0
51.
factor of
factor of 2
Possible rational zeros:
Using synthetic division, 3, and 5 are zeros.
Rational zeros: 3, 5, 32�1,
f �x� � �x � 1��x � 3��x � 5��2x � 3�
�1,
±452±152 ,±
92,±
52,±
32,±
12,
±45,±15,±9,±5,±3,±1,
q �
�45p �
f �x� � 2x4 � 17x3 � 35x2 � 9x � 45 52.
factor of
factor of 4
Possible rational zeros:
Using synthetic division, 1 and 2 are zeros.
Rational zeros: 2±12,±1,
f �x� � �x � 1��x � 1��x � 2��2x � 1��2x � 1�
�1,
±14±12,±1,±2,
q �
�2p �
f �x� � 4x5 � 8x4 � 5x3 � 10x2 � x � 2©
Hou
ghto
n M
ifflin
Com
pany
. All
right
s re
serv
ed.
120 Chapter 2 Polynomial and Rational Functions
63.
(a)
From the calculator we have and
(b)
(c)
The exact roots are x � 0, 3, 4, ±�2.
� x�x � 3��x � 4��x � �2 ��x � �2 �h�x� � x�x � 3��x � 4��x2 � 2�
4 1
1
�44
0
�20
�2
8�8
0
3 1
1
�73
�4
10�12
�2
14�6
8
�2424
0
x � ±1.414.x � 0, 3, 4
h�x� � x�x4 � 7x3 � 10x2 � 14x � 24�
h�x� � x5 � 7x4 � 10x3 � 14x2 � 24x
65.
4 variations in sign 4, 2 or 0 positive real zeros
0 variations in sign 0 negative real zeros ⇒ f ��x� � 2x4 � x3 � 6x2 � x � 5
⇒ f �x� � 2x4 � x3 � 6x2 � x � 5
67.
2 variations in sign 2 or 0 positive real zeros
1 variation in sign 1 negative real zero ⇒ g��x� � �4x3 � 5x � 8
⇒ g�x� � 4x3 � 5x � 8
64.
(a)
(b)
� �x � 3��x � 3��3x � 1��2x � 3�
g�x� � �x � 3��x � 3��6x2 � 11x � 3�
�3 6
6
7
�18
�11
�30
33
3
9
�9
0
3 6
6
�11
18
7
�51
21
�30
99
�90
9
�27
27
0
x � ±3.0, 1.5, 0.333
g�x� � 6x4 � 11x3 � 51x2 � 99x � 27
66.
3 sign changes or 1 positive zeros
1 sign change negative zero ⇒ 1
f ��x� � 3x4 � 5x3 � 6x2 � 8x � 3
⇒ 3
f �x� � 3x4 � 5x3 � 6x2 � 8x � 3
68.
1 sign change positive zero
No sign change no negative zeros ⇒
g��x� � �2x3 � 4x2 � 5
⇒ 1
g�x� � 2x3 � 4x2 � 5
59.
Using a graphing utility and synthetic division, 1,
and are rational zeros. Hence,
�52.
32,�2,�1,x � 1,
�x � 1��x � 1��x � 2��2x � 3��2x � 5� � 0 ⇒ �
52
32,�2,�1,
4x5 � 12x4 � 11x3 � 42x2 � 7x � 30 � 0 60.
Using a graphing utility and synthetic division, 1,and are rational zeros. Hence,
x � 1, �1, �1, 32, �52.
�x � 1��x � 1�2�2x � 3��2x � 5� � 0 ⇒�
52
32,�1,�1,
4x5 � 8x4 � 15x3 � 23x2 � 11x � 15 � 0
61.
(a) Zeros:
(b)is a zero.
(c)
� �t � 2��t � ��3 � 2���t � ��3 � 2�� h�t� � �t � 2��t2 � 4t � 1�
t � �2�2 1
1
�2�2
�4
�78
1
2�2
0
�2, 3.732, 0.268
h�t� � t3 � 2t2 � 7t � 2 62.
(a) Zeros: 6, 5.236, 0.764
(b)
� �s � 6��s � 3 � �5��s � 3 � �5�f�s� � �s � 6��s2 � 6s � 4�
6 1
1
�126
�6
40�36
4
�2424
0
f�s� � s3 � 12s2 � 40s � 24
©H
ough
ton
Miff
lin C
ompa
ny. A
ll rig
hts
rese
rved
.
