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

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.

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 [latex]x\to -\infty ,f\left(x\right)\to -\infty [/latex] and as [latex]x\to \infty ,f\left(x\right)\to -\infty [/latex].

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

6. [latex]f\left(x\right)={x}^{5}[/latex]

7. [latex]f\left(x\right)={\left({x}^{2}\right)}^{3}[/latex]

8. [latex]f\left(x\right)=x-{x}^{4}[/latex]

9. [latex]f\left(x\right)=\frac{{x}^{2}}{{x}^{2}-1}[/latex]

10. [latex]f\left(x\right)=2x\left(x+2\right){\left(x - 1\right)}^{2}[/latex]

11. [latex]f\left(x\right)={3}^{x+1}[/latex]

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

12. [latex]-3x{}^{4}[/latex]

13. [latex]7 - 2{x}^{2}[/latex]

14. [latex]-2{x}^{2}- 3{x}^{5}+ x - 6 [/latex]

15. [latex]x\left(4-{x}^{2}\right)\left(2x+1\right)[/latex]

16. [latex]{x}^{2}{\left(2x - 3\right)}^{2}[/latex]

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

17. [latex]f\left(x\right)={x}^{4}[/latex]

18. [latex]f\left(x\right)={x}^{3}[/latex]

19. [latex]f\left(x\right)=-{x}^{4}[/latex]

20. [latex]f\left(x\right)=-{x}^{9}[/latex]

21. [latex]f\left(x\right)=-2{x}^{4}- 3{x}^{2}+ x - 1[/latex]

22. [latex]f\left(x\right)=3{x}^{2}+ x - 2[/latex]

23. [latex]f\left(x\right)={x}^{2}\left(2{x}^{3}-x+1\right)[/latex]

24. [latex]f\left(x\right)={\left(2-x\right)}^{7}[/latex]

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

25. [latex]f\left(t\right)=2\left(t - 1\right)\left(t+2\right)\left(t - 3\right)[/latex]

26. [latex]g\left(n\right)=-2\left(3n - 1\right)\left(2n+1\right)[/latex]

27. [latex]f\left(x\right)={x}^{4}-16[/latex]

28. [latex]f\left(x\right)={x}^{3}+27[/latex]

29. [latex]f\left(x\right)=x\left({x}^{2}-2x - 8\right)[/latex]

30. [latex]f\left(x\right)=\left(x+3\right)\left(4{x}^{2}-1\right)[/latex]

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

31.

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

46. [latex]f\left(x\right)=-{x}^{3}[/latex]

47. [latex]f\left(x\right)={x}^{4}-5{x}^{2}[/latex]

48. [latex]f\left(x\right)={x}^{2}{\left(1-x\right)}^{2}[/latex]

49. [latex]f\left(x\right)=\left(x - 1\right)\left(x - 2\right)\left(3-x\right)[/latex]

50. [latex]f\left(x\right)=\frac{{x}^{5}}{10}-{x}^{4}[/latex]

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

51. [latex]f\left(x\right)={x}^{3}\left(x - 2\right)[/latex]

52. [latex]f\left(x\right)=x\left(x - 3\right)\left(x+3\right)[/latex]

53. [latex]f\left(x\right)=x\left(14 - 2x\right)\left(10 - 2x\right)[/latex]

54. [latex]f\left(x\right)=x\left(14 - 2x\right){\left(10 - 2x\right)}^{2}[/latex]

55. [latex]f\left(x\right)={x}^{3}-16x[/latex]

56. [latex]f\left(x\right)={x}^{3}-27[/latex]

57. [latex]f\left(x\right)={x}^{4}-81[/latex]

58. [latex]f\left(x\right)=-{x}^{3}+{x}^{2}+2x[/latex]

59. [latex]f\left(x\right)={x}^{3}-2{x}^{2}-15x[/latex]

60. [latex]f\left(x\right)={x}^{3}-0.01x[/latex]

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 y-intercept is [latex]\left(0,-4\right)[/latex]. The x-intercepts are [latex]\left(-2,0\right),\left(2,0\right)[/latex]. Degree is 2.

End behavior: [latex]\text{as }x\to -\infty ,f\left(x\right)\to \infty ,\text{ as }x\to \infty ,f\left(x\right)\to \infty [/latex].

62. The y-intercept is [latex]\left(0,9\right)[/latex]. The x-intercepts are [latex]\left(-3,0\right),\left(3,0\right)[/latex]. Degree is 2.

End behavior: [latex]\text{as }x\to -\infty ,f\left(x\right)\to -\infty ,\text{ as }x\to \infty ,f\left(x\right)\to -\infty\\ [/latex].

63. The y-intercept is [latex]\left(0,0\right)[/latex]. The x-intercepts are [latex]\left(0,0\right),\left(2,0\right)[/latex]. Degree is 3.

End behavior: [latex]\text{as }x\to -\infty ,f\left(x\right)\to -\infty ,\text{ as }x\to \infty ,f\left(x\right)\to \infty [/latex].

64. The y-intercept is [latex]\left(0,1\right)[/latex]. The x-intercept is [latex]\left(1,0\right)[/latex]. Degree is 3.

End behavior: [latex]\text{as }x\to -\infty ,f\left(x\right)\to \infty ,\text{ as }x\to \infty ,f\left(x\right)\to -\infty [/latex].

65. The y-intercept is [latex]\left(0,1\right)[/latex]. There is no x-intercept. Degree is 4.

End behavior: [latex]\text{as }x\to -\infty ,f\left(x\right)\to \infty ,\text{ as }x\to \infty ,f\left(x\right)\to \infty[/latex].

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 d, 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 m, the number of minutes elapsed.

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

69. An open box is to be constructed by cutting out square corners of 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 x.

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).