mathematics

5

343

Polynomial and Rational Functions

Figure 1 35-mm film, once the standard for capturing photographic images, has been made largely obsolete by digital photography. (credit “film”: modification of work by Horia Varlan; credit “memory cards”: modification of work by Paul Hudson)

ChAPTeR OUTlIne

5.1 Quadratic Functions 5.2 Power Functions and Polynomial Functions 5.3 graphs of Polynomial Functions 5.4 dividing Polynomials 5.5 Zeros of Polynomial Functions 5.6 Rational Functions 5.7 Inverses and Radical Functions 5.8 modeling Using variation

Introduction Digital photography has dramatically changed the nature of photography. No longer is an image etched in the emulsion on a roll of film. Instead, nearly every aspect of recording and manipulating images is now governed by mathematics. An image becomes a series of numbers, representing the characteristics of light striking an image sensor. When we open an image file, software on a camera or computer interprets the numbers and converts them to a visual image. Photo editing software uses complex polynomials to transform images, allowing us to manipulate the image in order to crop details, change the color palette, and add special effects. Inverse functions make it possible to convert from one file format to another. In this chapter, we will learn about these concepts and discover how mathematics can be used in such applications.

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SECTION 5.1 sectioN exercises 357

5.1 SeCTIOn exeRCISeS

veRbAl

1. Explain the advantage of writing a quadratic function in standard form.

2. How can the vertex of a parabola be used in solving real-world problems?

3. Explain why the condition of a ≠ 0 is imposed in the definition of the quadratic function.

4. What is another name for the standard form of a quadratic function?

5. What two algebraic methods can be used to find the horizontal intercepts of a quadratic function?

AlgebRAIC

For the following exercises, rewrite the quadratic functions in standard form and give the vertex. 6. f (x) = x 2 − 12x + 32 7. g(x) = x 2 + 2x − 3 8. f (x) = x 2 − x

9. f (x) = x 2 + 5x − 2 10. h(x) = 2x 2 + 8x − 10 11. k(x) = 3x 2 − 6x − 9

12. f (x) = 2x 2 − 6x 13. f (x) = 3x 2 − 5x − 1

For the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry.

14. y(x) = 2x 2 + 10x + 12 15. f(x) = 2x 2 − 10x + 4 16. f(x) = −x 2 + 4x + 3

17. f(x) = 4x 2 + x − 1 18. h(t) = −4t 2 + 6t − 1 19. f(x) = 1 __ 2 x 2 + 3x + 1

20. f(x) = −  1 __ 3 x 2 − 2x + 3

For the following exercises, determine the domain and range of the quadratic function.

21. f(x) = (x − 3)2 + 2 22. f(x) = −2(x + 3)2 − 6 23. f(x) = x 2 + 6x + 4

24. f(x) = 2x 2 − 4x + 2 25. k(x) = 3x 2 − 6x − 9

For the following exercises, use the vertex (h, k) and a point on the graph (x, y) to find the general form of the equation of the quadratic function.

26. (h, k) = (2, 0), (x, y) = (4, 4) 27. (h, k) = (−2, −1), (x, y) = (−4, 3) 28. (h, k) = (0, 1), (x, y) = (2, 5)

29. (h, k) = (2, 3), (x, y) = (5, 12) 30. (h, k) = (−5, 3), (x, y) = (2, 9) 31. (h, k) = (3, 2), (x, y) = (10, 1)

32. (h, k) = (0, 1), (x, y) = (1, 0) 33. (h, k) = (1, 0), (x, y) = (0, 1)

gRAPhICAl

For the following exercises, sketch a graph of the quadratic function and give the vertex, axis of symmetry, and intercepts.

34. f(x) = x 2 − 2x 35. f(x) = x 2 − 6x − 1 36. f(x) = x 2 − 5x − 6

37. f(x) = x 2 − 7x + 3 38. f(x) = −2x 2 + 5x − 8 39. f(x) = 4x 2 − 12x − 3

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358 CHAPTER 5 PolyNomial aNd ratioNal fuNctioNs

For the following exercises, write the equation for the graphed function. 40.

–4 –3 –2 –1 1 2 3 4 5 6

–5 –4 –3 –2 –1

1 2 3 4 5

x

y 41.

–6 –5 –4 –3 –2 –1 1 2 3 4

–2 –1

1 2 3 4 5 6 7 8

x

y 42.

–3 –2 –1 1 2 3 4 5 6 7

–3 –2 –1

1 2 3 4 5 6 7

x

y

43.

–6 –5 –4 –3 –2 –1 1 2 3 4

–7 –6 –5 –4 –3 –2 –1

1 2 3

x

y 44. 45.

–7 –6 –5 –4 –3 –2 –1 1 2 3

–5 –4 –3 –2 –1

1 2 3 4 5

x

y

nUmeRIC For the following exercises, use the table of values that represent points on the graph of a quadratic function. By determining the vertex and axis of symmetry, find the general form of the equation of the quadratic function.

46. 47. 48.

49. 50.

TeChnOlOgy For the following exercises, use a calculator to find the answer.

51. Graph on the same set of axes the functions f (x) = x2, f(x) = 2x 2, and f(x) = 1 __ 3 x

2. What appears to be the effect of changing the coefficient?

52. Graph on the same set of axes f(x) = x 2, f(x) = x 2 + 2 and f(x) = x 2, f(x) = x 2 + 5 and f(x) = x 2 − 3. What appears to be the effect of adding a constant?

53. Graph on the same set of axes f(x) = x 2, f(x) = (x − 2)2, f(x − 3)2, and f(x) = (x + 4)2. What appears to be the effect of adding or subtracting those numbers?

54. The path of an object projected at a 45 degree angle with initial velocity of 80 feet per second is given by the function h(x) = −32 ____ (80)2 x

2 + x where x is the

horizontal distance traveled and h(x) is the height in feet. Use the [TRACE] feature of your calculator to determine the height of the object when it has traveled 100 feet away horizontally.

55. A suspension bridge can be modeled by the quadratic function h(x) = 0.0001x 2 with −2000 ≤ x ≤ 2000 where ∣ x ∣ is the number of feet from the center and h(x) is height in feet. Use the [TRACE] feature of your calculator to estimate how far from the center does the bridge have a height of 100 feet.

–2 –1 1 2 3 4 5 6 7 8

–2 –1

1 2 3 4 5 6 7 8

x

y

x −2 −1 0 1 2

y 5 2 1 2 5 x −2 −1 0 1 2

y 1 0 1 4 9

x −2 −1 0 1 2

y −2 1 2 1 −2

x −2 −1 0 1 2

y −8 −3 0 1 0

x −2 −1 0 1 2

y 8 2 0 2 8

SECTION 5.1 sectioN exercises 359

exTenSIOnS For the following exercises, use the vertex of the graph of the quadratic function and the direction the graph opens to find the domain and range of the function.

56. Vertex (1, −2), opens up. 57. Vertex (−1, 2) opens down.

58. Vertex (−5, 11), opens down. 59. Vertex (−100, 100), opens up.

For the following exercises, write the equation of the quadratic function that contains the given point and has the same shape as the given function.

60. Contains (1, 1) and has shape of f(x) = 2x 2. Vertex is on the y-axis.

61. Contains (−1, 4) and has the shape of f(x) = 2x 2. Vertex is on the y-axis.

62. Contains (2, 3) and has the shape of f(x) = 3x 2. Vertex is on the y-axis.

63. Contains (1, −3) and has the shape of f(x) = −x 2. Vertex is on the y-axis.

64. Contains (4, 3) and has the shape of f(x) = 5x 2. Vertex is on the y-axis.

65. Contains (1, −6) has the shape of f(x) = 3x 2. Vertex has x-coordinate of −1.

ReAl-WORld APPlICATIOnS

66. Find the dimensions of the rectangular corral producing the greatest enclosed area given 200 feet of fencing.

67. Find the dimensions of the rectangular corral split into 2 pens of the same size producing the greatest possible enclosed area given 300 feet of fencing.

68. Find the dimensions of the rectangular corral producing the greatest enclosed area split into 3 pens of the same size given 500 feet of fencing.

69. Among all of the pairs of numbers whose sum is 6, find the pair with the largest product. What is the product?

70. Among all of the pairs of numbers whose difference is 12, find the pair with the smallest product. What is the product?

71. Suppose that the price per unit in dollars of a cell phone production is modeled by p = $45 − 0.0125x, where x is in thousands of phones produced, and the revenue represented by thousands of dollars is R = x ⋅ p. Find the production level that will maximize revenue.

72. A rocket is launched in the air. Its height, in meters above sea level, as a function of time, in seconds, is given by h(t) = −4.9t 2 + 229t + 234. Find the maximum height the rocket attains.

73. A ball is thrown in the air from the top of a building. Its height, in meters above ground, as a function of time, in seconds, is given by h(t) = −4.9t2 + 24t + 8. How long does it take to reach maximum height?

74. A soccer stadium holds 62,000 spectators. With a ticket price of $11, the average attendance has been 26,000. When the price dropped to $9, the average attendance rose to 31,000. Assuming that attendance is linearly related to ticket price, what ticket price would maximize revenue?

75. A farmer finds that if she plants 75 trees per acre, each tree will yield 20 bushels of fruit. She estimates that for each additional tree planted per acre, the yield of each tree will decrease by 3 bushels. How many trees should she plant per acre to maximize her harvest?

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372 CHAPTER 5 PolyNomial aNd ratioNal fuNctioNs

5.2 SeCTIOn exeRCISeS

veRbAl

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

AlgebRAIC

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

6. f (x) = x5 7. f (x) = (x2)3 8. f (x) = x − x4

9. f (x) = x 2 _____ x2 − 1

10. f (x) = 2x(x + 2)(x − 1)2 11. f (x) = 3x + 1

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

12. −3x 13. 7 − 2x2 14. −2x2 − 3x5 + x − 6

15. x(4 − x2)(2x + 1) 16. x 2 (2x − 3)2

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

17. f (x) = x4 18. f (x) = x3 19. f (x) = −x4

20. f (x) = −x9 21. f (x) = −2x4 − 3x2 + x − 1 22. f (x) = 3x2 + x − 2

23. f (x) = x2(2x3 − x + 1) 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) 26. g(n) = −2(3n − 1)(2n + 1) 27. f (x) = x4 − 16

28. f (x) = x3 + 27 29. f (x) = x(x2 − 2x − 8) 30. f (x) = (x + 3)(4x2 − 1)

gRAPhICAl

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

31.

2

x

y

–1–1

–2

–2

–3

–3

–4

–4

–5

–5

1

3

3

21 4

4

5

5

32.

2

x

y

–1–1

–2

–2

–3

–3

–4

–4

–5

–5

1

3

3

21 4

4

5

5

33.

2

x

y

–1–1

–2

–2

–3

–3

–4

–4

–5

–5

1

3

3

21 4

4

5

5

SECTION 5.2 sectioN exercises 373

34.

2

x

y

–1–1

–2

–2

–3

–3

–4

–4

–5

–5

1

3

3

21 4

4

5

5

35.

2

x

y

–1–1

–2

–2

–3

–3

–4

–4

–5

–5

1

3

3

21 4

4

5

5

36.

2

x

y

–1–1

–2

–2

–3

–3

–4

–4

–5

–5

1

3

3

21 4

4

5

5

37.

2

x

y

–1–1

–2

–2

–3

–3

–4

–4

–5 –6

–5

1

3

3

21 4

4

5

5 6

38.

2

x

y

–1–1

–2

–2

–3

–3

–4

–4

–5

–5

1

3

3

21 4

4

5

5 6

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.

x

y 40.

x

y 41.

x

y 42.

x

y

43.

x

y 44.

x

y 45.

x

y

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374 CHAPTER 5 PolyNomial aNd ratioNal fuNctioNs

nUmeRIC

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

46. f (x) = −x3 47. f (x) = x4 − 5x2 48. f (x) = x2(1 − x)2

49. f (x) = (x − 1)(x − 2)(3 − x) 50. f (x) = 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) = x3(x − 2) 52. f (x) = x(x − 3)(x + 3) 53. f (x) = x(14 − 2x)(10 − 2x)

54. f (x) = x(14 − 2x)(10 − 2x)2 55. f (x) = x3 − 16x 56. f (x) = x3 − 27

57. f (x) = x4 − 81 58. f (x) = −x3 + x2 + 2x 59. f (x) = x3 − 2x2 − 15x

60. f (x) = x3 − 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) → ∞.

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) → ∞.

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) → ∞.

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.

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).

390 CHAPTER 5 PolyNomial aNd ratioNal fuNctioNs

5.3 SeCTIOn exeRCISeS

veRbAl

1. What is the difference between an x-intercept and a zero of a polynomial function f ?

2. If a polynomial function of degree n has n distinct zeros, what do you know about the graph of the function?

3. Explain how the Intermediate Value Theorem can assist us in finding a zero of a function.

4. Explain how the factored form of the polynomial helps us in graphing it.

5. If the graph of a polynomial just touches the x-axis and then changes direction, what can we conclude about the factored form of the polynomial?

AlgebRAIC

For the following exercises, find the x- or t-intercepts of the polynomial functions.

6. C(t) = 2(t − 4)(t + 1)(t − 6) 7. C(t) = 3(t + 2)(t − 3)(t + 5) 8. C(t) = 4t(t − 2)2(t + 1)

9. C(t) = 2t(t − 3)(t + 1)2 10. C(t) = 2t4 − 8t3 + 6t2 11. C(t) = 4t4 + 12t3 − 40t2

12. f (x) = x4 − x2 13. f (x) = x3 + x2 − 20x 14. f (x) = x3 + 6x2 − 7x

15. f (x) = x3 + x2 − 4x − 4 16. f (x) = x3 + 2x2 − 9x − 18 17. f (x) = 2x3 − x2 − 8x + 4

18. f (x) = x6 − 7x3 − 8 19. f (x) = 2x4 + 6x2 − 8 20. f (x) = x3 − 3x2 − x + 3

21. f (x) = x6 − 2x4 − 3x2 22. f (x) = x6 − 3x4 − 4x2 23. f (x) = x5 − 5x3 + 4x

For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval.

