4.1 Section Exercises

Verbal

1. Explain the difference between the coefficient of a power function and its degree.

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2. If a polynomial function is in factored form, what would be a good first step in order to determine the degree of the function?

3. In general, explain the end behavior of a power function with odd degree if the leading coefficient is positive.

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As $\text{ }x\text{ }$ decreases without bound, so does $\text{ }f\left(x\right).\text{ }$ As $\text{ }x\text{ }$ increases without bound, so does $\text{ }f\left(x\right).$

4. What is the relationship between the degree of a polynomial function and the maximum number of turning points in its graph?

5. What can we conclude if, in general, the graph of a polynomial function exhibits the following end behavior? As $\text{ }x\to -\infty ,\text{ }f\left(x\right)\to -\infty \text{ }$ and as $\text{ }x\to \infty ,\text{ }f\left(x\right)\to -\infty .\text{ }$

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Algebraic

For the following exercises, identify the function as a power function, a polynomial function, or neither.

6. $f\left(x\right)={x}^{5}$

7. $f\left(x\right)={\left({x}^{2}\right)}^{3}$

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8. $f\left(x\right)=x-{x}^{4}$

9. $f\left(x\right)=\frac{{x}^{2}}{{x}^{2}-1}$

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10. $f\left(x\right)=2x\left(x+2\right){\left(x-1\right)}^{2}$

11. $f\left(x\right)={3}^{x+1}$

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For the following exercises, find the degree and leading coefficient for the given polynomial.

12. $-3x{}^{4}$

13. $7-2{x}^{2}$

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14. $-2{x}^{2}- 3{x}^{5}+ x-6$

15. $x\left(4-{x}^{2}\right)\left(2x+1\right)$

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16. ${x}^{2}{\left(2x-3\right)}^{2}$

For the following exercises, determine the end behavior of the functions.

17. $f\left(x\right)={x}^{4}$

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$\text{As}\text{ }x\to \infty ,\text{ }\text{ }f\left(x\right)\to \infty ,\text{ }\text{as}\text{ }x\to -\infty ,\text{ }f\left(x\right)\to \infty$

18. $f\left(x\right)={x}^{3}$

19. $f\left(x\right)=-{x}^{4}$

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$\text{As}\text{ }x\to -\infty ,\text{ }\text{ }f\left(x\right)\to -\infty ,\text{ }\text{as}\text{ }x\to \infty ,\text{ }f\left(x\right)\to -\infty$

20. $f\left(x\right)=-{x}^{9}$

21. $f\left(x\right)=-2{x}^{4}- 3{x}^{2}+ x-1$

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$\text{As}\text{ }x\to -\infty ,\text{ }\text{ }f\left(x\right)\to -\infty ,\text{ }\text{as}\text{ }x\to \infty ,\text{ }f\left(x\right)\to -\infty$

22. $f\left(x\right)=3{x}^{2}+ x-2$

23. $f\left(x\right)={x}^{2}\left(2{x}^{3}-x+1\right)$

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$\text{As}\text{ }x\to \infty ,\text{ }\text{ }f\left(x\right)\to \infty ,\text{ }\text{as}\text{ }x\to -\infty ,\text{ }f\left(x\right)\to -\infty$

24. $f\left(x\right)={\left(2-x\right)}^{7}$

For the following exercises, find the intercepts of the functions.

25. $f\left(t\right)=2\left(t-1\right)\left(t+2\right)\left(t-3\right)$

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y-intercept is $\text{ }\left(0,12\right),\text{ }$ t-intercepts are $\text{ }\left(1,0\right);\left(–2,0\right);\text{and }\left(3,0\right).$

26. $g\left(n\right)=-2\left(3n-1\right)\left(2n+1\right)$

27. $f\left(x\right)={x}^{4}-16$

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y-intercept is $\text{ }\left(0,-16\right).\text{ }$ x-intercepts are $\text{ }\left(2,0\right)\text{ }$ and $\text{ }\left(-2,0\right).$

28. $f\left(x\right)={x}^{3}+27$

29. $f\left(x\right)=x\left({x}^{2}-2x-8\right)$

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y-intercept is $\text{ }\left(0,0\right).\text{ }$ x-intercepts are $\text{ }\left(0,0\right),\left(4,0\right),\text{ }$ and $\text{ }\left(-2, 0\right).$

30. $f\left(x\right)=\left(x+3\right)\left(4{x}^{2}-1\right)$

Graphical

For the following exercises, determine the least possible degree of the polynomial function shown.

