College Algebra

Michael Sullivan

Chapter 5

Polynomial and Rational Functions

Educators


Problem 1

Answers are given at the end of these exercises If you get a wrong answer read the pages listed in red.
The intercepts of the equation $9 x^{2}+4 y=36$ are $(p p .159-160)$

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Problem 2

Answers are given at the end of these exercises If you get a wrong answer read the pages listed in red.
Is the expression $4 x^{3}-3.6 x^{2}-\sqrt{2}$ a polynomial? If so, what is its degree? (pp. 39-47)

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Problem 3

Answers are given at the end of these exercises If you get a wrong answer read the pages listed in red.
To graph $y=x^{2}-4,$ you would shift the graph of $y=x^{2}$______a distance of_____units.

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Problem 4

Answers are given at the end of these exercises If you get a wrong answer read the pages listed in red.
Use a graphing utility to approximate (rounded to two decimal places) the local maximum value and local minimum value of $f(x)=x^{3}-2 x^{2}-4 x+5,$ for $-3<x<3 .

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Problem 5

Answers are given at the end of these exercises If you get a wrong answer read the pages listed in red.
The $x$ -intercepts of the graph of a function $y=f(x)$ are the real solutions of the equation $f(x)=0$

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Problem 6

Answers are given at the end of these exercises If you get a wrong answer read the pages listed in red.
If $g(5)=0,$ what point is on the graph of $g ?$ What is the corresponding $x$ -intercept of the graph of $g ?

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Problem 7

Answers are given at the end of these exercises If you get a wrong answer read the pages listed in red.
The graph of every polynomial function is both_____and_____.

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Problem 8

Answers are given at the end of these exercises If you get a wrong answer read the pages listed in red.
If $r$ is a real zero of even multiplicity of a function $f,$ then the_____graph of $f$_____units.

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Problem 9

The graphs of power functions of the form $f(x)=x^{n},$ where $n$ is an even integer, always contain the points_____,_____and_____.

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Problem 10

Answers are given at the end of these exercises If you get a wrong answer read the pages listed in red.
If $r$ is a solution to the equation $f(x)=0,$ name three additional statements that can be made about $f$ and $r$ assuming $f$ is a polynomial function.

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Problem 11

Answers are given at the end of these exercises If you get a wrong answer read the pages listed in red.
The points at which a graph changes direction (from increasing to decreasing or decreasing to increasing) are called______.

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Problem 12

Answers are given at the end of these exercises If you get a wrong answer read the pages listed in red.
The graph of the function $f(x)=3 x^{4}-x^{3}+5 x^{2}-2 x-7 $will behave like the graph of_____for large values of $|x|$.

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Problem 13

Answers are given at the end of these exercises If you get a wrong answer read the pages listed in red.
$$\text { If } f(x)=-2 x^{5}+x^{3}-5 x^{2}+7, \text { then } \lim _{1 x} f(x)=$$_____$$\text { and } \lim f(x)=$$_____.

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Problem 14

Answers are given at the end of these exercises If you get a wrong answer read the pages listed in red.
Explain what the notation $\lim _{x \rightarrow \infty} f(x)=-\infty$ means.

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Problem 15

Determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not.
$$f(x)=4 x+x^{3}$$

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Problem 16

Determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not.
$$f(x)=5 x^{2}+4 x^{4}$$

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Problem 17

Determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not.
$$g(x)=\frac{1-x^{2}}{2}$$

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Problem 18

Determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not.
$$h(x)=3-\frac{1}{2} x$$

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Problem 19

Determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not.
$$f(x)=1-\frac{1}{x}$$

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Problem 20

Determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not.
$$f(x)=x(x-1)$$

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Problem 21

Determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not.
$$g(x)=x^{3 / 2}-x^{2}+2$$

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Problem 22

Determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not.
$$h(x)=\sqrt{x}(\sqrt{x}-1)$$

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Problem 23

Determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not.
$$F(x)=5 x^{4}-\pi x^{3}+\frac{1}{2}$$

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Problem 24

Determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not.
$$F(x)=\frac{x^{2}-5}{x^{3}}$$

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Problem 25

Determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not.
$$G(x)=2(x-1)^{2}\left(x^{2}+1\right)$$

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Problem 26

Determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not.
$$G(x)=-3 x^{2}(x+2)^{3}$$

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Problem 27

Use transformations of the graph of $y=x^{4}$ or $y=x^{5}$ to graph each function.
$$f(x)=(x+1)^{4}$$

