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Review Exercises

Determine whether the statement is true or false.

1. If $f (x) = (x + a) (x + b) (x - c)$ $f (x) = (x + a) (x + b) (x - c)$ , then $f (- b) = 0$ $f (- b) = 0$ . [4.3]
2. The graph of a rational function never crosses a vertical asymptote. [4.5]
3. For the function $g (x) = x^{4} - 8 x^{2} - 9$ $g (x) = x^{4} - 8 x^{2} - 9$ , the only possible rational zeros are 1, −1, 3, and −3. [4.4]
4. The graph of $P (x) = x^{6} - x^{8}$ $P (x) = x^{6} - x^{8}$ has at most 6 x-intercepts. [4.2]
5. The domain of the function

$f (x) = \frac{x - 4}{(x + 2) (x - 3)}$ $f (x) = \frac{x - 4}{(x + 2) (x - 3)}$

is $(- \infty, - 2) \cup (3, \infty)$ $(- \infty, - 2) \cup (3, \infty)$ . [4.5]

Determine the leading term, the leading coefficient, and the degree of the polynomial. Then classify the polynomial function as constant, linear, quadratic, cubic, or quartic. [4.1]

6. $f (x) = 7 x^{2} - 5 + 0.45 x^{4} - 3 x^{3}$ $f (x) = 7 x^{2} - 5 + 0.45 x^{4} - 3 x^{3}$
7. $h (x) = - 25$ $h (x) = - 25$
8. $g (x) = 6 - 0.5 x$ $g (x) = 6 - 0.5 x$
9. $f (x) = \frac{1}{3} x^{3} - 2 x + 3$ $f (x) = \frac{1}{3} x^{3} - 2 x + 3$

Use the leading-term test to describe the end behavior of the graph of the function. [4.1]

10. $f (x) = - \frac{1}{2} x^{4} + 3 x^{2} + x - 6$ $f (x) = - \frac{1}{2} x^{4} + 3 x^{2} + x - 6$
11. $f (x) = x^{5} + 2 x^{3} - x^{2} + 5 x + 4$ $f (x) = x^{5} + 2 x^{3} - x^{2} + 5 x + 4$

Find the zeros of the polynomial function and state the multiplicity of each. [4.1]

12. $g (x) = (x - \frac{2}{3}) {(x + 2)}^{3} {(x - 5)}^{2}$ $g (x) = (x - \frac{2}{3}) {(x + 2)}^{3} {(x - 5)}^{2}$
13. $f (x) = x^{4} - 26 x^{2} + 25$ $f (x) = x^{4} - 26 x^{2} + 25$
14. $h (x) = x^{3} + 4 x^{2} - 9 x - 36$ $h (x) = x^{3} + 4 x^{2} - 9 x - 36$
15. Interest Compounded Annually. When P dollars is invested at interest rate i, compounded annually, for t years, the investment grows to A dollars, where

$A = P {(1 + i)}^{t} .$ $A = P {(1 + i)}^{t} .$
1. Find the interest rate i if $6250 grows to $6760 in 2 years. [4.1]
2. Find the interest rate i if $1,000,000 grows to $1,215,506.25 in 4 years. [4.1]

Sketch the graph of the polynomial function.

16. $f (x) = - x^{4} + 2 x^{3}$ $f (x) = - x^{4} + 2 x^{3}$ [4.2]
17. $g (x) = {(x - 1)}^{3} {(x + 2)}^{2}$ $g (x) = {(x - 1)}^{3} {(x + 2)}^{2}$ [4.2]
18. $h (x) = x^{3} + 3 x^{2} - x - 3$ $h (x) = x^{3} + 3 x^{2} - x - 3$ [4.2]
19. $f (x) = x^{4} - 5 x^{3} + 6 x^{2} + 4 x - 8$ $f (x) = x^{4} - 5 x^{3} + 6 x^{2} + 4 x - 8$ [4.2], [4.3], [4.4]
20. $g (x) = 2 x^{3} + 7 x^{2} - 14 x + 5$ $g (x) = 2 x^{3} + 7 x^{2} - 14 x + 5$ [4.2], [4.4]

Using the intermediate value theorem, determine, if possible, whether the function f has a zero between a and b. [4.2]

