# 3.3E: Exercises

- Page ID
- 31083

## Section Exercises

### Verbal

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

Answer: The coefficient of the power function is the real number that is multiplied by the variable raised to a power. The degree is the highest power appearing in the function.

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.

Answer: As \(x\) decreases without bound, so does \(f(x)\). As \(x\) increases without bound, so does \(f(x)\).

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 x→−∞, \(f(x)→−∞\) and as x→∞, \(f(x)→−∞\).

Answer: The polynomial function is of even degree and leading coefficient is negative.

### Algebraic

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

6. \(f(x)=x^5\)

7. \(f(x)=(x^2)^3\)

Answer: Power function

8. \(f(x)=x−x^4\)

Exercise \(\PageIndex{9}\)

\(f(x)=\frac{x^2}{x^2−1}\)

**Answer**-
Neither

10. \(f(x)=2x(x+2)(x−1)^2\)

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

Answer: Neither

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

12. \(−3x^4\)

13. \(7−2x^2\)

Answer: Degree = 2, Coefficient = –2

14. \(−2x^2− 3x^5+ x−6\)

15. \(x(4−x^2)(2x+1)\)

Answer: Degree =4, Coefficient = –2

16. \(x^2(2x−3)^2\)

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

17. \(f(x)=x^4\)

Answer: As \(x→∞\), \(f(x)→∞\), as \(x→−∞\), \(f(x)→∞\)

18. \(f(x)=x^3\)

19. \(f(x)=−x^4\)

Answer: As \(x→−∞\), \(f(x)→−∞\), as \(x→∞\), \(f(x)→−∞\)

20. \(f(x)=−x^9\)

21. \(f(x)=−2x^4− 3x^2+ x−1\)

Answer: As \(x→−∞\), \(f(x)→−∞\), as \(x→∞\), \(f(x)→−∞\)

22. \(f(x)=3x^2+ x−2\)

23. \(f(x)=x^2(2x^3−x+1)\)

Answer: As \(x→∞\), \(f(x)→∞\), as \(x→−∞\), \(f(x)→−∞\)

24. \(f(x)=(2−x)^7\)

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

25. \(f(t)=2(t−1)(t+2)(t−3)\)

Answer: y-intercept is \((0,12)\), t-intercepts are \((1,0);(–2,0); \) and \((3,0)\).

26. \(g(n)=−2(3n−1)(2n+1)\)

Exercise \(\PageIndex{27}\)

\(f(x)=x^4−16\)

**Answer**-
y-intercept is \((0,−16).\) x-intercepts are \((2,0)\) and \((−2,0)\).

28. \(f(x)=x^3+27\)

29. \(f(x)=x(x^2−2x−8)\)

Answer: y-intercept is \((0,0)\). x-intercepts are \((0,0),(4,0),\) and \((−2, 0)\).

30. \(f(x)=(x+3)(4x^2−1)\)

### Graphical

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

31.

Answer: 3

32.

33.

Answer: 5

34.

35.

Answer: 3

36.

37.

Answer: 5

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.

39.

Answer: Yes. Number of turning points is 2. Least possible degree is 3.

40.

41.

Answer: Yes. Number of turning points is 1. Least possible degree is 2.

42.

43.

Answer: Yes. Number of turning points is 0. Least possible degree is 3.

Exercise \(\PageIndex{44}\)

**Answer**-
No (the graph is not smooth)

45.

Answer: Yes. Number of turning points is 0. Least possible degree is 1.

### Numeric

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

46. \(f(x)=−x^3\)

47. \(f(x)=x^4−5x^2\)

Answer:

\(x\) | \(f(x)\) |
---|---|

10 | 9,500 |

100 | 99,950,000 |

–10 | 9,500 |

–100 | 99,950,000 |

as \(x→−∞,\) \(f(x)→∞\), as \(x→∞,\) \(f(x)→∞\)

48. \(f(x)=x^2(1−x)^2\)

49. \(f(x)=(x−1)(x−2)(3−x)\)

Answer:

\(x\) | \(f(x)\) |
---|---|

10 | 9,500 |

100 | 99,950,000 |

–10 | 9,500 |

–100 | 99,950,000 |

as \(x→−∞,\) \(f(x)→∞\), as \(x→∞,\) \(f(x)→−∞\)

50. \(f(x)=\frac{x^5}{10}−x^4\)

### Technology

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

51. \(f(x)=x^3(x−2)\)

Answer:

The y-intercept is \((0, 0)\). The x-intercepts are \((0, 0), (2, 0).\) As \(x→−∞,\) \(f(x)→∞\), as \(x→∞,\) \(f(x)→∞\)

52. \(f(x)=x(x−3)(x+3)\)

53. \(f(x)=x(14−2x)(10−2x)\)

Answer:

The y-intercept is \((0,0)\) . The x-intercepts are \((0, 0), (5, 0), (7, 0)\). As \(x→−∞\), \(f(x)→−∞\), as \(x→∞,\) \(f(x)→∞\)

54. \(f(x)=x(14−2x)(10−2x)^2\)

55. \(f(x)=x^3−16x\)

Answer:

The y-intercept is (0, 0). The x-intercept is \((−4, 0), (0, 0), (4, 0)\). As \(x→−∞\), \(f(x)→−∞\), as \(x→∞,\) \(f(x)→∞\)

56. \(f(x)=x^3−27\)

57. \(f(x)=x^4−81\)

Answer:

The y-intercept is (0, −81). The x-intercept are \((3, 0), (−3, 0)\). As \(x→−∞,\) \(f(x)→∞\), as \(x→∞,\) \(f(x)→∞\)

58. \(f(x)=−x^3+x^2+2x\)

59. \(f(x)=x^3−2x^2−15x\)

Answer:

The y-intercept is \((0, 0)\). The x-intercepts are \((−3, 0), (0, 0), (5, 0).\) As \(x→−∞\), \(f(x)→−∞\), as \(x→∞,\) \(f(x)→∞\)

60. \(f(x)=x^3−0.01x\)

### 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 y-intercept is \((0,−4)\). The x-intercepts are \((−2,0), (2,0)\). Degree is 2.

End behavior: as \(x→−∞,\) \(f(x)→∞\), as \(x→∞,\) \(f(x)→∞\).

Answer: \(f(x)=x^2−4\)

62. The y-intercept is \((0,9)\). The x-intercepts are \((−3,0), (3,0)\). Degree is 2.

End behavior: as \(x→−∞,\) \(f(x)→−∞\), as \(x→∞,\) \(f(x)→−∞\).

63. The y-intercept is \((0,0)\). The x-intercepts are \((0,0), (2,0)\). Degree is 3.

End behavior: as \(x→−∞,\) \(f(x)→−∞\), as \(x→∞,\) \(f(x)→∞\).

Answer: \(f(x)=x^3−4x^2+4x\)

64. The y-intercept is \((0,1)\). The x-intercept is \((1,0)\). Degree is 3.

End behavior: as \(x→−∞\), \(f(x)→∞\), as \(x→∞\), \(f(x)→−∞\).

65. The y-intercept is \((0,1)\). There is no x-intercept. Degree is 4.

End behavior: as \(x→−∞,\) \(f(x)→∞\), as \(x→∞,\) \(f(x)→∞\).

Answer: \(f(x)=x^4+1\)

### 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 \(d\), 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.

Answer: \(V(m)=8m^3+36m^2+54m+27\)

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 \(x\).

Exercise \(\PageIndex{69}\)

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 \(x\).

**Answer**-
\(V(x)=4x^3−32x^2+64x\)

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).