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Problem 28

Use transformations of the graph of $y=x^{4}$ or $y=x^{5}$ to graph each function.
$$f(x)=(x-2)^{5}$$

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Problem 29

Use transformations of the graph of $y=x^{4}$ or $y=x^{5}$ to graph each function.
$$f(x)=x^{5}-3$$

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Problem 30

Use transformations of the graph of $y=x^{4}$ or $y=x^{5}$ to graph each function.
$$f(x)=x^{4}+2$$

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Problem 31

Use transformations of the graph of $y=x^{4}$ or $y=x^{5}$ to graph each function.
$$f(x)=\frac{1}{2} x^{4}$$

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Problem 32

Use transformations of the graph of $y=x^{4}$ or $y=x^{5}$ to graph each function.
$$f(x)=3 x^{5}$$

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Problem 33

Use transformations of the graph of $y=x^{4}$ or $y=x^{5}$ to graph each function.
$$f(x)=-x^{5}$$

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Problem 34

Use transformations of the graph of $y=x^{4}$ or $y=x^{5}$ to graph each function.
$$f(x)=-x^{4}$$

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Problem 35

Use transformations of the graph of $y=x^{4}$ or $y=x^{5}$ to graph each function.
$$f(x)=(x-1)^{5}+2$$

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Problem 36

Use transformations of the graph of $y=x^{4}$ or $y=x^{5}$ to graph each function.
$$f(x)=(x+2)^{4}-3$$

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Problem 37

Use transformations of the graph of $y=x^{4}$ or $y=x^{5}$ to graph each function.
$$f(x)=2(x+1)^{4}+1$$

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Problem 38

Use transformations of the graph of $y=x^{4}$ or $y=x^{5}$ to graph each function.
$$f(x)=\frac{1}{2}(x-1)^{5}-2$$

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Problem 39

Use transformations of the graph of $y=x^{4}$ or $y=x^{5}$ to graph each function.
$$f(x)=4-(x-2)^{5}$$

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Problem 40

Use transformations of the graph of $y=x^{4}$ or $y=x^{5}$ to graph each function.
$$f(x)=3-(x+2)^{4}$$

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Problem 41

Form a polynomial function whose real zeros and degree are given. Answers will vary depending on the choice of a leading coefficient.
Zeros: $-1,1,3 ;$ degree 3

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Problem 42

Form a polynomial function whose real zeros and degree are given. Answers will vary depending on the choice of a leading coefficient.
Zeros: $-2,2,3 ;$ degree 3

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Problem 43

Form a polynomial function whose real zeros and degree are given. Answers will vary depending on the choice of a leading coefficient.
Zeros: $-3,0,4 ;$ degree 3

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Problem 44

Form a polynomial function whose real zeros and degree are given. Answers will vary depending on the choice of a leading coefficient.
Zeros: $-4,0,2 ;$ degree 3

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Problem 45

Form a polynomial function whose real zeros and degree are given. Answers will vary depending on the choice of a leading coefficient.
Zeros: $-4,-1,2,3 ;$ degree 4

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Problem 46

Form a polynomial function whose real zeros and degree are given. Answers will vary depending on the choice of a leading coefficient.
Zeros: $-3,-1,2,5 ;$ degree 4

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Problem 47

Form a polynomial function whose real zeros and degree are given. Answers will vary depending on the choice of a leading coefficient.
Zeros: -1, multiplicity 1; 3, multiplicity 2; degree 3

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Problem 48

Form a polynomial function whose real zeros and degree are given. Answers will vary depending on the choice of a leading coefficient.
Zeros: $-2,$ multiplicity $2 ; 4,$ multiplicity $1 ;$ degree 3

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Problem 49

For each polynomial function:
(a) List each real zero and its multiplicity.
(b) Determine whether the graph crosses or touches the $x$ -axis at each $x$ -intercept.
(c) Determine the behavior of the graph near each $x$ -intercept (zero).
(d) Determine the maximum number of turning points on the graph.
(e) Determine the end behavior; that is, find the power function that the graph of fresembles for large values of $|x|$.
$$f(x)=3(x-7)(x+3)^{2}$$

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Problem 50

For each polynomial function:
(a) List each real zero and its multiplicity.
(b) Determine whether the graph crosses or touches the $x$ -axis at each $x$ -intercept.
(c) Determine the behavior of the graph near each $x$ -intercept (zero).
(d) Determine the maximum number of turning points on the graph.
(e) Determine the end behavior; that is, find the power function that the graph of fresembles for large values of $|x|$.
$$f(x)=4(x+4)(x+3)^{3}$$

