105
CHAPTER 2 Polynomial 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 © Houghton Mifflin Company. All rights reserved.

CHAPTER 2 Polynomial and Rational Functions€¦ · Section 2.1 Quadratic Functions 91 24. Vert ex: Intercepts: x 5 2 11 x2 10x 25 11 f x x2 10x 14 1.683, 0 , 8.317, 0 5, 11 −15

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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�

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  • 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

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  • 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

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  • 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

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  • 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©

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    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�

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  • 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©

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  • 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

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  • 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

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  • 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 