24. f (x) = x3 − 9x, between x = −4 and x = −2. 25. f (x) = x3 − 9x, between x = 2 and x = 4.

26. f (x) = x5 − 2x, between x = 1 and x = 2. 27. f (x) = −x4 + 4, between x = 1 and x = 3.

28. f (x) = −2x3 − x, between x = −1 and x = 1. 29. f (x) = x3 − 100x + 2, between x = 0.01 and x = 0.1

For the following exercises, find the zeros and give the multiplicity of each.

30. f (x) = (x + 2)3(x − 3)2 31. f (x) = x2(2x + 3)5(x − 4)2

32. f (x) = x3 (x − 1)3(x + 2) 33. f (x) = x2(x2 + 4x + 4)

34. f (x) = (2x + 1)3(9x2 − 6x + 1) 35. f (x) = (3x + 2)5(x2 − 10x + 25)

36. f (x) = x(4x2 − 12x + 9)(x2 + 8x + 16) 37. f (x) = x6 − x5 − 2x4

38. f (x) = 3x4 + 6x3 + 3x2 39. f (x) = 4x5 − 12x4 + 9x3

40. f (x) = 2x4(x3 − 4x2 + 4x) 41. f (x) = 4x4(9x4 − 12x3 + 4x2)

gRAPhICAl

For the following exercises, graph the polynomial functions. Note x- and y-intercepts, multiplicity, and end behavior.

42. f (x) = (x + 3)2(x − 2) 43. g(x) = (x + 4)(x − 1)2 44. h(x) = (x − 1)3(x + 3)2

45. k(x) = (x − 3)3(x − 2)2 46. m(x) = −2x(x − 1)(x + 3) 47. n(x) = −3x(x + 2)(x − 4)

SECTION 5.3 sectioN exercises 391

For the following exercises, use the graphs to write the formula for a polynomial function of least degree. 48.

2

x

f(x)

–1–1–2

–2

–3

–3

–4

–4

–5

–5

1

3

3

21 4

4

5

5

49.

2

x

f(x)

–1–1–2

–2

–3

–3

–4

–4

–5

–5

1

3

3

21 4

4

5

5

50.

2

x

f(x)

–1–1–2

–2

–3

–3

–4

–4

–5

–5

1

3

3

21 4

4

5

5

51.

2

x

f(x)

–1–1–2

–2

–3

–3

–4

–4

–5

–5

1

3

3

21 4

4

5

5

52.

2

x

f(x)

–1–1–2

–2

–3

–3

–4

–4

–5

–5

1

3

3

21 4

4

5

5

For the following exercises, use the graph to identify zeros and multiplicity.

53.

2

x

y

–1–1–2

–2

–3

–3

–4

–4

–5

–5

1

3

3

21 4

4

5

5

54.

2

x

y

–1–1–2

–2

–3

–3

–4

–4

–5

–5

1

3

3

21 4

4

5

5

55.

2

x

y

–1–1–2

–2

–3

–3

–4

–4

–5

–5

1

3

3

21 4

4

5

5

56.

2

x

y

–1–1–2

–2

–3

–3

–4

–4

–5

–5

1

3

3

21 4

4

5

5

For the following exercises, use the given information about the polynomial graph to write the equation.

57. Degree 3. Zeros at x = −2, x = 1, and x = 3. y-intercept at (0, −4).

58. Degree 3. Zeros at x = −5, x = −2, and x = 1. y-intercept at (0, 6)

59. Degree 5. Roots of multiplicity 2 at x = 3 and x = 1, and a root of multiplicity 1 at x = −3. y-intercept at (0, 9)

60. Degree 4. Root of multiplicity 2 at x = 4, and roots of multiplicity 1 at x = 1 and x = −2. y-intercept at (0, −3).

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392 CHAPTER 5 PolyNomial aNd ratioNal fuNctioNs

61. Degree 5. Double zero at x = 1, and triple zero at x = 3. Passes through the point (2, 15).

62. Degree 3. Zeros at x = 4, x = 3, and x = 2. y-intercept at (0, −24).

63. Degree 3. Zeros at x = −3, x = −2 and x = 1. y-intercept at (0, 12).

64. Degree 5. Roots of multiplicity 2 at x = −3 and x = 2 and a root of multiplicity 1 at x = −2. y-intercept at (0, 4).

65. Degree 4. Roots of multiplicity 2 at x = 1 _ 2

and roots of multiplicity 1 at x = 6 and x = −2. y-intercept at (0,18).

66. Double zero at x = −3 and triple zero at x = 0. Passes through the point (1, 32).

TeChnOlOgy For the following exercises, use a calculator to approximate local minima and maxima or the global minimum and maximum.

67. f (x) = x3 − x − 1 68. f (x) = 2x3 − 3x − 1 69. f (x) = x4 + x

70. f (x) = −x4 + 3x − 2 71. f (x) = x4 − x3 + 1

exTenSIOnS For the following exercises, use the graphs to write a polynomial function of least degree.

72.

x

f (x)

–2

–8

–4

–16

–6

–24

2 4

16

6

24

4 3

, 023 , 0 (0, 8)

73.

x

f (x) 6·107

5·107

4·107

3·107

2·107

1·107

–1·107

–2·107

–3·107

–300 –200 –100 0 100

(0, 50,000,000)

200 300 400 500 600 700

–4·107

–5·107

–6·107

–7·107

74. f(x)

x

2·105

1·105

–1·105 –400 –300 –200 –100 100

(100, 0)

(0, –90,000)

(–300, 0)

200

–2·105

–3·105

–4·105

ReAl-WORld APPlICATIOnS For the following exercises, write the polynomial function that models the given situation.

75. A rectangle has a length of 10 units and a width of 8 units. Squares of x by x units are cut out of each corner, and then the sides are folded up to create an open box. Express the volume of the box as a polynomial function in terms of x.

76. Consider the same rectangle of the preceding problem. Squares of 2x by 2x units are cut out of each corner. Express the volume of the box as a polynomial in terms of x.

77. A square has sides of 12 units. Squares x + 1 by x + 1 units are cut out of each corner, and then the sides are folded up to create an open box. Express the volume of the box as a function in terms of x.

78. A cylinder has a radius of x + 2 units and a height of 3 units greater. Express the volume of the cylinder as a polynomial function.

79. A right circular cone has a radius of 3x + 6 and a height 3 units less. Express the volume of the cone as a polynomial function. The volume of a cone is V = 1 _

3 πr 2h for radius r and height h.

400 CHAPTER 5 PolyNomial aNd ratioNal fuNctioNs

5.4 SeCTIOn exeRCISeS

veRbAl

1. If division of a polynomial by a binomial results in a remainder of zero, what can be conclude?

2. If a polynomial of degree n is divided by a binomial of degree 1, what is the degree of the quotient?

AlgebRAIC

For the following exercises, use long division to divide. Specify the quotient and the remainder.

3. (x2 + 5x − 1) ÷ (x − 1) 4. (2x2 − 9x − 5) ÷ (x − 5) 5. (3x2 + 23x + 14) ÷ (x + 7) 6. (4x2 − 10x + 6) ÷ (4x + 2) 7. (6x2 − 25x − 25) ÷ (6x + 5) 8. (−x2 − 1) ÷ (x + 1) 9. (2x2 − 3x + 2) ÷ (x + 2) 10. (x3 − 126) ÷ (x − 5) 11. (3x2 − 5x + 4) ÷ (3x + 1)

12. (x3 − 3x2 + 5x − 6) ÷ (x − 2) 13. (2x3 + 3x2 − 4x + 15) ÷ (x + 3)

For the following exercises, use synthetic division to find the quotient.

14. (3x3 − 2x2 + x − 4) ÷ (x + 3) 15. (2x3 − 6x2 − 7x + 6) ÷ (x − 4) 16. (6x3 − 10x2 − 7x − 15) ÷ (x + 1) 17. (4x3 − 12x2 − 5x − 1) ÷ (2x + 1) 18. (9x3 − 9x2 + 18x + 5) ÷ (3x − 1) 19. (3x3 − 2x2 + x − 4) ÷ (x + 3) 20. (−6x3 + x2 − 4) ÷ (2x − 3) 21. (2x3 + 7x2 − 13x − 3) ÷ (2x − 3) 22. (3x3 − 5x2 + 2x + 3) ÷ (x + 2) 23. (4x3 − 5x2 + 13) ÷ (x + 4) 24. (x3 − 3x + 2) ÷ (x + 2) 25. (x3 − 21x2 + 147x − 343) ÷ (x − 7) 26. (x3 − 15x2 + 75x − 125) ÷ (x − 5) 27. (9x3 − x + 2) ÷ (3x − 1) 28. (6x3 − x2 + 5x + 2) ÷ (3x + 1) 29. (x4 + x3 − 3x2 − 2x + 1) ÷ (x + 1) 30. (x4 − 3x2 + 1) ÷ (x − 1) 31. (x4 + 2x3 − 3x2 + 2x + 6) ÷ (x + 3) 32. (x4 − 10x3 + 37x2 − 60x + 36) ÷ (x − 2) 33. (x4 − 8x3 + 24x2 − 32x + 16) ÷ (x − 2) 34. (x4 + 5x3 − 3x2 − 13x + 10) ÷ (x + 5) 35. (x4 − 12x3 + 54x2 − 108x + 81) ÷ (x − 3) 36. (4x4 − 2x3 − 4x + 2) ÷ (2x − 1) 37. (4x4 + 2x3 − 4x2 + 2x + 2) ÷ (2x + 1)

For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization.

38. x − 2, 4x3 − 3x2 − 8x + 4 39. x − 2, 3x4 − 6x3 − 5x + 10 40. x + 3, −4x3 + 5x2 + 8

41. x − 2, 4x4 − 15x2 − 4 42. x − 1 __ 2 , 2x 4 − x3 + 2x − 1 43. x + 1 __ 3 , 3x

4 + x3 − 3x + 1

gRAPhICAl

For the following exercises, use the graph of the third-degree polynomial and one factor to write the factored form of the polynomial suggested by the graph. The leading coefficient is one.

44. Factor is x2 − x + 3

6

x

y

–2

–6

–4

–12

–6

–18

2 4

12

6

18

45. Factor is x2 + 2x + 4

6

x

y

–2

–6

–4

–12

–6

–18

2 4

12

6

18

46. Factor is x2 + 2x + 5

x

y

–2–4–6

–10

–20

–30

2

10

20

30

4 6

SECTION 5.4 sectioN exercises 401

47. Factor is x2 + x + 1

x

y

–2–4–6

–20

–40

–60

2

20

40

60

4 6

48. Factor is x2 + 2x + 2

x

y

–2–4–6

– 6

–12

–18

2

6

12

18

4 6

For the following exercises, use synthetic division to find the quotient and remainder.

49. 4x 3 − 33 _______ x − 2 50.

2x3 + 25 _______ x + 3 51. 3x3 + 2x − 5 __________ x − 1

52.  −4x 3 − x2 − 12 ____________ x + 4 53.

x4 − 22 ______ x + 2

TeChnOlOgy

For the following exercises, use a calculator with CAS to answer the questions.

54. Consider x k − 1 ______ x − 1 with k = 1, 2, 3. What do you

expect the result to be if k = 4? 55. Consider x

k + 1 ______ x + 1 for k = 1, 3, 5. What do you expect the result to be if k = 7?

56. Consider x 4 − k4 ______ x − k for k = 1, 2, 3. What do you

expect the result to be if k = 4? 57. Consider x

k _____ x + 1 with k = 1, 2, 3. What do you expect

the result to be if k = 4?

58. Consider x k _____ x − 1 with k = 1, 2, 3. What do you expect

the result to be if k = 4?

exTenSIOnS

For the following exercises, use synthetic division to determine the quotient involving a complex number.

59. x + 1 _____ x − i 60. x2 + 1 _____ x − i 61.

x + 1 _____ x + i

62. x 2 + 1 _____ x + i 63.

x3 + 1 _____ x − i

ReAl-WORld APPlICATIOnS

For the following exercises, use the given length and area of a rectangle to express the width algebraically.

64. Length is x + 5, area is 2x2 + 9x − 5. 65. Length is 2x + 5, area is 4x3 + 10x2 + 6x + 15 66. Length is 3x − 4, area is 6x4 − 8x3 + 9x2 − 9x − 4

For the following exercises, use the given volume of a box and its length and width to express the height of the box algebraically.

67. Volume is 12x3 + 20x2 − 21x − 36, length is 2x + 3, width is 3x − 4.

68. Volume is 18x3 − 21x2 − 40x + 48, length is 3x − 4, width is 3x − 4.

69. Volume is 10x3 + 27x2 + 2x − 24, length is 5x − 4, width is 2x + 3.

70. Volume is 10x3 + 30x2 − 8x − 24, length is 2, width is x + 3.

For the following exercises, use the given volume and radius of a cylinder to express the height of the cylinder  algebraically.

71. Volume is π(25x3 − 65x2 − 29x − 3), radius is 5x + 1. 72. Volume is π(4x3 + 12x2 − 15x − 50), radius is 2x + 5. 73. Volume is π(3x4 + 24x3 + 46x2 − 16x − 32),

radius is x + 4.

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412 CHAPTER 5 PolyNomial aNd ratioNal fuNctioNs

5.5 SeCTIOn exeRCISeS

veRbAl

1. Describe a use for the Remainder Theorem. 2. Explain why the Rational Zero Theorem does not guarantee finding zeros of a polynomial function.

3. What is the difference between rational and real zeros?

4. If Descartes’ Rule of Signs reveals a no change of signs or one sign of changes, what specific conclusion can be drawn?

5. If synthetic division reveals a zero, why should we try that value again as a possible solution?

AlgebRAIC

For the following exercises, use the Remainder Theorem to find the remainder.