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Problem 51

For each polynomial function:
(a) List each real zero and its multiplicity.
(b) Determine whether the graph crosses or touches the $x$ -axis at each $x$ -intercept.
(c) Determine the behavior of the graph near each $x$ -intercept (zero).
(d) Determine the maximum number of turning points on the graph.
(e) Determine the end behavior; that is, find the power function that the graph of fresembles for large values of $|x|$.
$$f(x)=4\left(x^{2}+1\right)(x-2)^{3}$$

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Problem 52

For each polynomial function:
(a) List each real zero and its multiplicity.
(b) Determine whether the graph crosses or touches the $x$ -axis at each $x$ -intercept.
(c) Determine the behavior of the graph near each $x$ -intercept (zero).
(d) Determine the maximum number of turning points on the graph.
(e) Determine the end behavior; that is, find the power function that the graph of fresembles for large values of $|x|$.
$$f(x)=2(x-3)\left(x^{2}+4\right)^{3}$$

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Problem 53

For each polynomial function:
(a) List each real zero and its multiplicity.
(b) Determine whether the graph crosses or touches the $x$ -axis at each $x$ -intercept.
(c) Determine the behavior of the graph near each $x$ -intercept (zero).
(d) Determine the maximum number of turning points on the graph.
(e) Determine the end behavior; that is, find the power function that the graph of fresembles for large values of $|x|$.
$$f(x)=-2\left(x+\frac{1}{2}\right)^{2}(x+4)^{3}$$

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Problem 54

For each polynomial function:
(a) List each real zero and its multiplicity.
(b) Determine whether the graph crosses or touches the $x$ -axis at each $x$ -intercept.
(c) Determine the behavior of the graph near each $x$ -intercept (zero).
(d) Determine the maximum number of turning points on the graph.
(e) Determine the end behavior; that is, find the power function that the graph of fresembles for large values of $|x|$.
$$f(x)=\left(x-\frac{1}{3}\right)^{2}(x-1)^{3}$$

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Problem 55

For each polynomial function:
(a) List each real zero and its multiplicity.
(b) Determine whether the graph crosses or touches the $x$ -axis at each $x$ -intercept.
(c) Determine the behavior of the graph near each $x$ -intercept (zero).
(d) Determine the maximum number of turning points on the graph.
(e) Determine the end behavior; that is, find the power function that the graph of fresembles for large values of $|x|$.
$$f(x)=(x-5)^{3}(x+4)^{2}$$

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Problem 56

For each polynomial function:
(a) List each real zero and its multiplicity.
(b) Determine whether the graph crosses or touches the $x$ -axis at each $x$ -intercept.
(c) Determine the behavior of the graph near each $x$ -intercept (zero).
(d) Determine the maximum number of turning points on the graph.
(e) Determine the end behavior; that is, find the power function that the graph of fresembles for large values of $|x|$.
$$f(x)=(x+\sqrt{3})^{2}(x-2)^{4}$$

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Problem 57

For each polynomial function:
(a) List each real zero and its multiplicity.
(b) Determine whether the graph crosses or touches the $x$ -axis at each $x$ -intercept.
(c) Determine the behavior of the graph near each $x$ -intercept (zero).
(d) Determine the maximum number of turning points on the graph.
(e) Determine the end behavior; that is, find the power function that the graph of fresembles for large values of $|x|$.
$$f(x)=3\left(x^{2}+8\right)\left(x^{2}+9\right)^{2}$$

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Problem 58

For each polynomial function:
(a) List each real zero and its multiplicity.
(b) Determine whether the graph crosses or touches the $x$ -axis at each $x$ -intercept.
(c) Determine the behavior of the graph near each $x$ -intercept (zero).
(d) Determine the maximum number of turning points on the graph.
(e) Determine the end behavior; that is, find the power function that the graph of fresembles for large values of $|x|$.
$$f(x)=-2\left(x^{2}+3\right)^{3}$$

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Problem 59

For each polynomial function:
(a) List each real zero and its multiplicity.
(b) Determine whether the graph crosses or touches the $x$ -axis at each $x$ -intercept.
(c) Determine the behavior of the graph near each $x$ -intercept (zero).
(d) Determine the maximum number of turning points on the graph.
(e) Determine the end behavior; that is, find the power function that the graph of fresembles for large values of $|x|$.
$$f(x)=-2 x^{2}\left(x^{2}-2\right)$$