31.

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32.

33.

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34.

35.

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36.

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38.

For the following exercises, determine whether the graph of the function provided is a graph of a polynomial function. If so, determine the number of turning points and the least possible degree for the function.

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Numeric

For the following exercises, make a table to confirm the end behavior of the function.

46. $f\left(x\right)=-{x}^{3}$

47. $f\left(x\right)={x}^{4}-5{x}^{2}$

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$x$	$f\left(x\right)$
10	9,500
100	99,950,000
–10	9,500
–100	99,950,000

$\text{as}\text{ }x\to -\infty ,\text{ }\text{ }f\left(x\right)\to \infty ,\text{ }\text{as}\text{ }x\to \infty ,\text{ }f\left(x\right)\to \infty$

48. $f\left(x\right)={x}^{2}{\left(1-x\right)}^{2}$

49. $f\left(x\right)=\left(x-1\right)\left(x-2\right)\left(3-x\right)$

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$x$	$f\left(x\right)$
10	–504
100	–941,094
–10	1,716
–100	1,061,106

$\text{as}\text{ }x\to -\infty ,\text{ }\text{ }f\left(x\right)\to \infty ,\text{ }\text{as}\text{ }x\to \infty ,\text{ }f\left(x\right)\to -\infty$

50. $f\left(x\right)=\frac{{x}^{5}}{10}-{x}^{4}$

Technology

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior.

51. $f\left(x\right)={x}^{3}\left(x-2\right)$

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The $\text{ }y\text{-}$ intercept is $\text{ }\left(0, 0\right).\text{ }$ The $\text{ }x\text{-}$ intercepts are $\text{ }\left(0, 0\right),\text{ }\left(2, 0\right).\text{ }$ $\text{As}\text{ }x\to -\infty ,\text{ }\text{ }f\left(x\right)\to \infty ,\text{ }\text{as}\text{ }x\to \infty ,\text{ }f\left(x\right)\to \infty$

52. $f\left(x\right)=x\left(x-3\right)\left(x+3\right)$

53. $f\left(x\right)=x\left(14-2x\right)\left(10-2x\right)$

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The $\text{ }y\text{-}$ intercept is $\text{ }\left(0,0\right)$ . The $\text{ }x\text{-}$ intercepts are $\text{ }\left(0, 0\right),\text{ }\left(5, 0\right),\text{ }\left(7, 0\right).\text{ }$ $\text{As}\text{ }x\to -\infty ,\text{ }\text{ }f\left(x\right)\to -\infty ,\text{ }\text{as}\text{ }x\to \infty ,\text{ }f\left(x\right)\to \infty$

54. $f\left(x\right)=x\left(14-2x\right){\left(10-2x\right)}^{2}$

55. $f\left(x\right)={x}^{3}-16x$

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The $\text{ }y\text{-}$ intercept is $\text{ }\left(0, 0\right).\text{ }$ The $\text{ }x\text{-}$ intercept is $\text{ }\left(-4, 0\right),\text{ }\left(0, 0\right),\text{ }\left(4, 0\right).\text{ }$ $As\text{ }x\to -\infty ,\text{ }\text{ }f\left(x\right)\to -\infty ,\text{ }\text{as}\text{ }x\to \infty ,\text{ }f\left(x\right)\to \infty$

56. $f\left(x\right)={x}^{3}-27$

57. $f\left(x\right)={x}^{4}-81$

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The $\text{ }y\text{-}$ intercept is $\text{ }\left(0, -81\right).\text{ }$ The $\text{ }x\text{-}$ intercept are $\text{ }\left(3, 0\right),\text{ }\left(-3, 0\right).\text{ }$ $\text{As}\text{ }x\to -\infty ,\text{ }\text{ }f\left(x\right)\to \infty ,\text{ }\text{as}\text{ }x\to \infty ,\text{ }f\left(x\right)\to \infty$

58. $f\left(x\right)=-{x}^{3}+{x}^{2}+2x$

59. $f\left(x\right)={x}^{3}-2{x}^{2}-15x$

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The $\text{ }y\text{-}$ intercept is $\text{ }\left(0, 0\right).\text{ }$ The $\text{ }x\text{-}$ intercepts are $\text{ }\left(-3, 0\right),\text{ }\left(0, 0\right),\text{ }\left(5, 0\right).\text{ }$ $\text{As}\text{ }x\to -\infty ,\text{ }\text{ }f\left(x\right)\to -\infty ,\text{ }\text{as}\text{ }x\to \infty ,\text{ }f\left(x\right)\to \infty$

60. $f\left(x\right)={x}^{3}-0.01x$

Extensions

For the following exercises, use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or –1. There may be more than one correct answer.