6. (x4 − 9x2 + 14) ÷ (x − 2) 7. (3x3 − 2x2 + x − 4) ÷ (x + 3)

8. (x4 + 5x3 − 4x − 17) ÷ (x + 1) 9. (−3x2 + 6x + 24) ÷ (x − 4)

10. (5x5 − 4x4 + 3x3 − 2x2 + x − 1) ÷ (x + 6) 11. (x4 − 1) ÷ (x − 4)

12. (3x3 + 4x2 − 8x + 2) ÷ (x − 3) 13. (4x3 + 5x2 − 2x + 7) ÷ (x + 2)

For the following exercises, use the Factor Theorem to find all real zeros for the given polynomial function and one factor.

14. f (x) = 2x3 − 9x2 + 13x − 6; x − 1 15. f (x) = 2x3 + x2 − 5x + 2; x + 2

16. f (x) = 3x3 + x2 − 20x + 12; x + 3 17. f (x) = 2x3 + 3x2 + x + 6; x + 2

18. f (x) = −5x3 + 16x2 − 9; x − 3 19. x3 + 3x2 + 4x + 12; x + 3

20. 4x3 − 7x + 3; x − 1 21. 2x3 + 5x2 − 12x − 30, 2x + 5

For the following exercises, use the Rational Zero Theorem to find all real zeros.

22. x3 − 3x2 − 10x + 24 = 0 23. 2x3 + 7x2 − 10x − 24 = 0 24. x3 + 2x2 − 9x − 18 = 0

25. x3 + 5x2 − 16x − 80 = 0 26. x3 − 3x2 − 25x + 75 = 0 27. 2x3 − 3x2 − 32x − 15 = 0

28. 2x3 + x2 − 7x − 6 = 0 29. 2x3 − 3x2 − x + 1 = 0 30. 3x3 − x2 − 11x − 6 = 0

31. 2x3 − 5x2 + 9x − 9 = 0 32. 2x3 − 3x2 + 4x + 3 = 0 33. x4 − 2x3 − 7x2 + 8x + 12 = 0

34. x4 + 2x3 − 9x2 − 2x + 8 = 0 35. 4x4 + 4x3 − 25x2 − x + 6 = 0 36. 2x4 − 3x3 − 15x2 + 32x − 12 = 0

37. x4 + 2x3 − 4x2 − 10x − 5 = 0 38. 4x3 − 3x + 1 = 0 39. 8x4 + 26x3 + 39x2 + 26x + 6

For the following exercises, find all complex solutions (real and non-real).

40. x3 + x2 + x + 1 = 0 41. x3 − 8x2 + 25x − 26 = 0 42. x3 + 13x2 + 57x + 85 = 0

43. 3x3 − 4x2 + 11x + 10 = 0 44. x4 + 2x3 + 22x2 + 50x − 75 = 0 45. 2x3 − 3x2 + 32x + 17 = 0

gRAPhICAl

Use Descartes’ Rule to determine the possible number of positive and negative solutions. Then graph to confirm which of those possibilities is the actual combination.

46. f (x) = x3 − 1 47. f (x) = x4 − x2 − 1 48. f (x) = x3 − 2x2 − 5x + 6

49. f (x) = x3 − 2x2 + x − 1 50. f (x) = x4 + 2x3 − 12x2 + 14x − 5 51. f (x) = 2x3 + 37x2 + 200x + 300

SECTION 5.5 sectioN exercises 413

52. f (x) = x3 − 2x2 − 16x + 32 53. f (x) = 2x4 − 5x3 − 5x2 + 5x + 3 54. f (x) = 2x4 − 5x3 − 14x2 + 20x + 8

55. f (x) = 10x4 − 21x2 + 11

nUmeRIC

For the following exercises, list all possible rational zeros for the functions.

56. f (x) = x4 + 3x3 − 4x + 4 57. f (x) = 2x3 + 3x2 − 8x + 5 58. f (x) = 3x 3 + 5x2 − 5x + 4

59. f (x) = 6x4 − 10x2 + 13x + 1 60. f (x) = 4x5 − 10x4 + 8x3 + x2 − 8

TeChnOlOgy

For the following exercises, use your calculator to graph the polynomial function. Based on the graph, find the rational zeros. All real solutions are rational.

61. f (x) = 6x3 − 7x2 + 1 62. f (x) = 4x3 − 4x2 − 13x − 5

63. f (x) = 8x3 − 6x2 − 23x + 6 64. f (x) = 12x4 + 55x3 + 12x2 − 117x + 54

65. f (x) = 16x4 − 24x3 + x2 − 15x + 25

exTenSIOnS

For the following exercises, construct a polynomial function of least degree possible using the given information. 66. Real roots: −1, 1, 3 and (2, f (2)) = (2, 4) 67. Real roots: −1, 1 (with multiplicity 2 and 1) and

(2, f (2)) = (2, 4)

68. Real roots: −2, 1 __ 2 (with multiplicity 2) and (−3, f (−3)) = (−3, 5)

69. Real roots: −  1 __ 2 , 0, 1 __ 2 and (−2, f (−2)) = (−2, 6)

70. Real roots: −4, −1, 1, 4 and (−2, f (−2)) = (−2, 10)

ReAl-WORld APPlICATIOnS

For the following exercises, find the dimensions of the box described.

71. The length is twice as long as the width. The height is 2 inches greater than the width. The volume is 192 cubic inches.

72. The length, width, and height are consecutive whole numbers. The volume is 120 cubic inches.

73. The length is one inch more than the width, which is one inch more than the height. The volume is 86.625 cubic inches.

74. The length is three times the height and the height is one inch less than the width. The volume is 108 cubic inches.

75. The length is 3 inches more than the width. The width is 2 inches more than the height. The volume is 120 cubic inches.

For the following exercises, find the dimensions of the right circular cylinder described.

76. The radius is 3 inches more than the height. The volume is 16π cubic meters.

77. The height is one less than one half the radius. The volume is 72π cubic meters.

78. The radius and height differ by one meter. The radius is larger and the volume is 48π cubic meters.

79. The radius and height differ by two meters. The height is greater and the volume is 28.125π cubic meters.

80. The radius is 1 __ 3 meter greater than the height. The

volume is 98 ___ 9π π cubic meters.

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SECTION 5.6 sectioN exercises 431

5.6 SeCTIOn exeRCISeS

veRbAl

1. What is the fundamental difference in the algebraic representation of a polynomial function and a rational function?

2. What is the fundamental difference in the graphs of polynomial functions and rational functions?

3. If the graph of a rational function has a removable discontinuity, what must be true of the functional rule?

4. Can a graph of a rational function have no vertical asymptote? If so, how?

5. Can a graph of a rational function have no x-intercepts? If so, how?

AlgebRAIC

For the following exercises, find the domain of the rational functions.

6. f (x) = x − 1 _____ x + 2 7. f (x) = x + 1 _____ x2 − 1 8. f (x) =

x2 + 4 _________ x2 − 2x − 8

9. f (x) = x 2 + 4x − 3 _________ x4 − 5x2 + 4

For the following exercises, find the domain, vertical asymptotes, and horizontal asymptotes of the functions.

10. f (x) = 4 ____ x − 1 11. f (x) = 2 _____ 5x + 2

12. f (x) = x _____ x2 − 9

13. f (x) = x __________ x2 + 5x − 36 14. f (x) = 3 + x ______ x3 − 27 15. f (x) =

3x − 4 _______ x3 − 16x

16. f (x) = x 2 − 1 ___________ x3 + 9x2 + 14x 17. f (x) =

x + 5 ______ x2 − 25 18. f (x) = x − 4 _____ x − 6

19. f (x) = 4 − 2x ______ 3x − 1

For the following exercises, find the x- and y-intercepts for the functions.

20. f (x) = x + 5 _____ x2 + 4 21. f (x) = x _____ x2 − x 22. f (x) =

x2 + 8x + 7 ___________ x2 + 11x + 30

23. f (x) = x 2 + x + 6 ___________ x2 − 10x + 24 24. f (x) =

94 − 2x2 _______ 3x2 − 12

For the following exercises, describe the local and end behavior of the functions.

25. f (x) = x _____ 2x + 1 26. f (x) = 2x _____ x − 6 27. f (x) =

−2x _____ x − 6

28. f (x) = x 2 − 4x + 3 _________ x2 − 4x − 5 29. f (x) =

2x2 − 32 ___________ 6x2 + 13x − 5

For the following exercises, find the slant asymptote of the functions.

30. f (x) = 24x 2 + 6x ________ 2x + 1 31. f (x) =

4x2 − 10 _______ 2x − 4 32. f (x) = 81x2 − 18 ________ 3x − 2

33. f (x) = 6x 3 − 5x _______ 3x2 + 4 34. f (x) =

x2 + 5x + 4 _________ x − 1

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432 CHAPTER 5 PolyNomial aNd ratioNal fuNctioNs

gRAPhICAl

For the following exercises, use the given transformation to graph the function. Note the vertical and horizontal asymptotes.

35. The reciprocal function shifted up two units. 36. The reciprocal function shifted down one unit and left three units.

37. The reciprocal squared function shifted to the right 2 units.

38. The reciprocal squared function shifted down 2 units and right 1 unit.

For the following exercises, find the horizontal intercepts, the vertical intercept, the vertical asymptotes, and the horizontal or slant asymptote of the functions. Use that information to sketch a graph.

39. p(x) = 2x − 3 ______ x + 4 40. q(x) = x − 5 _____ 3x − 1 41. s(x) =

4 ______ (x − 2)2

42. r(x) = 5 ______ (x + 1)2 43. f (x) = 3x 2 − 14x − 5 __ 3x 2 + 8x − 16

44. g(x) = 2x 2 + 7x − 15 __ 3x 2 − 14 + 15

45. a(x) = x 2 + 2x − 3 _

x 2 − 1 46. b(x) = x

2 − x − 6 _ x 2 − 4

47. h(x) = 2x 2 + x − 1 _ x − 4

48. k(x) = 2x 2 − 3x − 20 __ x − 5 49. w(x) =

(x − 1)(x + 3)(x − 5) __

(x + 2)2(x − 4) 50. z(x) =

(x + 2)2(x − 5) __

(x − 3)(x + 1)(x + 4)

For the following exercises, write an equation for a rational function with the given characteristics.

51. Vertical asymptotes at x = 5 and x = −5, x-intercepts at (2, 0) and (−1, 0), y-intercept at (0, 4)

52. Vertical asymptotes at x = −4 and x = −1, x-intercepts at (1, 0) and (5, 0), y-intercept at (0, 7)

53. Vertical asymptotes at x = −4 and x = −5, x-intercepts at (4, 0) and (−6, 0), horizontal asymptote at y = 7

54. Vertical asymptotes at x = −3 and x = 6, x-intercepts at (−2, 0) and (1, 0), horizontal asymptote at y = −2

55. Vertical asymptote at x = −1, double zero at x = 2, y-intercept at (0, 2)

56. Vertical asymptote at x = 3, double zero at x = 1, y-intercept at (0, 4)

For the following exercises, use the graphs to write an equation for the function.

57. y

x –2–4–6–8–10 –1

–2 –3 –4 –5

0 642

1 2 3 4 5

8 10

58. y

x –2–4–6–8–10 –1

–2 –3 –4 –5

642

1 2 3 4 5

8 10

59. y

x –2–4–6–8–10 –1

–2 –3 –4 –5

642

1 2 3 4 5

8 10

SECTION 5.6 sectioN exercises 433

60. y

x –2–4–6–8–10 –1

–2 –3 –4 –5

642

1 2 3 4 5

8 10

61. y

x –2–4–6–8–10 –1

–2 –3 –4 –5

642

1 2 3 4 5

8 10

62. y

x –2–4–6–8–10 –1

–2 –3 –4 –5

642

1 2 3 4 5

8 10

63. y

x –2–4–6–8–10 –1

–2 –3 –4 –5

642

1 2 3 4 5

8 10

64. y

x –2–4–6–8–10 –1

–2 –3 –4 –5

642

1 2 3 4 5

8 10

nUmeRIC

For the following exercises, make tables to show the behavior of the function near the vertical asymptote and reflecting the horizontal asymptote

65. f (x) = 1 _____ x − 2 66. f (x) = x _____ x − 3 67. f (x) =

2x _____ x + 4

68. f (x) = 2x ______ (x − 3)2 69. f (x) = x2 _________ x2 + 2x + 1

TeChnOlOgy

For the following exercises, use a calculator to graph f (x). Use the graph to solve f (x) > 0.

70. f (x) = 2 _____ x + 1 71. f (x) = 4 _____ 2x − 3 72. f (x) =

2 ___________ (x − 1)(x + 2)

73. f (x) = x + 2 ___________ (x − 1)(x − 4) 74. f (x) = (x + 3)2 ____________ (x − 1)2(x + 1)

exTenSIOnS

For the following exercises, identify the removable discontinuity.

75. f (x) = x 2 − 4 _____ x − 2 76. f (x) =

x3 + 1 _____ x + 1 77. f (x) = x2 + x − 6 ________ x − 2

78. f (x) = 2x 2 + 5x − 3 __________ x + 3 79. f (x) =

x3 + x2 ______ x + 1

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434 CHAPTER 5 PolyNomial aNd ratioNal fuNctioNs

ReAl-WORld APPlICATIOnS

For the following exercises, express a rational function that describes the situation.

80. A large mixing tank currently contains 200 gallons of water, into which 10 pounds of sugar have been mixed. A tap will open, pouring 10 gallons of water per minute into the tank at the same time sugar is poured into the tank at a rate of 3 pounds per minute. Find the concentration (pounds per gallon) of sugar in the tank after t minutes.

81. A large mixing tank currently contains 300 gallons of water, into which 8 pounds of sugar have been mixed. A tap will open, pouring 20 gallons of water per minute into the tank at the same time sugar is poured into the tank at a rate of 2 pounds per minute. Find the concentration (pounds per gallon) of sugar in the tank after t minutes.

For the following exercises, use the given rational function to answer the question.