21. $f (x) = 4 x^{2} - 5 x - 3;$ $f (x) = 4 x^{2} - 5 x - 3;$ $a = 1, b = 2$ $a = 1, b = 2$
22. $f (x) = x^{3} - 4 x^{2} + \frac{1}{2} x + 2;$ $f (x) = x^{3} - 4 x^{2} + \frac{1}{2} x + 2;$ $a = - 1, b = 1$ $a = - 1, b = 1$

In each of the following, a polynomial $P (x)$ $P (x)$ and a divisor $d (x)$ $d (x)$ are given. Use long division to find the quotient $Q (x)$ $Q (x)$ and the remainder $R (x)$ $R (x)$ when $P (x)$ $P (x)$ is divided by $d (x)$ $d (x)$ . Express $P (x)$ $P (x)$ in the form $d (x) \cdot Q (x) + R (x)$ $d (x) \cdot Q (x) + R (x)$ . [4.3]

23. $\begin{array}{rcl} P (x) & = & 6 x^{3} - 2 x^{2} + 4 x - 1, \\ d (x) & = & x - 3 \end{array}$ $\begin{array}{rcl} P (x) & = & 6 x^{3} - 2 x^{2} + 4 x - 1, \\ d (x) & = & x - 3 \end{array}$
24. $\begin{array}{rcl} P (x) & = & x^{4} - 2 x^{3} + x + 5, \\ d (x) & = & x + 1 \end{array}$ $\begin{array}{rcl} P (x) & = & x^{4} - 2 x^{3} + x + 5, \\ d (x) & = & x + 1 \end{array}$

Use synthetic division to find the quotient and the remainder. [4.3]

25. $(x^{3} + 2 x^{2} - 13 x + 10) \div (x - 5)$ $(x^{3} + 2 x^{2} - 13 x + 10) \div (x - 5)$
26. $(x^{4} + 3 x^{3} + 3 x^{2} + 3 x + 2) \div (x + 2)$ $(x^{4} + 3 x^{3} + 3 x^{2} + 3 x + 2) \div (x + 2)$
27. $(x^{5} - 2 x) \div (x + 1)$ $(x^{5} - 2 x) \div (x + 1)$

Use synthetic division to find the indicated function value. [4.3]

28. $f (x) = x^{3} + 2 x^{2} - 13 x + 10;$ $f (x) = x^{3} + 2 x^{2} - 13 x + 10;$ $f (- 2)$ $f (- 2)$
29. $f (x) = x^{4} - 16;$ $f (x) = x^{4} - 16;$ $f (- 2)$ $f (- 2)$
30. $f (x) = x^{5} - 4 x^{4} + x^{3} - x^{2} + 2 x - 100;$ $f (x) = x^{5} - 4 x^{4} + x^{3} - x^{2} + 2 x - 100;$ $f (- 10)$ $f (- 10)$

Using synthetic division, determine whether the given numbers are zeros of the polynomial function. [4.3]

31. $- i, - 5; f (x) = x^{3} - 5 x^{2} + x - 5$ $- i, - 5; f (x) = x^{3} - 5 x^{2} + x - 5$
32. $- 1, - 2; f (x) = x^{4} - 4 x^{3} - 3 x^{2} + 14 x - 8$ $- 1, - 2; f (x) = x^{4} - 4 x^{3} - 3 x^{2} + 14 x - 8$
33. $\frac{1}{3}$ $\frac{1}{3}$ , 1; $f (x) = x^{3} - \frac{4}{3} x^{2} - \frac{5}{3} x + \frac{2}{3}$ $f (x) = x^{3} - \frac{4}{3} x^{2} - \frac{5}{3} x + \frac{2}{3}$
34. 2, $- \sqrt{3};$ $- \sqrt{3};$ $f (x) = x^{4} - 5 x^{2} + 6$ $f (x) = x^{4} - 5 x^{2} + 6$

Factor the polynomial $f (x)$ $f (x)$ . Then solve the equation $f (x) = 0$ $f (x) = 0$ . [4.3], [4.4]