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Problem 60

For each polynomial function:
(a) List each real zero and its multiplicity.
(b) Determine whether the graph crosses or touches the $x$ -axis at each $x$ -intercept.
(c) Determine the behavior of the graph near each $x$ -intercept (zero).
(d) Determine the maximum number of turning points on the graph.
(e) Determine the end behavior; that is, find the power function that the graph of fresembles for large values of $|x|$.
$$f(x)=4 x\left(x^{2}-3\right)$$

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Problem 61

Identify which of the graphs could be the graph of a polynomial function. For those that could, list the real zeros and state the least degree the polynomial can have. For those that could not, say why not.
(THE GRAPH CANNOT COPY)

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Problem 62

Identify which of the graphs could be the graph of a polynomial function. For those that could, list the real zeros and state the least degree the polynomial can have. For those that could not, say why not.
(THE GRAPH CANNOT COPY)

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Problem 63

Identify which of the graphs could be the graph of a polynomial function. For those that could, list the real zeros and state the least degree the polynomial can have. For those that could not, say why not.
(THE GRAPH CANNOT COPY)

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Problem 64

Identify which of the graphs could be the graph of a polynomial function. For those that could, list the real zeros and state the least degree the polynomial can have. For those that could not, say why not.
(THE GRAPH CANNOT COPY)

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Problem 65

Construct a polynomial function that might have the given graph. (More than one answer may be possible.)
(THE GRAPH CANNOT COPY)

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Problem 66

Construct a polynomial function that might have the given graph. (More than one answer may be possible.)
(THE GRAPH CANNOT COPY)

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Problem 67

Construct a polynomial function that might have the given graph. (More than one answer may be possible.)
(THE GRAPH CANNOT COPY)

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Problem 68

Construct a polynomial function that might have the given graph. (More than one answer may be possible.)
(THE GRAPH CANNOT COPY)

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Problem 69

Analyze each polynomial function by following Steps 1 through 6 on page 333 .
$$f(x)=x^{2}(x-3)$$

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Problem 70

Analyze each polynomial function by following Steps 1 through 6 on page 333 .
$$f(x)=x(x+2)^{2}$$

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Problem 71

Analyze each polynomial function by following Steps 1 through 6 on page 333 .
$$f(x)=(x+4)(x-2)^{2}$$

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Problem 72

Analyze each polynomial function by following Steps 1 through 6 on page 333 .
$$f(x)=(x-1)(x+3)^{2}$$

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Problem 73

Analyze each polynomial function by following Steps 1 through 6 on page 333 .
$$f(x)=-2(x+2)(x-2)^{3}$$

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Problem 74

Analyze each polynomial function by following Steps 1 through 6 on page 333 .
$$f(x)=-\frac{1}{2}(x+4)(x-1)^{3}$$

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Problem 75

Analyze each polynomial function by following Steps 1 through 6 on page 333 .
$$f(x)=(x+1)(x-2)(x+4)$$

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Problem 76

Analyze each polynomial function by following Steps 1 through 6 on page 333 .
$$f(x)=(x-1)(x+4)(x-3)$$

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Problem 77

Analyze each polynomial function by following Steps 1 through 6 on page 333 .
$$f(x)=x^{2}(x-2)(x+2)$$

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Problem 78

Analyze each polynomial function by following Steps 1 through 6 on page 333 .
$$f(x)=x^{2}(x-3)(x+4)$$

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Problem 79

Analyze each polynomial function by following Steps 1 through 6 on page 333 .
$$f(x)=(x+1)^{2}(x-2)^{2}$$

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Problem 80

Analyze each polynomial function by following Steps 1 through 6 on page 333 .
$$f(x)=(x+1)^{3}(x-3)$$

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Problem 81

Analyze each polynomial function by following Steps 1 through 6 on page 333 .
$$f(x)=x^{2}(x-3)(x+1)$$

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Problem 82

Analyze each polynomial function by following Steps 1 through 6 on page 333 .
$$f(x)=x^{2}(x-3)(x-1)$$

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Problem 83

Analyze each polynomial function by following Steps 1 through 6 on page 333 .
$$f(x)=(x+2)^{2}(x-4)^{2}$$

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Problem 84

Analyze each polynomial function by following Steps 1 through 6 on page 333 .
$$f(x)=(x-2)^{2}(x+2)(x+4)$$

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Problem 85

Analyze each polynomial function by following Steps 1 through 6 on page 333 .
$$f(x)=x^{2}(x-2)\left(x^{2}+3\right)$$