82. The concentration C of a drug in a patient’s bloodstream t hours after injection in given by C(t) = 2t _

3 + t2 . What happens to the concentration of

the drug as t increases?

83. The concentration C of a drug in a patient’s bloodstream t hours after injection is given by C(t) = 100t _

2t2 + 75 . Use a calculator to approximate the

time when the concentration is highest.

For the following exercises, construct a rational function that will help solve the problem. Then, use a calculator to answer the question.

84. An open box with a square base is to have a volume of 108 cubic inches. Find the dimensions of the box that will have minimum surface area. Let x = length of the side of the base.

85. A rectangular box with a square base is to have a volume of 20 cubic feet. The material for the base costs 30 cents/square foot. The material for the sides costs 10 cents/square foot. The material for the top costs 20 cents/square foot. Determine the dimensions that will yield minimum cost. Let x = length of the side of the base.

86. A right circular cylinder has volume of 100 cubic inches. Find the radius and height that will yield minimum surface area. Let x = radius.

87. A right circular cylinder with no top has a volume of 50 cubic meters. Find the radius that will yield minimum surface area. Let x = radius.

88. A right circular cylinder is to have a volume of 40 cubic inches. It costs 4 cents/square inch to construct the top and bottom and 1 cent/square inch to construct the rest of the cylinder. Find the radius to yield minimum cost. Let x = radius.

444 CHAPTER 5 PolyNomial aNd ratioNal fuNctioNs

5.7 SeCTIOn exeRCISeS

veRbAl

1. Explain why we cannot find inverse functions for all polynomial functions.

2. Why must we restrict the domain of a quadratic function when finding its inverse?

3. When finding the inverse of a radical function, what restriction will we need to make?

4. The inverse of a quadratic function will always take what form?

AlgebRAIC

For the following exercises, find the inverse of the function on the given domain.

5. f (x) = (x − 4)2, [4, ∞) 6. f (x) = (x + 2)2, [−2, ∞) 7. f (x) = (x + 1)2 − 3, [−1, ∞)

8. f (x) = 3x 2 + 5, (−∞, 0] 9. f (x) = 12 − x 2, [0, ∞) 10. f (x) = 9 − x 2, [0, ∞)

11. f (x) = 2x 2 + 4, [0, ∞)

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

12. f (x) = x 3 + 5 13. f (x) = 3x 3 + 1 14. f (x) = 4 − x 3

15. f (x) = 4 − 2x 3

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

16. f (x) = √ —

2x + 1 17. f (x) = √ —

3 − 4x 18. f (x) = 9 + √ —

4x − 4

19. f (x) = √ —

6x − 8 + 5 20. f (x) = 9 + 2 3 √ — x 21. f (x) = 3 − 3 √

— x

22. f (x) = 2 _____ x + 8 23. f (x) = 3 _____ x − 4 24. f (x) =

x + 3 _____ x + 7

25. f (x) = x − 2 _____ x + 7 26. f (x) = 3x + 4 ______ 5 − 4x 27. f (x) =

5x + 1 ______ 2 − 5x

28. f (x) = x 2 + 2x, [−1, ∞) 29. f (x) = x 2 + 4x + 1, [−2, ∞) 30. f (x) = x 2 − 6x + 3, [3, ∞)

gRAPhICAl

For the following exercises, find the inverse of the function and graph both the function and its inverse.

31. f (x) = x 2 + 2, x ≥ 0 32. f (x) = 4 − x 2, x ≥ 0 33. f (x) = (x + 3)2, x ≥ −3

34. f (x) = (x − 4)2, x ≥ 4 35. f (x) = x 3 + 3 36. f (x) = 1 − x 3

37. f (x) = x 2 + 4x, x ≥ −2 38. f (x) = x 2 − 6x + 1, x ≥ 3 39. f (x) =   2 __ x

40. f (x) = 1 __ x2 , x ≥ 0

For the following exercises, use a graph to help determine the domain of the functions.

41. f (x) = √ _____________

(x + 1)(x − 1)

__ x 42. f (x) = √ _____________

(x + 2)(x − 3)

__ x − 1

43. f (x) = √ ________

x(x + 3)

_ x − 4

44. f (x) = √ ___________

x 2 − x − 20 _

x − 2 45. f (x) = √

______

9 − x 2 _ x + 4

SECTION 5.7 sectioN exercises 445

TeChnOlOgy

For the following exercises, use a calculator to graph the function. Then, using the graph, give three points on the graph of the inverse with y-coordinates given.

46. f (x) = x3 − x − 2, y = 1, 2, 3 47. f (x) = x3 + x − 2, y = 0, 1, 2 48. f (x) = x3 + 3x − 4, y = 0, 1, 2

49. f (x) = x3 + 8x − 4, y = −1, 0, 1 50. f (x) = x4 + 5x + 1, y = −1, 0, 1

exTenSIOnS

For the following exercises, find the inverse of the functions with a, b, c positive real numbers.

51. f (x) = ax3 + b 52. f (x) = x2 + bx 53. f (x) = √ —

ax2 + b

54. f (x) = 3 √ —

ax + b 55. f (x) = ax + b ______ x + c

ReAl-WORld APPlICATIOnS

For the following exercises, determine the function described and then use it to answer the question.

56. An object dropped from a height of 200 meters has a height, h(t), in meters after t seconds have lapsed, such that h(t) = 200 − 4.9t 2. Express t as a function of height, h, and find the time to reach a height of 50 meters.

57. An object dropped from a height of 600 feet has a height, h(t), in feet after t seconds have elapsed, such that h(t) = 600 − 16t 2. Express t as a function of height h, and find the time to reach a height of 400 feet.

58. The volume, V, of a sphere in terms of its radius, r, is given by V(r) = 4 __ 3 πr

3. Express r as a function of V, and find the radius of a sphere with volume of 200 cubic feet.

59. The surface area, A, of a sphere in terms of its radius, r, is given by A(r) = 4πr 2. Express r as a function of V, and find the radius of a sphere with a surface area of 1000 square inches.

60. A container holds 100 ml of a solution that is 25 ml acid. If n ml of a solution that is 60% acid is added, the function C(n) = 25 + 0.6n ________ 100 + n gives the concentration, C, as a function of the number of ml added, n. Express n as a function of C and determine the number of mL that need to be added to have a solution that is 50% acid.

61. The period T, in seconds, of a simple pendulum as a function of its length l, in feet, is given by T(l) = 2π √

____

l ____ 32.2 . Express l as a function of T and

determine the length of a pendulum with period of 2 seconds.

62. The volume of a cylinder, V, in terms of radius, r, and height, h, is given by V = πr 2h. If a cylinder has a height of 6 meters, express the radius as a function of V and find the radius of a cylinder with volume of 300 cubic meters.

63. The surface area, A, of a cylinder in terms of its radius, r, and height, h, is given by A = 2πr2 + 2πrh. If the height of the cylinder is 4 feet, express the radius as a function of V and find the radius if the surface area is 200 square feet.

64. The volume of a right circular cone, V, in terms of its radius, r, and its height, h, is given by V = 1 _

3 πr 2h.

Express r in terms of h if the height of the cone is 12 feet and find the radius of a cone with volume of 50 cubic inches.

65. Consider a cone with height of 30 feet. Express the radius, r, in terms of the volume, V, and find the radius of a cone with volume of 1000 cubic feet.

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SECTION 5.8 sectioN exercises 451

5.8 SeCTIOn exeRCISeS

veRbAl 1. What is true of the appearance of graphs that

reflect a direct variation between two variables? 2. If two variables vary inversely, what will an equation

representing their relationship look like? 3. Is there a limit to the number of variables that

can jointly vary? Explain.

AlgebRAIC For the following exercises, write an equation describing the relationship of the given variables.

4. y varies directly as x and when x = 6, y = 12. 5. y varies directly as the square of x and when x = 4, y = 80. 6. y varies directly as the square root of x and when

x = 36, y = 24. 7. y varies directly as the cube of x and when x = 36, y = 24.

8. y varies directly as the cube root of x and when x = 27, y = 15.

9. y varies directly as the fourth power of x and when x = 1, y = 6.

10. y varies inversely as x and when x = 4, y = 2. 11. y varies inversely as the square of x and when x = 3, y = 2. 12. y varies inversely as the cube of x and when

x = 2, y = 5. 13. y varies inversely as the fourth power of x and when

x = 3, y = 1.

14. y varies inversely as the square root of x and when x = 25, y = 3.

15. y varies inversely as the cube root of x and when x = 64, y = 5.

16. y varies jointly with x and z and when x = 2 and z = 3, y = 36.

17. y varies jointly as x, z, and w and when x = 1, z = 2, w = 5, then y = 100.

18. y varies jointly as the square of x and the square of z and when x = 3 and z = 4, then y = 72.

19. y varies jointly as x and the square root of z and when x = 2 and z = 25, then y = 100.

20. y varies jointly as the square of x the cube of z and the square root of w. When x = 1, z = 2, and w = 36, then y = 48.

21. y varies jointly as x and z and inversely as w. When x = 3, z = 5, and w = 6, then y = 10.

22. y varies jointly as the square of x and the square root of z and inversely as the cube of w. When x = 3, z = 4, and w = 3, then y = 6.

23. y varies jointly as x and z and inversely as the square root of w and the square of t. When x = 3, z = 1, w = 25, and t = 2, then y = 6.

nUmeRIC For the following exercises, use the given information to find the unknown value.

24. y varies directly as x. When x = 3, then y = 12. Find y when x = 20.

25. y varies directly as the square of x. When x = 2, then y = 16. Find y when x = 8.

26. y varies directly as the cube of x. When x = 3, then y = 5. Find y when x = 4.

27. y varies directly as the square root of x. When x = 16, then y = 4. Find y when x = 36.

28. y varies directly as the cube root of x. When x = 125, then y = 15. Find y when x = 1,000.

29. y varies inversely with x. When x = 3, then y = 2. Find y when x = 1.

30. y varies inversely with the square of x. When x = 4, then y = 3. Find y when x = 2.

31. y varies inversely with the cube of x. When x = 3, then y = 1. Find y when x = 1.

32. y varies inversely with the square root of x. When x = 64, then y = 12. Find y when x = 36.

33. y varies inversely with the cube root of x. When x = 27, then y = 5. Find y when x = 125.

34. y varies jointly as x and z. When x = 4 and z = 2, then y = 16. Find y when x = 3 and z = 3.

35. y varies jointly as x, z, and w. When x = 2, z = 1, and w = 12, then y = 72. Find y when x = 1, z = 2, and w = 3.

36. y varies jointly as x and the square of z. When x = 2 and z = 4, then y = 144. Find y when x = 4 and z = 5.

37. y varies jointly as the square of x and the square root of z. When x = 2 and z = 9, then y = 24. Find y when x = 3 and z = 25.

38. y varies jointly as x and z and inversely as w. When x = 5, z = 2, and w = 20, then y = 4. Find y when x = 3 and z = 8, and w = 48.

39. y varies jointly as the square of x and the cube of z and inversely as the square root of w. When x = 2, z = 2, and w = 64, then y = 12. Find y when x = 1, z = 3, and w = 4.

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452 CHAPTER 5 PolyNomial aNd ratioNal fuNctioNs

40. y varies jointly as the square of x and of z and inversely as the square root of w and of t. When x = 2, z = 3, w = 16, and t = 3, then y = 1. Find y when x = 3, z = 2, w = 36, and t = 5.

TeChnOlOgy For the following exercises, use a calculator to graph the equation implied by the given variation.

41. y varies directly with the square of x and when x = 2, y = 3.

42. y varies directly as the cube of x and when x = 2, y = 4.

43. y varies directly as the square root of x and when x = 36, y = 2.

44. y varies inversely with x and when x = 6, y = 2.

45. y varies inversely as the square of x and when x = 1, y = 4.

exTenSIOnS For the following exercises, use Kepler’s Law, which states that the square of the time, T, required for a planet to orbit the Sun varies directly with the cube of the mean distance, a, that the planet is from the Sun.

46. Using the Earth’s time of 1 year and mean distance of 93 million miles, find the equation relating T and a.

47. Use the result from the previous exercise to determine the time required for Mars to orbit the Sun if its mean distance is 142 million miles.

48. Using Earth’s distance of 150 million kilometers, find the equation relating T and a.

49. Use the result from the previous exercise to determine the time required for Venus to orbit the Sun if its mean distance is 108 million kilometers.

50. Using Earth’s distance of 1 astronomical unit (A.U.), determine the time for Saturn to orbit the Sun if its mean distance is 9.54 A.U.

ReAl-WORld APPlICATIOnS For the following exercises, use the given information to answer the questions.

51. The distance s that an object falls varies directly with the square of the time, t, of the fall. If an object falls 16 feet in one second, how long for it to fall 144 feet?

52. The velocity v of a falling object varies directly to the time, t, of the fall. If after 2 seconds, the velocity of the object is 64 feet per second, what is the velocity after 5 seconds?

53. The rate of vibration of a string under constant tension varies inversely with the length of the string. If a string is 24 inches long and vibrates 128 times per second, what is the length of a string that vibrates 64 times per second?

54. The volume of a gas held at constant temperature varies indirectly as the pressure of the gas. If the volume of a gas is 1200 cubic centimeters when the pressure is 200 millimeters of mercury, what is the volume when the pressure is 300 millimeters of mercury?

55. The weight of an object above the surface of the Earth varies inversely with the square of the distance from the center of the Earth. If a body weighs 50 pounds when it is 3960 miles from Earth’s center, what would it weigh it were 3970 miles from Earth’s center?

56. The intensity of light measured in foot-candles varies inversely with the square of the distance from the light source. Suppose the intensity of a light bulb is 0.08 foot- candles at a distance of 3 meters. Find the intensity level at 8 meters.

57. The current in a circuit varies inversely with its resistance measured in ohms. When the current in a circuit is 40 amperes, the resistance is 10 ohms. Find the current if the resistance is 12 ohms.