35. $f (x) = x^{3} + 2 x^{2} - 7 x + 4$ $f (x) = x^{3} + 2 x^{2} - 7 x + 4$
36. $f (x) = x^{3} + 4 x^{2} - 3 x - 18$ $f (x) = x^{3} + 4 x^{2} - 3 x - 18$
37. $f (x) = x^{4} - 4 x^{3} - 21 x^{2} + 100 x - 100$ $f (x) = x^{4} - 4 x^{3} - 21 x^{2} + 100 x - 100$
38. $f (x) = x^{4} - 3 x^{2} + 2$ $f (x) = x^{4} - 3 x^{2} + 2$

Find a polynomial function of degree 3 with the given numbers as zeros. [4.4]

39. $- 4, - 1, 2$ $- 4, - 1, 2$
40. $- 3, 1 - i, 1 + i$ $- 3, 1 - i, 1 + i$
41. $\frac{1}{2}, 1 - \sqrt{2}, 1 + \sqrt{2}$ $\frac{1}{2}, 1 - \sqrt{2}, 1 + \sqrt{2}$
42. Find a polynomial function of degree 4 with −5 as a zero of multiplicity 3 and $\frac{1}{2}$ $\frac{1}{2}$ as a zero of multiplicity 1. [4.4]
43. Find a polynomial function of degree 5 with −3 as a zero of multiplicity 2, 2 as a zero of multiplicity 1, and 0 as a zero of multiplicity 2. [4.4]

Suppose that a polynomial function of degree 5 with rational coefficients has the given zeros. Find the other zero(s). [4.4]

44. $- \frac{2}{3}, \sqrt{5}, i$ $- \frac{2}{3}, \sqrt{5}, i$
45. $0, 1 + \sqrt{3}, - \sqrt{3}$ $0, 1 + \sqrt{3}, - \sqrt{3}$
46. $- \sqrt{2}, \frac{1}{2}, 1, 2$ $- \sqrt{2}, \frac{1}{2}, 1, 2$

Find a polynomial function of lowest degree with rational coefficients and the following as some of its zeros. [4.4]

47. $\sqrt{11}$ $\sqrt{11}$
48. $- i, 6$ $- i, 6$
49. $- 1, 4, 1 + i$ $- 1, 4, 1 + i$
50. $\sqrt{5}, - 2 i$ $\sqrt{5}, - 2 i$
51. $\frac{1}{3}, 0, - 3$ $\frac{1}{3}, 0, - 3$

List all possible rational zeros. [4.4]

52. $h (x) = 4 x^{5} - 2 x^{3} + 6 x - 12$ $h (x) = 4 x^{5} - 2 x^{3} + 6 x - 12$
53. $g (x) = 3 x^{4} - x^{3} + 5 x^{2} - x + 1$ $g (x) = 3 x^{4} - x^{3} + 5 x^{2} - x + 1$
54. $f (x) = x^{3} - 2 x^{2} + x - 24$ $f (x) = x^{3} - 2 x^{2} + x - 24$

For each polynomial function, (a) find the rational zeros and then the other zeros; that is, solve $f (x) = 0;$ $f (x) = 0;$ and (b) factor $f (x)$ $f (x)$ into linear factors. [4.4]

55. $f (x) = 3 x^{5} + 2 x^{4} - 25 x^{3} - 28 x^{2} + 12 x$ $f (x) = 3 x^{5} + 2 x^{4} - 25 x^{3} - 28 x^{2} + 12 x$
56. $f (x) = x^{3} - 2 x^{2} - 3 x + 6$ $f (x) = x^{3} - 2 x^{2} - 3 x + 6$
57. $f (x) = x^{4} - 6 x^{3} + 9 x^{2} + 6 x - 10$ $f (x) = x^{4} - 6 x^{3} + 9 x^{2} + 6 x - 10$
58. $f (x) = x^{3} + 3 x^{2} - 11 x - 5$ $f (x) = x^{3} + 3 x^{2} - 11 x - 5$
59. $f (x) = 3 x^{3} - 8 x^{2} + 7 x - 2$ $f (x) = 3 x^{3} - 8 x^{2} + 7 x - 2$
60. $f (x) = x^{5} - 8 x^{4} + 20 x^{3} - 8 x^{2} - 32 x + 32$ $f (x) = x^{5} - 8 x^{4} + 20 x^{3} - 8 x^{2} - 32 x + 32$
61. $f (x) = x^{6} + x^{5} - 28 x^{4} - 16 x^{3} + 192 x^{2}$ $f (x) = x^{6} + x^{5} - 28 x^{4} - 16 x^{3} + 192 x^{2}$
62. $f (x) = 2 x^{5} - 13 x^{4} + 32 x^{3} - 38 x^{2} + 22 x - 5$ $f (x) = 2 x^{5} - 13 x^{4} + 32 x^{3} - 38 x^{2} + 22 x - 5$