61. The $\text{ }y-$ intercept is $\text{ }\left(0,-4\right).\text{ }$ The $\text{ }x-$ intercepts are $\text{ }\left(-2,0\right),\text{ }\left(2,0\right).\text{ }$ Degree is 2.

End behavior: $\text{ }\text{as}\text{ }x\to -\infty ,\text{ }\text{ }f\left(x\right)\to \infty ,\text{ }\text{as}\text{ }x\to \infty ,\text{ }f\left(x\right)\to \infty .$

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$f\left(x\right)={x}^{2}-4$

62. The $\text{ }y-$ intercept is $\text{ }\left(0,9\right).\text{ }$ The $\text{ }x\text{-}$ intercepts are $\text{ }\left(-3,0\right),\text{ }\left(3,0\right).\text{ }$ Degree is 2.

End behavior: $\text{ }\text{as}\text{ }x\to -\infty ,\text{ }\text{ }f\left(x\right)\to -\infty ,\text{ }\text{as}\text{ }x\to \infty ,\text{ }f\left(x\right)\to -\infty .$

63. The $\text{ }y-$ intercept is $\text{ }\left(0,0\right).\text{ }$ The $\text{ }x-$ intercepts are $\text{ }\left(0,0\right),\text{ }\left(2,0\right).\text{ }$ Degree is 3.

End behavior: $\text{ }\text{as}\text{ }x\to -\infty ,\text{ }\text{ }f\left(x\right)\to -\infty ,\text{ }\text{as}\text{ }x\to \infty ,\text{ }f\left(x\right)\to \infty .$

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$f\left(x\right)={x}^{3}-4{x}^{2}+4x$

64. The $\text{ }y-$ intercept is $\text{ }\left(0,1\right).\text{ }$ The $\text{ }x-$ intercept is $\text{ }\left(1,0\right).\text{ }$ Degree is 3.

End behavior: $\text{ }\text{as}\text{ }x\to -\infty ,\text{ }\text{ }f\left(x\right)\to \infty ,\text{ }\text{as}\text{ }x\to \infty ,\text{ }f\left(x\right)\to -\infty .$

65. The $\text{ }y-$ intercept is $\text{ }\left(0,1\right).\text{ }$ There is no $\text{ }x-$ intercept. Degree is 4.

End behavior: $\text{ }\text{as}\text{ }x\to -\infty ,\text{ }\text{ }f\left(x\right)\to \infty ,\text{ }\text{as}\text{ }x\to \infty ,\text{ }f\left(x\right)\to \infty .$

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$f\left(x\right)={x}^{4}+1$

Real-World Applications

For the following exercises, use the written statements to construct a polynomial function that represents the required information.

66. An oil slick is expanding as a circle. The radius of the circle is increasing at the rate of 20 meters per day. Express the area of the circle as a function of $\text{ }d,\text{ }$ the number of days elapsed.

67. A cube has an edge of 3 feet. The edge is increasing at the rate of 2 feet per minute. Express the volume of the cube as a function of $\text{ }m,\text{ }$ the number of minutes elapsed.

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$V\left(m\right)=8{m}^{3}+36{m}^{2}+54m+27$

68. A rectangle has a length of 10 inches and a width of 6 inches. If the length is increased by $\text{ }x\text{ }$ inches and the width increased by twice that amount, express the area of the rectangle as a function of $\text{ }x.$

69. An open box is to be constructed by cutting out square corners of $\text{ }x-$ inch sides from a piece of cardboard 8 inches by 8 inches and then folding up the sides. Express the volume of the box as a function of $\text{ }x.$

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$V\left(x\right)=4{x}^{3}-32{x}^{2}+64x$

70. A rectangle is twice as long as it is wide. Squares of side 2 feet are cut out from each corner. Then the sides are folded up to make an open box. Express the volume of the box as a function of the width ( $x$ ).

Polynomial and Rational Functions