58. The force exerted by the wind on a plane surface varies jointly with the square of the velocity of the wind and with the area of the plane surface. If the area of the surface is 40 square feet surface and the wind velocity is 20 miles per hour, the resulting force is 15 pounds. Find the force on a surface of 65 square feet with a velocity of 30 miles per hour.

59. The horsepower (hp) that a shaft can safely transmit varies jointly with its speed (in revolutions per minute (rpm)) and the cube of the diameter. If the shaft of a certain material 3 inches in diameter can transmit 45 hp at 100 rpm, what must the diameter be in order to transmit 60 hp at 150 rpm?

60. The kinetic energy K of a moving object varies jointly with its mass m and the square of its velocity v. If an object weighing 40 kilograms with a velocity of 15 meters per second has a kinetic energy of 1000 joules, find the kinetic energy if the velocity is increased to 20 meters per second.

CHAPTER 5 review 453

ChAPTeR 5 RevIeW

Key Terms arrow notation a way to represent the local and end behavior of a function by using arrows to indicate that an input

or output approaches a value axis of symmetry a vertical line drawn through the vertex of a parabola, that opens up or down, around which the

parabola is symmetric; it is defined by x = −   b __ 2a . coefficient a nonzero real number multiplied by a variable raised to an exponent constant of variation the non-zero value k that helps define the relationship between variables in direct or inverse

variation continuous function a function whose graph can be drawn without lifting the pen from the paper because there are

no breaks in the graph degree the highest power of the variable that occurs in a polynomial Descartes’ Rule of Signs a rule that determines the maximum possible numbers of positive and negative real zeros

based on the number of sign changes of f (x) and f (−x) direct variation the relationship between two variables that are a constant multiple of each other; as one quantity

increases, so does the other Division Algorithm given a polynomial dividend f (x) and a non-zero polynomial divisor d(x) where the degree of

d(x) is less than or equal to the degree of f (x), there exist unique polynomials q(x) and r(x) such that f (x) = d(x) q(x) + r(x) where q(x) is the quotient and r(x) is the remainder. The remainder is either equal to zero or has degree strictly less than d(x).

end behavior the behavior of the graph of a function as the input decreases without bound and increases without bound Factor Theorem k is a zero of polynomial function f (x) if and only if (x − k) is a factor of f (x) Fundamental Theorem of Algebra a polynomial function with degree greater than 0 has at least one complex zero general form of a quadratic function the function that describes a parabola, written in the form f (x) = ax 2 + bx + c,

where a, b, and c are real numbers and a ≠ 0. global maximum highest turning point on a graph; f (a) where f (a) ≥ f (x) for all x. global minimum lowest turning point on a graph; f (a) where f (a) ≤ f (x) for all x. horizontal asymptote a horizontal line y = b where the graph approaches the line as the inputs increase or decrease

without bound. Intermediate Value Theorem for two numbers a and b in the domain of f, if a < b and f (a) ≠ f (b), then the function f

takes on every value between f (a) and f (b); specifically, when a polynomial function changes from a negative value to a positive value, the function must cross the x-axis

inverse variation the relationship between two variables in which the product of the variables is a constant inversely proportional a relationship where one quantity is a constant divided by the other quantity; as one quantity

increases, the other decreases invertible function any function that has an inverse function imaginary number a number in the form bi where i = √

— −1

joint variation a relationship where a variable varies directly or inversely with multiple variables leading coefficient the coefficient of the leading term leading term the term containing the highest power of the variable Linear Factorization Theorem allowing for multiplicities, a polynomial function will have the same number of factors

as its degree, and each factor will be in the form (x − c), where c is a complex number multiplicity the number of times a given factor appears in the factored form of the equation of a polynomial; if a

polynomial contains a factor of the form (x − h)p, x = h is a zero of multiplicity p.

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454 CHAPTER 5 PolyNomial aNd ratioNal fuNctioNs

polynomial function a function that consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power.

power function a function that can be represented in the form f (x) = kxp where k is a constant, the base is a variable, and the exponent, p, is a constant

rational function a function that can be written as the ratio of two polynomials Rational Zero Theorem the possible rational zeros of a polynomial function have the form

p _ q where p is a factor of

the constant term and q is a factor of the leading coefficient. Remainder Theorem if a polynomial f (x) is divided by x − k, then the remainder is equal to the value f (k) removable discontinuity a single point at which a function is undefined that, if filled in, would make the function

continuous; it appears as a hole on the graph of a function roots in a given function, the values of x at which y = 0, also called zeros smooth curve a graph with no sharp corners standard form of a quadratic function the function that describes a parabola, written in the form f (x) = a(x − h)2 + k,

where (h, k) is the vertex. synthetic division a shortcut method that can be used to divide a polynomial by a binomial of the form x − k

term of a polynomial function any aix i of a polynomial function in the form f (x) = anx

n + ... + a2x 2 + a1x + a0

turning point the location at which the graph of a function changes direction varies directly a relationship where one quantity is a constant multiplied by the other quantity varies inversely a relationship where one quantity is a constant divided by the other quantity vertex the point at which a parabola changes direction, corresponding to the minimum or maximum value of the

quadratic function vertex form of a quadratic function another name for the standard form of a quadratic function vertical asymptote a vertical line x = a where the graph tends toward positive or negative infinity as the inputs approach a zeros in a given function, the values of x at which y = 0, also called roots

Key equations

general form of a quadratic function f (x) = ax 2 + bx + c

standard form of a quadratic function f (x) = a(x − h)2 + k

general form of a polynomial function f (x) = an x n + ... + a2 x

2 + a1x + a0

Division Algorithm f (x) = d(x)q(x) + r(x) where q(x) ≠ 0

Rational Function f (x) = P(x) ____ Q(x) =    ap x

p + ap − 1x

p−1 + ... + a1x + a0 ___

bq x q + bq − 1 x

q−1 + ... + b1 x + b0 , Q(x) ≠ 0

Direct variation y = kx n, k is a nonzero constant.

Inverse variation y = k _ xn , k is a nonzero constant.

CHAPTER 5 review 455

Key Concepts

5.1 Quadratic Functions • A polynomial function of degree two is called a quadratic function.

• The graph of a quadratic function is a parabola. A parabola is a U-shaped curve that can open either up or down.

• The axis of symmetry is the vertical line passing through the vertex. The zeros, or x-intercepts, are the points at which the parabola crosses the x-axis. The y-intercept is the point at which the parabola crosses the y-axis. See Example 1, Example 7, and Example 8.

• Quadratic functions are often written in general form. Standard or vertex form is useful to easily identify the vertex of a parabola. Either form can be written from a graph. See Example 2.

• The vertex can be found from an equation representing a quadratic function. See Example 3.

• The domain of a quadratic function is all real numbers. The range varies with the function. See Example 4.

• A quadratic function’s minimum or maximum value is given by the y-value of the vertex.

• The minimum or maximum value of a quadratic function can be used to determine the range of the function and to solve many kinds of real-world problems, including problems involving area and revenue. See Example 5 and Example 6.

• The vertex and the intercepts can be identified and interpreted to solve real-world problems. See Example 9.

5.2 Power Functions and Polynomial Functions • A power function is a variable base raised to a number power. See Example 1.

• The behavior of a graph as the input decreases beyond bound and increases beyond bound is called the end behavior.

• The end behavior depends on whether the power is even or odd. See Example 2 and Example 3.

• A polynomial function is the sum of terms, each of which consists of a transformed power function with positive whole number power. See Example 4.

• The degree of a polynomial function is the highest power of the variable that occurs in a polynomial. The term containing the highest power of the variable is called the leading term. The coefficient of the leading term is called the leading coefficient. See Example 5.

• The end behavior of a polynomial function is the same as the end behavior of the power function represented by the leading term of the function. See Example 6 and Example 7.

• A polynomial of degree n will have at most n x-intercepts and at most n − 1 turning points. See Example 8, Example 9, Example 10, Example 11, and Example 12.

5.3 Graphs of Polynomial Functions • Polynomial functions of degree 2 or more are smooth, continuous functions. See Example 1.

• To find the zeros of a polynomial function, if it can be factored, factor the function and set each factor equal to zero. See Example 2, Example 3, and Example 4.

• Another way to find the x-intercepts of a polynomial function is to graph the function and identify the points at which the graph crosses the x-axis. See Example 5.

• The multiplicity of a zero determines how the graph behaves at the x-intercepts. See Example 6.

• The graph of a polynomial will cross the horizontal axis at a zero with odd multiplicity.

• The graph of a polynomial will touch the horizontal axis at a zero with even multiplicity.

• The end behavior of a polynomial function depends on the leading term.

• The graph of a polynomial function changes direction at its turning points.

• A polynomial function of degree n has at most n − 1 turning points. See Example 7.

• To graph polynomial functions, find the zeros and their multiplicities, determine the end behavior, and ensure that the final graph has at most n − 1 turning points. See Example 8 and Example 10.

• Graphing a polynomial function helps to estimate local and global extremas. See Example 11.

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456 CHAPTER 5 PolyNomial aNd ratioNal fuNctioNs

• The Intermediate Value Theorem tells us that if f (a) and f (b) have opposite signs, then there exists at least one value c between a and b for which f (c) = 0. See Example 9.

5.4 Dividing Polynomials • Polynomial long division can be used to divide a polynomial by any polynomial with equal or lower degree. See

Example 1 and Example 2.

• The Division Algorithm tells us that a polynomial dividend can be written as the product of the divisor and the quotient added to the remainder.

• Synthetic division is a shortcut that can be used to divide a polynomial by a binomial in the form x − k. See Example 3, Example 4, and Example 5.

• Polynomial division can be used to solve application problems, including area and volume. See Example 6.

5.5 Zeros of Polynomial Functions • To find f (k), determine the remainder of the polynomial f (x) when it is divided by x − k. this is known as the

Remainder Theorem. See Example 1.

• According to the Factor Theorem, k is a zero of f (x) if and only if (x − k) is a factor of f (x). See Example 2.

• According to the Rational Zero Theorem, each rational zero of a polynomial function with integer coefficients will be equal to a factor of the constant term divided by a factor of the leading coefficient. See Example 3 and Example 4.

• When the leading coefficient is 1, the possible rational zeros are the factors of the constant term.

• Synthetic division can be used to find the zeros of a polynomial function. See Example 5.

• According to the Fundamental Theorem, every polynomial function has at least one complex zero. See Example 6.

• Every polynomial function with degree greater than 0 has at least one complex zero.

• Allowing for multiplicities, a polynomial function will have the same number of factors as its degree. Each factor will be in the form (x − c), where c is a complex number. See Example 7.

• The number of positive real zeros of a polynomial function is either the number of sign changes of the function or less than the number of sign changes by an even integer.

• The number of negative real zeros of a polynomial function is either the number of sign changes of f (−x) or less than the number of sign changes by an even integer. See Example 8.

• Polynomial equations model many real-world scenarios. Solving the equations is easiest done by synthetic division. See Example 9.

5.6 Rational Functions • We can use arrow notation to describe local behavior and end behavior of the toolkit functions f (x) =   1 _ x and f (x) =

1 _ x2

. See Example 1.

• A function that levels off at a horizontal value has a horizontal asymptote. A function can have more than one vertical asymptote. See Example 2.

• Application problems involving rates and concentrations often involve rational functions. See Example 3.

• The domain of a rational function includes all real numbers except those that cause the denominator to equal zero. See Example 4.

• The vertical asymptotes of a rational function will occur where the denominator of the function is equal to zero and the numerator is not zero. See Example 5.

• A removable discontinuity might occur in the graph of a rational function if an input causes both numerator and denominator to be zero. See Example 6.

• A rational function’s end behavior will mirror that of the ratio of the leading terms of the numerator and denominator functions. See Example 7, Example 8, Example 9, and Example 10.

• Graph rational functions by finding the intercepts, behavior at the intercepts and asymptotes, and end behavior. See Example 11.

• If a rational function has x-intercepts at x = x1, x2, …, xn, vertical asymptotes at x = v1, v2, ..., vm, and no xi = any vj, then the function can be written in the form

CHAPTER 5 review 457

f (x) = a (x − x1)

p 1(x − x2)

p 2...(x − xn)

p n ___

(x − v1) q

1(x − v2) q

2...(x − vm) q

n

See Example 12.

5.7 Inverses and Radical Functions • The inverse of a quadratic function is a square root function.

• If f −l is the inverse of a function f, then f is the inverse of the function f −l. See Example 1.

• While it is not possible to find an inverse of most polynomial functions, some basic polynomials are invertible. See Example 2.

• To find the inverse of certain functions, we must restrict the function to a domain on which it will be one-to-one. See Example 3 and Example 4.

• When finding the inverse of a radical function, we need a restriction on the domain of the answer. See Example 5 and Example 7.

• Inverse and radical and functions can be used to solve application problems. See Example 6 and Example 8.

5.8 Modeling Using Variation • A relationship where one quantity is a constant multiplied by another quantity is called direct variation. See Example 1.

• Two variables that are directly proportional to one another will have a constant ratio.

• A relationship where one quantity is a constant divided by another quantity is called inverse variation. See Example 2.

• Two variables that are inversely proportional to one another will have a constant multiple. See Example 3.

• In many problems, a variable varies directly or inversely with multiple variables. We call this type of relationship joint variation. See Example 4.

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458 CHAPTER 5 PolyNomial aNd ratioNal fuNctioNs

ChAPTeR 5 RevIeW exeRCISeS

QUAdRATIC FUnCTIOnS

For the following exercises, write the quadratic function in standard form. Then, give the vertex and axes intercepts. Finally, graph the function.

1. f (x) = x2 − 4x − 5 2. f (x) = −2x2 − 4x

For the following problems, find the equation of the quadratic function using the given information.

3. The vertex is (−2, 3) and a point on the graph is (3, 6). 4. The vertex is (−3, 6.5) and a point on the graph is (2, 6).

For the following exercises, complete the task.

5. A rectangular plot of land is to be enclosed by fencing. One side is along a river and so needs no fence. If the total fencing available is 600 meters, find the dimensions of the plot to have maximum area.