What does Descartes’ rule of signs tell you about the number of positive real zeros and the number of negative real zeros of each of the following polynomial functions? [4.4]

63. $f (x) = 2 x^{6} - 7 x^{3} + x^{2} - x$ $f (x) = 2 x^{6} - 7 x^{3} + x^{2} - x$
64. $h (x) = - x^{8} + 6 x^{5} - x^{3} + 2 x - 2$ $h (x) = - x^{8} + 6 x^{5} - x^{3} + 2 x - 2$
65. $g (x) = 5 x^{5} - 4 x^{2} + x - 1$ $g (x) = 5 x^{5} - 4 x^{2} + x - 1$

Graph the function. Be sure to label all the asymptotes. List the domain and the x- and y-intercepts. [4.5]

66. $f (x) = \frac{x^{2} - 5}{x + 2}$ $f (x) = \frac{x^{2} - 5}{x + 2}$
67. $f (x) = \frac{5}{{(x - 2)}^{2}}$ $f (x) = \frac{5}{{(x - 2)}^{2}}$
68. $f (x) = \frac{x^{2} + x - 6}{x^{2} - x - 20}$ $f (x) = \frac{x^{2} + x - 6}{x^{2} - x - 20}$
69. $f (x) = \frac{x - 2}{x^{2} - 2 x - 15}$ $f (x) = \frac{x - 2}{x^{2} - 2 x - 15}$

In Exercises 70 and 71, find a rational function that satisfies the given conditions. Answers may vary, but try to give the simplest answer possible. [4.5]

70. Vertical asymptotes $x = - 2, x = 3$ $x = - 2, x = 3$
71. Vertical asymptotes $x = - 2, x = 3;$ $x = - 2, x = 3;$ horizontal asymptote $y = 4;$ $y = 4;$ x-intercept (−3, 0)
72. Medical Dosage. The function

$N (t) = \frac{0.7 t + 2000}{8 t + 9}, t \geq 5,$ $N (t) = \frac{0.7 t + 2000}{8 t + 9}, t \geq 5,$

gives the body concentration $N (t)$ $N (t)$ , in parts per million, of a certain dosage of medication after time t, in hours.
1. Find the horizontal asymptote of the graph and complete the following:
  
  $N (t) \to as t \to \infty .$ $N (t) \to as t \to \infty .$ [4.5]
2. Explain the meaning of the answer to part (a) in terms of the application. [4.5]

Solve. [4.6]

73. $x^{2} - 9 < 0$ $x^{2} - 9 < 0$
74. $2 x^{2} > 3 x + 2$ $2 x^{2} > 3 x + 2$
75. $(1 - x) (x + 4) (x - 2) \leq 0$ $(1 - x) (x + 4) (x - 2) \leq 0$
76. $\frac{x - 2}{x + 3} < 4$ $\frac{x - 2}{x + 3} < 4$
77. Height of a Rocket. The function

$S (t) = - 16 t^{2} + 80 t + 224$ $S (t) = - 16 t^{2} + 80 t + 224$

gives the height S, in feet, of a model rocket launched with a velocity of $80 ft / \sec$ $80 ft / \sec$ from a hill that is 224 ft high, where t is the time, in seconds.
1. Determine when the rocket reaches the ground. [4.1]
2. On what interval is the height greater than 320 ft? [4.1], [4.6]
78. Population Growth. The population P, in thousands, of Novi is given by

$P (t) = \frac{8000 t}{4 t^{2} + 10},$ $P (t) = \frac{8000 t}{4 t^{2} + 10},$

where t is the time, in months. Find the interval on which the population was 400,000 or greater. [4.6]
79. Which of the following is the domain of the function