Section 2.3 Real Zeros of Polynomial Functions 121
72. (a)
2 sign changes or 2 positive zeros
2 sign changes or 2 negative zeros
(b)
(c)
(d) Zeros: ±2, ±12
−6
−15
6
9
±14, ±12, ±1, ±2, ±4
⇒ 0
f ��x� � 4x4 � 17x2 � 4
⇒ 0
f �x� � 4x4 � 17x2 � 471.
(a) has 3 variations in sign 3 or 1 positivereal zeros.
has 1variation in sign 1 negative real zero.
(b) Possible rational zeros:
(c)
(d) Real zeros: �12, 1, 2, 4
−4
−8
8
16
±8±4,±2,±1,±12,
⇒ f ��x� � �2x4 � 13x3 � 21x2 � 2x � 8
⇒ f �x�f �x� � �2x4 � 13x3 � 21x2 � 2x � 8
73.
(a) has 2 variations in sign 2 or 0 positivereal zeros.
has 1variation in sign 1 negative real zero.
(b) Possible rational zeros:
(c)
(d) Real zeros: 1, 34
, �18
−4
−2
4
6
±3±32,±34,±
38,±
316,±
332,±1,±
12,
±14,±18,±
116,±
132,
⇒ f ��x� � �32x3 � 52x2 � 17x � 3
⇒ f �x�f �x� � 32x3 � 52x2 � 17x � 3 74. (a)
1 sign change 1 positive zero
2 sign changes or 2 negative zeros
(b)
(c)
(d) Zeros: �2, 18
±�145
8
−8
−24
8
8
±34, ±94
±1, ±2, ±3, ±6, ±9, ±18, ±12, ±32, ±
92, ±
14
⇒ 0
f ��x� � �4x3 � 7x2 � 11x � 18
⇒
f �x� � 4x3 � 7x2 � 11x � 18
70. (a)
3 sign changes or 1 positive zeros
0 sign changes negative zeros
(b)
(c)
(d) Zeros: 23, 2, 4
−6
−4
12
8
±13, ±23, ±
43, ±
83, ±
163 ; ±1, ±2, ±4, ±8, ±16
⇒ 0
f ��x� � 3x3 � 20x2 � 36x � 16
⇒ 3
f �x� � �3x3 � 20x2 � 36x � 1669.
(a) has 1 variation in sign 1 positive realzero.
has 2 variationsin sign 2 or 0 negative real zeros.
(b) Possible rational zeros:
(c)
(d) Real zeros: �2, �1, 2
−6
−7
6
1
±1, ±2, ±4
⇒ f ��x� � �x3 � x2 � 4x � 4
⇒ f �x�f �x� � x3 � x2 � 4x � 4
©H
ough
ton
Miff
lin C
ompa
ny. A
ll rig
hts
rese
rved
.
122 Chapter 2 Polynomial and Rational Functions
77.
5 is an upper bound.
is a lower bound.
Real zeros: 2�2,
�3
�3 1
1
�4�3
�7
021
21
16�63
�47
�16141
125
5 1
5
�425
21
0105
105
16525
541
�162705
2689
f �x� � x4 � 4x3 � 16x � 16 78.
3 is an upper bound.
is a lower bound.
Real zeros: 0.380, 1.435
�4
�4 2
2
0�8
�8
032
32
�8�128
�136
3544
547
3 2
2
06
6
018
18
�854
46
3138
141
f �x� � 2x4 � 8x � 3
79.
The rational zeros are and ±2.±32
� 14�2x � 3��2x � 3��x � 2��x � 2�
� 14�4x2 � 9��x2 � 4�
� 14�4x4 � 25x2 � 36�
P�x� � x4 � 254 x2 � 9
81.
The rational zeros are 14 and ±1.
� 14�4x � 1��x � 1��x � 1�
� 14�4x � 1��x2 � 1�
� 14�x2�4x � 1� � 1�4x � 1��
� 14�4x3 � x2 � 4x � 1�
f �x� � x3 � 14x2 � x �14
80.
Possible rational zeros:
Rational zeros: �3, 12, 4
� 12�x � 4��2x � 1��x � 3�
f �x� � 12�x � 4��2x2 � 5x � 3�
4 2
2
�3
8
5
�23
20
�3
12
�12
0
±12, ±32±1, ±2, ±3, ±4, ±6, ±12,
f �x� � 12�2x3 � 3x2 � 23x � 12�
82.