6. An object projected from the ground at a 45 degree angle with initial velocity of 120 feet per second has height, h, in terms of horizontal distance traveled, x, given by h(x) = −32 _____ (120)2 x

2 + x. Find the maximum

height the object attains.

POWeR FUnCTIOnS And POlynOmIAl FUnCTIOnS

For the following exercises, determine if the function is a polynomial function and, if so, give the degree and leading coefficient.

7. f (x) = 4x5 − 3x3 + 2x − 1 8. f (x) = 5x + 1 − x2 9. f (x) = x2(3 − 6x + x2)

For the following exercises, determine end behavior of the polynomial function.

10. f (x) = 2x4 + 3x3 − 5x2 + 7 11. f (x) = 4x3 − 6x2 + 2 12. f (x) = 2x2(1 + 3x − x2)

gRAPhS OF POlynOmIAl FUnCTIOnS

For the following exercises, find all zeros of the polynomial function, noting multiplicities.

13. f (x) = (x + 3)2(2x − 1)(x + 1)3 14. f (x) = x5 + 4x4 + 4x3 15. f (x) = x3 − 4x2 + x − 4

For the following exercises, based on the given graph, determine the zeros of the function and note multiplicity.

16.

x

y

–1–2–4–5 –3 –4 –8

–12 –16 –20

321

4 8

12 16 20

4 5

17.

x

y

–2–4–8–10 –6 –2 –4 –6 –8

–10

642

2 4 6 8

10

8 10

CHAPTER 5 review 459

18. Use the Intermediate Value Theorem to show that at least one zero lies between 2 and 3 for the function f (x) = x3 − 5x + 1

dIvIdIng POlynOmIAlS

For the following exercises, use long division to find the quotient and remainder.

19. x 3 − 2x2 + 4x + 4 ______________ x − 2 20.

3x4 − 4x2 + 4x + 8 _______________ x + 1

For the following exercises, use synthetic division to find the quotient. If the divisor is a factor, then write the factored form.

21. x 3 − 2x2 + 5x − 1 ______________ x + 3 22.

x3 + 4x + 10 __________ x − 3

23. 2x 3 + 6x2 − 11x − 12 _________________ x + 4 24.

3x4 + 3x3 + 2x + 2 _______________ x + 1

ZeROS OF POlynOmIAl FUnCTIOnS

For the following exercises, use the Rational Zero Theorem to help you solve the polynomial equation.

25. 2x3 − 3x2 − 18x − 8 = 0 26. 3x3 + 11x2 + 8x − 4 = 0

27. 2x4 − 17x3 + 46x2 − 43x + 12 = 0 28. 4x4 + 8x3 + 19x2 + 32x + 12 = 0

For the following exercises, use Descartes’ Rule of Signs to find the possible number of positive and negative solutions.

29. x3 − 3x2 − 2x + 4 = 0 30. 2x4 − x3 + 4x2 − 5x + 1 = 0

RATIOnAl FUnCTIOnS

For the following exercises, find the intercepts and the vertical and horizontal asymptotes, and then use them to sketch a graph of the function.

31. f (x) = x + 2 _____ x − 5 32. f (x) = x2 + 1 _____ x2 − 4

33. f (x) = 3x 2 − 27 ________ x2 + x − 2 34. f (x) =

x + 2 _____ x2 − 9

For the following exercises, find the slant asymptote.

35. f (x) = x 2 − 1 _____ x + 2 36. f (x) =

2x3 − x2 + 4 __________ x2 + 1

InveRSeS And RAdICAl FUnCTIOnS For the following exercises, find the inverse of the function with the domain given.

37. f (x) = (x − 2)2, x ≥ 2 38. f (x) = (x + 4)2 − 3, x ≥ −4 39. f (x) = x2 + 6x − 2, x ≥ −3

40. f (x) = 2x3 − 3 41. f (x) = √ —

4x + 5 − 3 42. f (x) = x − 3 _____ 2x + 1

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460 CHAPTER 5 PolyNomial aNd ratioNal fuNctioNs

mOdelIng USIng vARIATIOn

For the following exercises, find the unknown value.

43. y varies directly as the square of x. If when x = 3, y = 36, find y if x = 4.

44. y varies inversely as the square root of x. If when x = 25, y = 2, find y if x = 4.

45. y varies jointly as the cube of x and as z. If when x = 1 and z = 2, y = 6, find y if x = 2 and z = 3.

46. y varies jointly as x and the square of z and inversely as the cube of w. If when x = 3, z = 4, and w = 2, y = 48, find y if x = 4, z = 5, and w = 3.

For the following exercises, solve the application problem.

47. The weight of an object above the surface of the earth varies inversely with the distance from the center of the earth. If a person weighs 150 pounds when he is on the surface of the earth (3,960 miles from center), find the weight of the person if he is 20 miles above the surface.

48. The volume V of an ideal gas varies directly with the temperature T and inversely with the pressure P. A cylinder contains oxygen at a temperature of 310 degrees K and a pressure of 18 atmospheres in a volume of 120 liters. Find the pressure if the volume is decreased to 100 liters and the temperature is increased to 320 degrees K.

461CHAPTER 5 Practice test

ChAPTeR 5 PRACTICe TeST

Give the degree and leading coefficient of the following polynomial function.

1. f (x) = x3(3 − 6x2 − 2x2)

Determine the end behavior of the polynomial function.

2. f (x) = 8x3 − 3x2 + 2x − 4 3. f (x) = −2x2(4 − 3x − 5x2)

Write the quadratic function in standard form. Determine the vertex and axes intercepts and graph the function.

4. f (x) = x2 + 2x − 8

Given information about the graph of a quadratic function, find its equation.

5. Vertex (2, 0) and point on graph (4, 12).

Solve the following application problem.

6. A rectangular field is to be enclosed by fencing. In addition to the enclosing fence, another fence is to divide the field into two parts, running parallel to two sides. If 1,200 feet of fencing is available, find the maximum area that can be enclosed.

Find all zeros of the following polynomial functions, noting multiplicities.

7. f (x) = (x − 3)3(3x − 1)(x − 1)2 8. f (x) = 2x6 − 12x5 + 18x4

Based on the graph, determine the zeros of the function and multiplicities.

9.

x

y

–1–2–4–5 –3 –25 –50 –75

–100 –125

321

25 50 75

100 125

4 5

Use long division to find the quotient.

10. 2x 3 + 3x − 4 __________ x + 2

Use synthetic division to find the quotient. If the divisor is a factor, write the factored form.

11. x 4 + 3x2 − 4 __________ x − 2 12.

2x3 + 5x2 − 7x − 12 ________________ x + 3

Use the Rational Zero Theorem to help you find the zeros of the polynomial functions.

13. f (x) = 2x3 + 5x2 − 6x − 9 14. f (x) = 4x4 + 8x3 + 21x2 + 17x + 4

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462 CHAPTER 5 PolyNomial aNd ratioNal fuNctioNs

15. f (x) = 4x4 + 16x3 + 13x2 − 15x − 18 16. f (x) = x5 + 6x4 + 13x3 + 14x2 + 12x + 8

Given the following information about a polynomial function, find the function.

17. It has a double zero at x = 3 and zeroes at x = 1 and x = −2. It’s y-intercept is (0, 12).

18. It has a zero of multiplicity 3 at x =   1 _ 2 and another zero at x = −3. It contains the point (1, 8).

Use Descartes’ Rule of Signs to determine the possible number of positive and negative solutions.

19. 8x3 − 21x2 + 6 = 0

For the following rational functions, find the intercepts and horizontal and vertical asymptotes, and sketch a graph.

20. f (x) = x + 4 _________ x2 − 2x − 3 21. f (x) = x2 + 2x − 3 _________ x2 − 4

Find the slant asymptote of the rational function.

22. f (x) = x 2 + 3x − 3 _________ x − 1

Find the inverse of the function.

23. f (x) = √ —

x − 2 + 4 24. f (x) = 3x3 − 4

25. f (x) = 2x + 3 ______ 3x − 1

Find the unknown value.

26. y varies inversely as the square of x and when x = 3, y = 2. Find y if x = 1.

27. y varies jointly with x and the cube root of z. If when x = 2 and z = 27, y = 12, find y if x = 5 and z = 8.

Solve the following application problem.

28. The distance a body falls varies directly as the square of the time it falls. If an object falls 64 feet in 2 seconds, how long will it take to fall 256 feet?

ODD ANSWERS C-11

39. Hawaii 41. During the year 1933 43. $105,620 45. a. 696 people b. 4 years c. 174 people per year d. 305 people e. P(t) = 305 + 174t f. 2,219 people 47. a. C(x) = 0.15x + 10 b. The flat monthly fee is $10 and there is a $0.15 fee for each additional minute used. c. $113.05 49. a. P(t) = 190t + 4,360 b. 6,640 moose 51. a. R(t)= −2.1t + 16 b. 5.5 billion cubic feet c. During the year 2017 53. More than 133 minutes 55. More than $42,857.14 worth of jewelry 57. More than $66,666.67 in sales

Section 4.3 1. When our model no longer applies, after some value in the domain, the model itself doesn’t hold. 3. We predict a value outside the domain and range of the data. 5. The closer the number is to 1, the less scattered the data, the closer the number is to 0, the more scattered the data. 7. 61.966 years 9. No 11. No 13. Interpolation, about 60° F 15. C 17. B

19.

0 0

2

2 4 6 8 10

4

6

8

10

21.

0 0

2

2 4 6 8 10

4

6

8

10

23. Yes, trend appears linear; during 2016 25. y = 1.640x + 13.800, r = 0.987 27. y = −0.962x + 26.86, r = −0.965 29. y = −1.981x + 60.197; r = −0.998 31. y = 0.121x − 38.841, r = 0.998 33. (−2, −6), (1, −12), (5, −20), (6, −22), (9, −28) 35. (189.8, 0) If the company sells 18,980 units, its profits will be zero dollars. 37. y = 0.00587x + 1985.41 39. y = 20.25x − 671.5 41. y = −10.75x + 742.50

Chapter 4 Review exercises 1. Yes 3. Increasing 5. y = −3x + 26 7. 3 9. y = 2x − 2 11. Not linear 13. Parallel 15. (−9, 0); (0, −7) 17. Line 1: m −2, Line 2: m = −2, parallel 19. y = −0.2x + 21

21.

x

y

–1–1 –2 –3

–5 –4

–6

–2–3–4–5–6 321 4 5 6

3 2 1

4 5 6

23. 250 25. 118,000 27. y = −300x + 11,500 29. a. 800 b. 100 students per year c. P(t) = 100t + 1700 31. 18,500 33. y = $91, 625

35. Extrapolation 37.

Chapter 4 practice test 1. Yes 3. Increasing 5. y = −1.5x − 6 7. y = −2x − 1 9. No 11. Perpendicular 13. (−7, 0); (0, −2) 15. y = −0.25x + 12

17. Slope = −1 and y-intercept = 6

19. 150 21. 165,000 23. y = 875x + 10,625 25. a. 375 b. dropped an average of 46.875, or about 47 people per year c. y = −46.875t + 1250

27. 29. Early in 2018 31. y = 0.00455x + 1979.5 33. r = 0.999

ChapteR 5

Section 5.1 1. When written in that form, the vertex can be easily identified. 3. If a = 0 then the function becomes a linear function. 5. If possible, we can use factoring. Otherwise, we can use the quadratic formula. 7. g(x) = (x + 1)2 − 4; vertex: (−1, −4)

9. f (x)=  x + 5 _ 2  2 − 33 _ 4 ; vertex:  −

5 _ 2 , − 33 _ 4 

11. k(x) = 3(x − 1)2 − 12; vertex: (1, −12)

13. f (x) = 3  x − 5 _ 6  2 − 37 _ 12 ; vertex: 

5 _ 6

, − 37 _ 12  15. Minimum is − 17 _ 2 and occurs at

5 _ 2 ; axis of symmetry: x = 5 _ 2

17. Minimum is − 17 _ 16

and occurs at − 1 _ 8 ; axis of symmetry: x = − 1 _ 8

19. Minimum is − 7 _ 2 and occurs at −3; axis of symmetry: x = −3

x

y

6,900 6,800 6,700 6,600 6,500 6,400 6,300 6,200 6,100 6,000 5,900 5,800 5,700 5,600

1985 1990 1995 2000 Year

Po pu

la tio

n

2005 20100

0 0

20 40 60 80

100 120

2 4 6 8 10 x

y

39. Midway through 2023 41. y = −1.294x + 49.412; r = −0.974 43. Early in 2027 45. 7, 660

x

y

–1–1 –2 –3 –4

–2–3 321 4 5 6 7 8 9

3 2 1

4 5 6 7 8

0 0 5

10 15 20 25 30 35

2 4 6 8 10 12 x

y

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ODD ANSWERSC-12

21. Domain: (−∞, ∞); range: [2, ∞) 23. Domain: (−∞, ∞); range: [−5, ∞) 25. Domain: (−∞, ∞); range: [−12, ∞) 27. f (x) = x 2 + 4x + 3 29. f (x) = x 2 − 4x + 7 31. f (x) = − 1 __ 49 x

2 + 6 __ 49 x + 89 __ 49 33. f (x) = x

2 − 2x + 1

35. Vertex: (3, −10), axis of symmetry: x = 3, intercepts: (3 + √

— 10 , 0) and (3 − √

— 10 , 0)

x

y

−3 −6 −9

−15 −12

−5−10 5 10

9 6 3

12 15

37. Vertex:  7 _ 2 , − 37 _ 4  , axis of

symmetry: x = 7 _ 2 , intercepts:

 7 + √ —

37 _ 2 , 0  and  7 − √ —

37 _ 2 , 0 

x

y

−3 −6 −9

−15 −12

−5−10 5 10

9 6 3

12 15

39. Vertex:  3 _ 2 , −12  , axis of symmetry: x = 3 _ 2 , intercept:

 3 + 2 √ — 3 _ 2 , 0  and  3 − 2 √

— 3 _ 2 , 0 

x

y

−3 −6 −9

−15 −12

−1−2−3−4−5 2 3 4 51

9 6 3

12 15

41. f (x) = x 2 + 2x + 3 43. f (x) = −3 x 2 − 6x − 1

45. f (x) = − 1 __ 4 x 2 − x + 2

47. f (x) = x 2 + 2x + 1 49. f (x) = −x 2 + 2x 51. The value stretches or compresses the width of the graph. The greater the value, the narrower the graph. 53. The graph is shifted to the right or left (a horizontal shift). 55. The suspension bridge has 1,000 feet distance from the center. 57. Domain: (−∞, ∞); range: (−∞, 2]

59. Domain: (−∞, ∞); range: [100, ∞) 61. f (x) = 2x 2 + 2 63. f (x) = −x 2 − 2 65. f (x) = 3x 2 + 6x − 15 67. 75 feet by 50 feet 69. 3 and 3; product is 9 71. The revenue reaches the maximum value when 1800 thousand phones are produced. 73. 2.449 seconds 75. 41 trees per acre

Section 5.2 1. 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. 3. As x decreases without bound, so does f (x). As x increases without bound, so does f (x). 5. The polynomial function is of even degree and leading coefficient is negative. 7. Power function 9. Neither 11. Neither 13. Degree: 2, coefficient: −2 15. Degree: 4, coefficient: −2 17. As x → ∞, f (x) → ∞, as x → −∞, f (x) → ∞ 19. As x → −∞, f (x) → −∞, as x → ∞, f (x) → −∞ 21. As x → −∞, f (x) → −∞, as x → ∞, f (x) → −∞ 23. As x → ∞, f (x) → ∞, as x → −∞, f (x) → −∞ 25. y-intercept is (0, 12), t-intercepts are (1, 0), (−2, 0), and (3, 0) 27. y-intercept is (0, −16), x-intercepts are (2, 0), and (−2, 0) 29. y-intercept is (0, 0), x-intercepts are (0, 0), (4, 0), and (−2, 0) 31. 3 33. 5 35. 3 37. 5 39. Yes, 2 turning points, least possible degree: 3 41. Yes, 1 turning point, least possible degree: 2

43. Yes, 0 turning points, least possible degree: 1 45. Yes, 0 turning points, least possible degree: 1 47. As x → −∞, f (x ) → ∞, as x → ∞, f (x) → ∞

x f (x) 10 9,500

100 99,950,000 −10 9,500

−100 99,950,000

49. As x → −∞, f (x ) → ∞, as x → ∞, f (x) → −∞

x f (x) 10 −504

100 −941,094 −10 1,716

−100 1,061,106 51. y-intercept: (0, 0); x-intercepts: (0, 0) and (2, 0); as x → −∞, f (x ) → ∞, as x → ∞, f (x) → ∞

2

x

y

−1 −1−2

−2

−3

−3

−4

−4

−5−6

−5

1

3

3

21 4

4

5 6

5

53. y-intercept: (0, 0); x-intercepts: (0, 0), (5, 0), (7, 0); as x → −∞, f (x ) → −∞, as x → ∞, f (x) → ∞

80

x

y

−1−2

40

3

120

21 4

160 200

−80 −40

−120 −160 −200

5 6 7 8 9 10

55. y-intercept: (0, 0); x-intercepts: (−4, 0), (0, 0), (4, 0); as x → −∞, f (x ) → −∞, as x → ∞, f (x) → ∞

200

x

y

−2−4−6−8−10

100

6

300

42 8

400 500

10

−200 −100

−300 −400 −500

57. y-intercept: (0, −81); x-intercepts: (−3, 0), and (3, 0); as x → −∞, f (x ) → ∞, as x → ∞, f (x) → ∞

40

x

y

−1−2−3−4−5−6

20

3

60

21 4

80

5 6

100

−40 −20

−60 −80

−100

59. y-intercept: (0, 0); x-intercepts: (−3, 0), (0, 0), (5, 0); as x → −∞, f (x ) → −∞, as x → ∞, f (x) → ∞

61. f (x ) = x 2 − 4 63. f (x ) = x 3 − 4x 2 + 4x 65. f (x ) = x 4 + 1 67. V(m ) = 8m 3 + 36m 2 + 54m + 27 69. V(x ) = 4x 3 − 32x 2 + 64x

20

x

y

−1−2−3−4−5

10

3

30

21 4

40

5 6

50

−20 −10

−30 −40 −50

Section 5.3 1. The x-intercept is where the graph of the function crosses the x-axis, and the zero of the function is the input value for which f (x) = 0. 3. If we evaluate the function at a and at b and the sign of the function value changes, then we know a zero exists between a and b. 5. There will be a factor raised to an even power. 7. (−2, 0), (3, 0), (−5, 0) 9. (3, 0), (−1, 0), (0, 0) 11. (0, 0), (−5, 0), (2, 0) 13. (0, 0), (−5, 0), (4, 0)

15. (2, 0), (−2, 0), (−1, 0) 17. (−2, 0), (2, 0),  1 _ 2 , 0 

ODD ANSWERS C-13

19. (1, 0), (−1, 0) 21. (0, 0),  √— 3 , 0  ,  − √— 3 , 0  23. (0, 0), (1, 0), (−1, 0), (2, 0), (−2, 0) 25. f (2) = −10, f (4) = 28; sign change confirms 27. f (1) = 3, f (3) = −77; sign change confirms 29. f (0.01) = 1.000001, f (0.1) = −7.999; sign change confirms 31. 0 with multiplicity 2, − 3 _ 2 multiplicity 5, 4 multiplicity 2

33. 0 with multiplicity 2, −2 with multiplicity 2 35. − 2 _ 3 with multiplicity 5, 5 with multiplicity 2 37. 0 with

multiplicity 4, 2 with multiplicity 1, −1 with multiplicity 1

39. 3 _ 2 with multiplicity 2, 0 with multiplicity 3 41. 0 with

multiplicity 6, 2 _ 3 with multiplicity 2 43. x-intercept: (1, 0) with multiplicity 2, (−4, 0) with multiplicity 1; y-intercept: (0, 4); as x → −∞, g (x) → −∞, as x → ∞, g (x) → ∞

x

g(x)

−4 −8

−1−2−3−4−5 2 3 4 51

12 16 20

8 4

45. x-intercept: (3, 0) with multiplicity 3, (2, 0) with multiplicity 2; y-intercept: (0, −108); as x → −∞, k (x) → −∞, as x → ∞, k (x) → ∞

x

k(x)

−12 −24 −36 −48 −60 −72 −84 −96

−108 −120

−1−2−3−4−5−6 2 3 4 5 61

12 24

47. x-intercepts: (0, 0), (−2, 0), (4, 0) with multiplicity 1; y-intercept: (0, 0); as x → −∞, n (x) → ∞, as x → ∞, n (x) → −∞

x

n(x)

−15

−45 −30

−60 −75

−1−2−3−4−5 2 3 4 51

75 60 45 30 15

49. f (x) = − 2 _ 9 (x − 3)(x + 1)(x + 3)

51. f (x) = 1 _ 4 (x + 2) 2(x − 3)

53. −4, −2, 1, 3 with multiplicity 1

55. −2, 3 each with multiplicity 2

57. f (x) = − 2 _ 3 (x + 2)(x − 1)(x − 3)

59. f (x) = 1 _ 3 (x − 3) 2(x − 1)2(x + 3)

61. f (x) = −15(x − 1)2(x − 3)3

63. f (x) = −2(x + 3)(x + 2)(x − 1)

65. f (x) = − 3 _ 2 (2x − 1) 2(x − 6)(x + 2)

67. Local max: (−0.58, −0.62); local min: (0.58, −1.38) 69. Global min: (−0.63, −0.47) 71. Global min: (0.75, −1.11)

73. f (x) = (x − 500)2(x + 200) 75. f (x) = 4x 3 − 36x 2 + 80x

77. f (x) = 4x 3 − 36x 2 + 60x + 100

79. f (x) = 1 _ π (9x 3 + 45x 2 + 72x + 36)

Section 5.4 1. The binomial is a factor of the polynomial. 3. x + 6 + 5 _ x − 1 , quotient: x + 6, remainder: 5

5. 3x + 2, quotient: 3x + 2, remainder: 0 7. x − 5, quotient: x − 5, remainder: 0 9. 2x − 7 + 16 _ x + 2 , quotient: 2x − 7,

remainder 16 11. x − 2 + 6 _ 3x + 1 , quotient: x − 2, remainder: 6

13. 2x 2 − 3x + 5, quotient: 2x 2 − 3x + 5, remainder: 0

15. 2x 2 + 2x + 1 + 10 _ x − 4 17. 2x 2 − 7x + 1 − 2 _ 2x + 1

19. 3x 2 − 11x + 34 − 106 _ x + 3 21. x 2 + 5x + 1

23. 4x 2 − 21x + 84 − 323 _ x + 4 25. x 2 − 14x + 49

27. 3x 2 + x + 2 _ 3x − 1 29. x 3 − 3x + 1 31. x 3 − x 2 + 2

33. x 3 − 6x 2 + 12x − 8 35. x 3 − 9x 2 + 27x − 27 37. 2x 3 − 2x + 2 39. Yes, (x − 2)(3x 3 − 5) 41. Yes, (x − 2)(4x 3 + 8x 2 + x + 2) 43. No 45. (x − 1)(x 2 + 2x + 4) 47. (x − 5)(x 2 + x + 1) 49. Quotient: 4x 2 + 8x + 16, remainder: −1 51. Quotient is 3x 2 + 3x + 5, remainder: 0 53. Quotient is x 3 − 2x 2 + 4x − 8, remainder: −6 55. x 6 − x 5 + x 4 − x 3 + x 2 − x + 1

57. x 3 − x 2 + x − 1 + 1 _ x + 1 59. 1 + 1 + i _ x − i

61. 1 + 1 − i _ x + i

63. x 2 + ix − 1 + 1 − i _ x − i

65. 2x 2 + 3 67. 2x + 3

69. x + 2 71. x − 3 73. 3x 2 − 2

Section 5.5 1. The theorem can be used to evaluate a polynomial. 3. Rational zeros can be expressed as fractions whereas real zeros include irrational numbers. 5. Polynomial functions can have repeated zeros, so the fact that number is a zero doesn’t preclude it being a zero again. 7. −106 9. 0 11. 255

13. −1 15. −2, 1, 1 _ 2 17. −2 19. −3

21. − 5 _ 2 , √ — 6 , − √

— 6 23. 2, −4, − 3 _ 2 25. 4, −4, −5

27. 5, −3, − 1 _ 2 29. 1 _ 2 ,

1 + √ — 5 _ 2 ,

1 − √ — 5 _ 2

31. 3 _ 2 33. 2, 3, −1, −2 35. 1 _ 2 , −

1 _ 2 , 2, −3

37. −1, −1, √ — 5 , − √

— 5 39. − 3 _ 4 , −

1 _ 2 41. 2, 3 + 2i, 3 − 2i

43. − 2 _ 3 , 1 + 2i, 1 − 2i 45. − 1 _ 2 , 1 + 4i, 1 − 4i

47. 1 positive, 1 negative

2

x

f (x)

−1 −1−2

−2

−3

−3

−4

−4

−5−6

−5

1

3

3

21 4

4

5 6

5

49. 1 positive, 0 negative

2

x

f (x)

−1−2−3−4−5−6

1

3

3

21 4

4 5

−2 −1

−3 −4 −5

5 6

51. 0 positive, 3 negative

40

x

f (x)

−2 −20−4

−40

−6

−60

−8

−80 −100

−10−12−14

20

6

60

42 8

80 100

10

53. 2 positive, 2 negative

16

x

f (x)

−1−2−3−4−5−6

8

3

24

21 4

32 40

−16 −8

−24 −32 −40

5 6

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ODD ANSWERSC-14

55. 2 positive, 2 negative

6

x

f (x)

−1−2−3−4−5−6

3

3

9

21 4

12 15

−6 −3

−9 −12 −15

5 6

57. ± 1 _ 2 , ±1, ±5, ± 5 _ 2

59. ±1, ± 1 _ 2 , ± 1 _ 3 , ±

1 _ 6

61. 1, 1 _ 2 , − 1 _ 3

63. 2, 1 _ 4 , − 3 _ 2 65.