$g (x) = \frac{x^{2} + 2 x - 3}{x^{2} - 5 x + 6 ?}$ $g (x) = \frac{x^{2} + 2 x - 3}{x^{2} - 5 x + 6 ?}$ [4.5]
1. $(- \infty, 2) \cup (2, 3) \cup (3, \infty)$ $(- \infty, 2) \cup (2, 3) \cup (3, \infty)$
2. $(- \infty, - 3) \cup (- 3, 1) \cup (1, \infty)$ $(- \infty, - 3) \cup (- 3, 1) \cup (1, \infty)$
3. $(- \infty, 2) \cup (3, \infty)$ $(- \infty, 2) \cup (3, \infty)$
4. $(- \infty, - 3) \cup (1, \infty)$ $(- \infty, - 3) \cup (1, \infty)$
80. Which of the following lists the vertical asymptotes of the function

$f (x) = \frac{x - 4}{(x + 1) (x - 2) (x + 4)} ?$ $f (x) = \frac{x - 4}{(x + 1) (x - 2) (x + 4)} ?$ [4.5]
1. $x = 1, x = - 2$ $x = 1, x = - 2$ , and $x = 4$ $x = 4$
2. $x = - 1, x = 2, x = - 4$ $x = - 1, x = 2, x = - 4$ , and $x = 4$ $x = 4$
3. $x = - 1, x = 2$ $x = - 1, x = 2$ , and $x = - 4$ $x = - 4$
4. $x = 4$ $x = 4$
81. The graph of $f (x) = - \frac{1}{2} x^{4} + x^{3} + 1$ $f (x) = - \frac{1}{2} x^{4} + x^{3} + 1$ is which of the following? [4.2]

Synthesis

Solve.

82. $x^{2} \geq 5 - 2 x$ $x^{2} \geq 5 - 2 x$ [4.6]
83. $| 1 - \frac{1}{x^{2}} | < 3$ $| 1 - \frac{1}{x^{2}} | < 3$ [4.6]
84. $x^{4} - 2 x^{3} + 3 x^{2} - 2 x + 2 = 0$ $x^{4} - 2 x^{3} + 3 x^{2} - 2 x + 2 = 0$ [4.4]
85. ${(x - 2)}^{- 3} < 0$ ${(x - 2)}^{- 3} < 0$ [4.6]
86. Express $x^{3} - 1$ $x^{3} - 1$ as a product of linear factors. [4.4]
87. Find k such that $x + 3$ $x + 3$ is a factor of $x^{3} + k x^{2} + k x - 15$ $x^{3} + k x^{2} + k x - 15$ . [4.3]
88. When $x^{2} - 4 x + 3 k$ $x^{2} - 4 x + 3 k$ is divided by $x + 5$ $x + 5$ , the remainder is 33. Find the value of k. [4.3]

Find the domain of the function. [4.5]

89. $f (x) = \sqrt{x^{2} + 3 x - 10}$ $f (x) = \sqrt{x^{2} + 3 x - 10}$
90. $f (x) = \sqrt{x^{2} - 3.1 x + 2.2} + 1.75$ $f (x) = \sqrt{x^{2} - 3.1 x + 2.2} + 1.75$
91. $f (x) = \frac{1}{\sqrt{5 - | 7 x + 2 |}}$ $f (x) = \frac{1}{\sqrt{5 - | 7 x + 2 |}}$

Collaborative Discussion and Writing

92. Explain the difference between a polynomial function and a rational function. [4.1], [4.5]
93. Is it possible for a third-degree polynomial with rational coefficients to have no real zeros? Why or why not? [4.4]
94. Explain and contrast the three types of asymptotes considered for rational functions. [4.5]
95. If $P (x)$ $P (x)$ is an even function, and by Descartes’ rule of signs, $P (x)$ $P (x)$ has one positive real zero, how many negative real zeros does $P (x)$ $P (x)$ have? Explain. [4.4]
96. Explain why the graph of a rational function cannot have both a horizontal asymptote and an oblique asymptote. [4.5]
97. Under what circumstances would a quadratic inequality have a solution set that is a closed interval? [4.6]

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Table of Contents for Review Exercises

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Table of Contents for
Review Exercises