Possible rational zeros:
Rational zeros: �2, �13, 12
� 16�z � 2��3z � 1��2z � 1�
f �x� � 16�z � 2��6z2 � z � 1�
�2 6
6
11
�12
�1
�3
2
�1
�2
2
0
±1, ±2, ±12, ±13, ±
23, ±
16
f �z� � 16�6z3 � 11z2 � 3z � 2�
75.
4 is an upper bound.
is a lower bound.
Real zeros: 1.937, 3.705
�1
�1 1
1
�4�1
�5
05
5
0�5
�5
155
20
4 1
1
�44
0
00
0
00
0
150
15
f �x� � x4 � 4x3 � 15 76.
4 is an upper bound.
is a lower bound.
Real zeros: 0.611, 3.041�2.152,
�3
�3 2
2
�3�6
�9
�1227
15
8�45
�37
4 2
2
�38
5
�1220
8
832
40
f �x� � 2x3 � 3x2 � 12x � 8
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Section 2.3 Real Zeros of Polynomial Functions 123
87.
Using the graph and synthetic division, is a zero:
is a zero of the cubic, so
For the quadratic term, use the Quadratic Formula.
The real zeros are 1, 2 ± �3.�12,
x �4 ± �16 � 4
2� 2 ± �3
y � �2x � 1��x � 1��x2 � 4x � 1�.x � 1
y � �x � 12��2x3 � 10x2 � 10x � 2�
�12 2
2
�9
�1
�10
5
5
10
3
�5
�2
�1
1
0
�1�2
y � 2x4 � 9x3 � 5x2 � 3x � 1 88.
Using the graph and synthetic division, 1 and are zeros:
Using the Quadratic Formula:
The real zeros are 1, 3 ± 2�2.�2,
x �6 ± �36 � 4
2� 3 ± 2�2
y � �x � 1��x � 2��x2 � 6x � 1�
�2
y � x4 � 5x3 � 7x2 � 13x � 2
89.
Using the graph and synthetic division, and are zeros:
Using the Quadratic Formula:
The real zeros are 4 ± �17.3�2,�1,
x �8 ± �64 � 4
2� 4 ± �17
y � ��x � 1��2x � 3��x2 � 8x � 1�
3�2�1
y � �2x4 � 17x3 � 3x2 � 25x � 3 90.
Using the graph and synthetic division, 2 and are zeros:
Using the Quadratic Formula:
The real zeros are 2, 2 ± �6.�1,
x �4 ± �16 � 8
2� 2 ± �6
y � ��x � 2��x � 1��x2 � 4x � 2�
�1
y � �x4 � 5x3 � 10x � 4
83.
Rational zeros: 1
Irrational zeros: 0
Matches (d).
�x � 1�
� �x � 1��x2 � x � 1�
f�x� � x3 � 1
85.
Rational zeros: 3
Irrational zeros: 0
Matches (b).
�x � 0, ±1�
f �x� � x3 � x � x�x � 1��x � 1�
84.
Rational zeros: 0
Irrational zeros:
Matches (a).
1, �x � 3�2 �
� �x � 3�2 ��x2 � 3�2x � 3�4 � f �x� � x3 � 2
86.
Rational zeros:
Irrational zeros:
Matches (c).
2, �x � ±�2 �1, �x � 0�
� x�x � �2 ��x � �2 � � x�x2 � 2�
f �x� � x3 � 2x
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124 Chapter 2 Polynomial and Rational Functions
94.
(a) (b) The second air-fuel ratio of 16.89 can be obtainedby finding the second point where the curves y and
intersect.
(c) Solve or
By synthetic division:
(d) The positive zero of the quadratic can be found using the Quadratic Formula.
x �75.75 � ���75.75�2 � 4��5.05��2720.75�
2��5.05�� 16.89
�5.05x2 � 75.75x � 2720.75
15 �5.05
�5.05
0�75.75
�75.75
3857�1136.25
2720.75
�40,811.2540,811.25
0
�5.05x3 � 3857x � 40,811.25 � 0.�5.05x3 � 3857x � 38,411.25 � 2400
y1 � 2400
130
18
2700
y � �5.05x3 � 3857x � 38,411.25, 13 ≤ x ≤ 18
93. (a) Combined length and width:
Volume
(b)
Dimensions with maximum volume:20 � 20 � 40
00
30
18,000
� 4x2�30 � x�
� x2�120 � 4x�
� l � w � h � x2y
4x � y � 120 ⇒ y � 120 � 4x(c)
Using the Quadratic Formula, or
The value of is not possible because it is negative.15 � 15�5
2
15 ± 15�52
.x � 15
�x � 15��x2 