5 _ 4

67. f (x) = 4 _ 9 (x 3 + x 2 − x − 1)

69. f (x) = − 1 __ 5 (4x 3 − x)

71. 8 by 4 by 6 inches

73. 5.5 by 4.5 by 3.5 inches 75. 8 by 5 by 3 inches 77. Radius: 6 meters; height: 2 meters 79. Radius: 2.5 meters, height: 4.5 meters

Section 5.6 1. The rational function will be represented by a quotient of polynomial functions. 3. The numerator and denominator must have a common factor. 5. Yes. The numerator of the formula of the functions would have only complex roots and/or factors common to both the numerator and denominator. 7. All reals except x = −1, 1 9. All reals except x = −1, 1, −2, 2

11. Vertical asymptote: x = − 2 _ 5 ; horizontal asymptote: y = 0;

domain: all reals except x = − 2 _ 5 13. Vertical asymptotes:

x = 4, −9; horizontal asymptote: y = 0; domain: all reals except x = 4, −9 15. Vertical asymptotes: x = 0, 4, −4; horizontal asymptote: y = 0; domain: all reals except x = 0, 4, −4 17. Vertical asymptotes: x = −5; horizontal asymptote: y = 0; domain: all reals except x = 5, −5

19. Vertical asymptote: x = 1 _ 3 ; horizontal asymptote: y = − 2 _ 3 ;

domain: all reals except x = 1 _ 3 21. None

23. x-intercepts: none, y-intercept:  0, 1 _ 4  25. Local behavior: x → − 1 _ 2

+, f (x) → −∞, x → − 1 _ 2 −

, f (x) → ∞

End behavior: x → ±∞, f (x) → 1 _ 2

27. Local behavior: x → 6+, f (x) → −∞, x → 6−, f (x) → ∞ End behavior: x → ±∞, f (x) → −2

29. Local behavior: x → − 1 _ 3 +

, f (x) → ∞, x → − 1 _ 3 −

, f (x) → −∞,

x → − 5 _ 2 −

, f (x) → ∞, x → − 5 _ 2 +

, f (x) → −∞

End behavior: x → ±∞, f (x) → 1 _ 3

31. y = 2x + 4 33. y = 2x 35. Vertical asymptote at x = 0, horizontal asymptote at y = 2

2

x

y

−1 −1 −2 −3 −4 −5

−2−3−4−5

1

3

3

21 4

4

5

5

x = 3 y = −4

37. Vertical asymptote at x = 2, horizontal asymptote at y = 0

4

x

y

−2 −2 −4 −6 −8

−10

−4−6−8−10

2

6

6

42 8

8

10

10

y = 0 x = 2

39. Vertical asymptote at x = −4; horizontal asymptote

at y = 2;  3 _ 2 , 0  ,  0, − 3 _ 4 

4

x −2 −2 −4 −6 −8

−10

−4−6−8−10

2

6

6

42 8

8

10

10

y = 2

x = −4

p(x)

41. Vertical asymptote at x = 2; horizontal asymptote at y = 0; (0, 1)

2

x

s(x)

−2 −1 −2

−4−6−8−10

1

6

3

42 8

4

10

5 6 7 8 9

10 12

y = 0

x = 2

43. Vertical asymptote at x = −4, 4 _ 3 ; horizontal asymptote at y = 1; (5, 0),  − 1 _ 3 , 0  ,  0,

5 _ 16



4

x

f(x)

−2 −2 −4 −6 −8

−10 −12

−4−6−8−10−12

2

6

6

42 8

8

10 12

10 12

x = 43 y = 1

x = −4

45. Vertical asymptote at x = −1; horizontal asymptote at y = 1; (−3, 0), (0, 3)

6

x

a(x)

−3 −3 −6 −9

−12 −15

−6−9−12−15

3

9

9

63 12

12

15

15

y = 1

x =−1

47. Vertical asymptote at x = 4; slant asymptote at y = 2x + 9; (−1, 0),  1 _ 2 , 0  ,  0, 1 _ 4 

20

x

h(x)

−10 −10 −20 −30 −40 −50

−20−30−40−50

10

30

10 20 30 40 50

40 50

y = 2x + 9

x = 4

49. Vertical asymptote at x = −2, 4; horizontal asymptote at y = 1; (−3, 0), (0, 3)

−10

4

x

w(x)

−2 −2 −4 −6 −8

−4−6−8−10−12

2

6

6

42 8

8

10 12

10 12 x = 4

y = 1

x = −2

51. f (x) = 50 x 2 − x − 2 ________ x2 − 25 53. f (x) = 7

x2 + 2x − 24 __ x2 + 9x + 20

55. f (x) = 1 _ 2 ⋅ x2 − 4x + 4 __ x + 1 57. f (x) = 4

x − 3 __ x2 − x − 12

59. f (x) = −9 x − 2 __ x 2 − 9

61. f (x) = 1 _ 3 ∙ x2 + x − 6 _ 3x − 1

63. f (x) = −6 (x − 1) 2 ____________ (x + 3)(x − 2)2

65. Vertical asymptote at x = 2; horizontal asymptote at y = 0

x 2.01 2.001 2.0001 1.99 1.999 y 100 1,000 10,000 −100 −1,000

x 10 100 1,000 10,000 100,000 y 0.125 0.0102 0.001 0.0001 0.00001

ODD ANSWERS C-15

67. Vertical asymptote at x = −4; horizontal asymptote at y = 2 x −4.1 −4.01 −4.001 −3.99 −3.999 y 82 802 8,002 −798 −7998

x 10 100 1,000 10,000 100,000 y 1.4286 1.9331 1.992 1.9992 1.999992

69. Vertical asymptote at x = −1; horizontal asymptote at y = 1 x −0.9 −0.99 −0.999 −1.1 −1.01 y 81 9,801 998,001 121 10,201

x 10 100 1,000 10,000 100,000 y 0.82645 0.9803 0.998 0.9998

71.  3 _ 2 , ∞ 

4

x

f(x)

–2–2 –4 –6 –8

–10

–4–6–8–10

2

6

6

42 8

8

10

10

73. (−∞, 1) ∪ (4, ∞)

4

x

f(x)

–2–2 –4 –6 –8

–10

–4–6–8–10

2

6

6

42 8

8

10

10

75. (2, 4) 77. (2, 5) 79. (−1, 1) 81. C(t) = 8 + 2t _ 300 + 20t

83. After about 6.12 hours 85. 2 by 2 by 5 feet 87. radius 2.52 meters

Section 5.7 1. It can be too difficult or impossible to solve for x in terms of y. 3. We will need a restriction on the domain of the answer. 5. f −1 (x) = √

— x + 4 7. f −1(x) = √

— x + 3 −1

9. f −1(x) = √ —

12 − x 11. f −1(x) = ± √ ______

x − 4 _ 2

13. f −1(x) = 3 √ —

x − 1 _ 3 15. f −1(x) =

3 √ —

4 − x _ 2

17. f −1(x) = 3 − x 2 ______ 4 , [0, ∞) 19. f

−1(x) = (x − 5) 2 + 8 __________ 6

21. f −1(x) = (3 − x)3 23. f −1(x) = 4x + 3 ______ x 25. f −1(x) = 7x + 2 ______ 1 − x

27. f −1(x) = 2x − 1 ______ 5x + 5 29. f −1(x) = √

— x + 3 − 2

31. f −1(x) = √ —

x − 2

4

x

y

−2−4−6−8−10

2

6

6

42 8

8

10

10

−4 −2

−6 −8

−10

33. f −1(x) = √ — x − 3

6

x

y

−3−6−9−12−15

3

9

9

63 12

12

15

15

−6 −3

−9 −12 −15

35. f −1(x) = 3 √ —

x − 3

x

y

3 4

4

5

5

2

2

1

1

6

3

6

−3−6

−3 −4

−4

−5

−5

−2

−2 −1−1

−6

37. f −1 (x) = √ —

x + 4 − 2

x

y

6 8 102 4

6 8

10

4 2

−6−8−10 −4−2

−6 −8

−10

−2 −4

39.

4

x

y

−2−4−6−8−10

2

6

6

42 8

8

10

10

−4 −2

−6 −8

−10

41. [−1, 0) ∪ [1, ∞)

2

x

f (x)

−1−2−3−4−5

1

3

3

21 4

4

5

5

−2 −1

−3 −4 −5

43. [−3, 0 ] ∪ (4, ∞)

16

x 8

24 32 40

168 24 32 40

−16 −8

−24 −32 −40

−16−8−24−32−40

f (x) 45. [−∞, −4) ⋅ [−3, 3]

16

x 8

24 32 40

168 24 32 40

−16 −8

−24 −32 −40

−16−8−24−32−40

f (x)

47. (−2, 0), (0, 1), (8, 2)

x

y

3 4 51 2

24 32 40

16 8

−3−4−5 −2 –1

−24 −32 −40

−8 −16

49. (−13, −1), (−4, 0), (5, 1)

x

y

3 4 51 2

24 32 40

16 8

−3−4−5 −2 –1

−24 −32 −40

−8 −16

51. f −1(x) = 3 √ ______

x − b _ a 53. f −1 (x) = √

_______

x 2 − b _ a

55. f −1(x) = cx − b _ a − x 57. t(h) = √ ________

600 − h _ 16

, 3.54 seconds

59. r (A) = √ ___

A _ 4π , ≈ 8.92 in. 61. l(T) = 32.2  T __ 2π 

2 , ≈ 3.26 ft

63. r(A) = √ _______

A + 8π _ 2π −2, 3.99 ft 65. r(V) = √ ___

V _ 10π , ≈ 5.64 ft

Section 5.8 1. The graph will have the appearance of a power function. 3. No. Multiple variables may jointly vary. 5. y = 5x 2

7. y = 10x 3 9. y = 6x 4 11. y = 18 _ x 2

13. y = 81 _ x 4

15. y = 20 _ 3 √

— x 17. y = 10xzw 19. y = 10x √

— z

21. y = 4 xz _ w 23. y = 40 xz _

√ —

w t2 25. y = 256

27. y = 6 29. y = 6 31. y = 27 33. y = 3

35. y = 18 37. y = 90 39. y = 81 _ 2

This OpenStax book is available for free at http://cnx.org/content/col11758/latest

ODD ANSWERSC-16

41. y = 3 _ 4 x 2

30

x

y

−2−4−6−8−10

15

6

45

42 8

60

10

75

−30 −15

−45 −60 −75

43. y = 1 __ 3 √ — x

4

x

y

−5−10−15−20−25

2

15

6

105 20

8

25

10

−4 −2

−6 −8

−10

45. y = 4 __ x2

4

x

y

−2−4−6−8−10

2

6

6

42 8

8

10

10

−4 −2

−6 −8

−10

47. ≈ 1.89 years 49. ≈ 0.61 years 51. 3 seconds 53. 48 inches 55. ≈ 49.75 pounds 57. ≈ 33.33 amperes 59. ≈ 2.88 inches

Chapter 5 Review exercises 1. f (x) = (x − 2)2 −9; vertex: (2, −9); intercepts: (5, 0), (−1, 0), (0, −5)

x

f(x)

−1 −2 −4 −6

−10 −8

−2−3−4−5 321 4 5 6 7

6 4 2

8 10

3. f (x) = 3 _ 25 (x + 2) 2 + 3

5. 300 meters by 150 meters, the longer side parallel to the river 7. Yes; degree: 5, leading coefficient: 4 9. Yes; degree: 4; leading coefficient: 1

11. As x → −∞, f (x) → −∞, as x → ∞, f (x) → ∞

13. −3 with multiplicity 2, − 1 _ 2 with multiplicity 1, −1 with multiplicity 3 15. 4 with multiplicity 1 17. 1 _ 2 with multiplicity 1, 3 with multiplicity 3 19. x 2 + 4 with remainder is 12 21. x 2 − 5x + 20 − 61 _ x + 3

23. 2x 2 − 2x − 3, so factored form is (x + 4)(2x 2 − 2x − 3)

25.  −2, 4, − 1 _ 2  27.  1, 3, 4, 1 _ 2 

29. 2 or 0 positive, 1 negative

31. Intercepts: (−2, 0),  0, − 2 _ 5  , asymptotes: x = 5 and y = 1

x −5

−10 −15

−25 −20

5 10 15 20 25 30−5−10−15−20−25

15 10

5

20 25

y

y = 1

x = 5

33. Intercepts: (3, 0), (−3, 0),  0, 27 _ 2  ; asymptotes: x = 1, −2 and y = 3

y

x −8

−16 −24

−40 −32

3 6 9 12 15−3−6−9−12−15

24 16

8

32 40

x = −2

y = 3

x = 1

35. y = x − 2 37. f −1(x) = √ — x + 2 39. f −1(x) = √

— x + 11 − 3

41. f −1(x) = (x + 3)2 − 5

__ 4 , x ≥ −3 43. y = 64 45. y = 72

47. ≈ 148.5 pounds

Chapter 5 practice test

1. Degree: 5, leading coefficient: −2 3. As x → −∞, f (x) → ∞, as x → ∞, f (x) → ∞ 5. f (x) = 3(x − 2)2

7. 3 with multiplicity 3, 1 _ 3 with multiplicity 1, 1with multiplicity 2

9. − 1 _ 2 with multiplicity 3, 2 with multiplicity 2

11. x 3 + 2x 2 + 7x + 14 + 26 _ x − 2 13.  −3, −1, 3 _ 2 

15. 1, −2, and − 3 _ 2 (multiplicity 2)

17. f (x) = − 2 _ 3 (x − 3) 2(x − 1)(x + 2) 19. 2 or 0 positive, 1 negative

21. (−3, 0), (1, 0),  0, 3 _ 4  ; asymptotes x = −2, 2 and y = 1

y

x −2 −4 −6

−10 −8

2 4 6 8 10−2−4−6−8−10

6 4 2

8 10

x = −2 x = 2

y = 1

23. f −1(x) = (x − 4)2 + 2, x ≥ 4

25. f −1(x) = x + 3 _ 3x − 2

27. y = 20

ChapteR 6

Section 6.1 1. Linear functions have a constant rate of change. Exponential functions increase based on a percent of the original. 3. When interest is compounded, the percentage of interest earned to principal ends up being greater than the annual percentage rate for the investment account. Thus, the annual percentage rate does not necessarily correspond to the real interest earned, which is the very definition of nominal. 5. Exponential; the population decreases by a proportional rate. 7. Not exponential; the charge decreases by a constant amount each visit, so the statement represents a linear function. 9. Forest B 11. After 20 years forest A will have 43 more trees than forest B. 13. Answers will vary. Sample response: For a number of years, the population of forest A will increasingly exceed forest B, but because forest B actually grows at a faster rate, the population will eventually become larger than forest A and will remain that way as long as the population growth models hold. Some factors that might influence the long-term validity of the exponential growth model are drought, an epidemic that culls the population, and other environmental and biological factors. 15. Exponential growth; the growth factor, 1.06, is greater than 1. 17. Exponential decay; the decay factor, 0.97, is between 0 and 1.

19. f (x) = 2000(0.1)x 21. f (x) =  1 __ 6  − 35–  1 __ 6 

x 5– ≈ 2.93(0.699)x

23. Linear 25. Neither 27. Linear 29. $10,250

31. $13,268.58 33. P = A(t) ⋅  1 + r _ n  −nt

35. $4,569.10 37. 4% 39. Continuous growth; the growth rate is greater than 0. 41. Continuous decay; the growth rate is less than 0.

43. $669.42 45. f (−1) = −4 47. f (−1) ≈ −0.2707 49. f (3) ≈ 483.8146 51. y = 3 ⋅ 5x 53. y ≈ 18 ⋅ 1.025x

55. y ≈ 0.2 ⋅ 1.95x