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Problem 86

Analyze each polynomial function by following Steps 1 through 6 on page 333 .
$$f(x)=x^{2}\left(x^{2}+1\right)(x+4)$$

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Problem 87

Analyze each polynomial function $f$ by following Steps 1 through 8 on page 336.
$$f(x)=x^{3}+0.2 x^{2}-1.5876 x-0.31752$$

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Problem 88

Analyze each polynomial function $f$ by following Steps 1 through 8 on page
$$f(x)=x^{3}-0.8 x^{2}-4.6656 x+3.73248$$

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Problem 89

Analyze each polynomial function $f$ by following Steps 1 through 8 on page
$$f(x)=x^{3}+2.56 x^{2}-3.31 x+0.89$$

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Problem 90

Analyze each polynomial function $f$ by following Steps 1 through 8 on page
$$f(x)=x^{3}-2.91 x^{2}-7.668 x-3.8151$$

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Problem 91

Analyze each polynomial function $f$ by following Steps 1 through 8 on page 336.
$$f(x)=x^{4}-2.5 x^{2}+0.5625$$

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Problem 92

Analyze each polynomial function $f$ by following Steps 1 through 8 on page 336.
$$f(x)=x^{4}-18.5 x^{2}+50.2619$$

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Problem 93

Analyze each polynomial function $f$ by following Steps 1 through 8 on page 336.
$$f(x)=2 x^{4}-\pi x^{3}+\sqrt{5} x-4$$

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Problem 94

Analyze each polynomial function $f$ by following Steps 1 through 8 on page 336.
$$f(x)=-1.2 x^{4}+0.5 x^{2}-\sqrt{3} x+2$$

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Problem 95

In Problems $95-102$, analyze each polynomial function by following Steps 1 through 6 on page 333 . [Hint: You will need to first factor the polynomial].
$$f(x)=4 x-x^{3}$$

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Problem 96

Analyze each polynomial function by following Steps 1 through 6 on page 333 . [Hint: You will need to first factor the polynomial].
$$f(x)=x-x^{3}$$

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Problem 97

Analyze each polynomial function by following Steps 1 through 6 on page 333 . [Hint: You will need to first factor the polynomial].
$$f(x)=x^{3}+x^{2}-12 x$$

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Problem 98

Analyze each polynomial function by following Steps 1 through 6 on page 333 . [Hint: You will need to first factor the polynomial].
$$f(x)=x^{3}+2 x^{2}-8 x$$

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Problem 99

Analyze each polynomial function by following Steps 1 through 6 on page 333 . [Hint: You will need to first factor the polynomial].
$$f(x)=2 x^{4}+12 x^{3}-8 x^{2}-48 x$$

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Problem 100

Analyze each polynomial function by following Steps 1 through 6 on page 333 . [Hint: You will need to first factor the polynomial].
$$f(x)=4 x^{3}+10 x^{2}-4 x-10$$

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Problem 101

Construct a polynomial function $f$ with the given characteristics.
Zeros: $-3,1,4 ;$ degree $3 ; y$ -intercept: 36

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Problem 104

Construct a polynomial function $f$ with the given characteristics.
Zeros: $-4,-1,2 ;$ degree $3 ; y$ -intercept: 16

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Problem 105

Construct a polynomial function $f$ with the given characteristics.
Zeros: $-5$ (multiplicity 2 ); 2 (multiplicity 1 ): 4 (multiplicity 1 ); degree $4 ;$ contains the point $(3,128)$

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Problem 106

Construct a polynomial function $f$ with the given characteristics.
Zeros: $-4 \text { (multiplicity } 1) ; 0$ (multiplicity 3 ); 2 (multiplicity 1 ):
degree 5 ; contains the point $(-2,64)$

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Problem 107

Construct a polynomial function $f$ with the given characteristics.
$-G(x)=(x+3)^{2}(x-2)$
(a) Identify the $x$ -intercepts of the graph of $G$
(b) What are the $x$ -intercepts of the graph of $y=G(x+3) ?$

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Problem 108

Construct a polynomial function $f$ with the given characteristics.
$h(x)=(x+2)(x-4)^{3}$
(a) Identify the $x$ -intercepts of the graph of $h$
$\begin{array}{lllllllll}\text { (b) What } & \text { are } & \text { the } & x \text { -intercepts } & \text { of } & \text { the } & \text { graph } & \text { of }\end{array}$ $y=h(x-2) ?$

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