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1 Quadratic Functions and Models 135 In Exercises 29–36, use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and -intercepts. Then check your results algebraically by writing the quadratic function in standard form. x 50. Vertex: 51. Vertex: 52. Vertex: 4 ; 5 2, 3 2, 0; 5 6, 6; point: 2, 4 7 2, 16 point: 61 10, 3 2 3 point: 29. 31. 33. 35. f x x 2 2x 3 gx x 2 8x 11 f x 2x 2 16x 31 gx 1 x 2 4x 2 2 30. 32. 34. 36. f x x2 x 30 f x x2 10x 14 f x 4x2 24x 41 f x 3 5 x 2 6x 5 In Exercises 37– 42, find the standard form of the quadratic function. 37. 39. 41. y 8 6 (1, 0) 38. y 2 (−1, 0) (0, 1) (1, 0) −2 2 4 x (0, 1) −2 (−1, 4) (−3, 0) −4 −2 x x 4 2 y (1, 0) 2 2 −2 −4 y 2 (−2, 2) (−3, 0) −6 −4 x 2 (−1, 0) −6 40. −4 −6 y 6 (0, 3) 2 −6 −4 (−2, −1) x 2 42. y 8 6 4 2 (2, 0) (3, 2) −2 2 4 6 x In Exercises 43–52, write the standard form of the equation of the parabola that has the indicated vertex and whose graph passes through the given point. 43. Vertex: 44. Vertex: 45. Vertex: 46. Vertex: 47. Vertex: 48. Vertex: 49. Vertex: point: point: 0, 9 2, 3 point: point: point: 1, 2 0, 2 7, 15 2, 5; 4, 1; 3, 4; 2, 3; 5, 12; 2, 2; 1 ; 4, 3 2 point: 2, 0 point: 1, 0 In Exercises 53–56, determine -intercept(s) of the graph visually. Then
find the Graphical Reasoning the x x -intercepts algebraically to confirm your results. 53. y x 2 16 y −8 −4 x 8 55. y x 2 4x 5 y 54. y x 2 6x 56. y 2x 2 5x 3 x −6 −4 8 −4 −4 −8 y 2 −2 −4 x 2 58. 57. In Exercises 57–64, use a graphing utility to graph the quadratic function. Find the -intercepts of the graph and x compare them with the solutions of the corresponding f x 0. quadratic equation when f x x 2 4x f x 2 x2 10x f x x 2 9x 18 f x x2 8x 20 f x 2x 2 7x 30 f x 4x2 25x 21 f x 1 x 2 6x 7 2 x2 12x 45 f x 7 10 60. 59. 62. 64. 63. 61. x In Exercises 65–70, find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given -intercepts. (There are many correct answers.) 1, 0, 3, 0 0, 0, 10, 0 3, 0, 1 5, 0, 5, 0 4, 0, 8, 0 2, 0, 2, 0 68. 70. 5 2, 0 65. 69. 66. 67. 333202_0201.qxd 12/7/05 9:10 AM Page 136 136 Chapter 2 Polynomial and Rational Functions In Exercises 71–74, find two positive real numbers whose product is a maximum. 71. The sum is 110. 72. The sum is S. (c) Use the result of part (b) to write the area of the rectangular region as a function of What dimensions will produce a maximum area of the rectangle? x. A 77. Path of a Diver The path of a diver is given by 73. The sum of the first and twice the second is 24. 74. The sum of the first and three times the second is 42. y 4 9 x 2 24 9 x 12 75. Numerical, Graphical, and Analytical Analysis A rancher has 200 feet of fencing to enclose two adjacent rectangular corrals (see figure). x x y (a) Write the area of the corral as a function of A x. (b) Create a
table showing possible values of and the corresponding areas of the corral. Use the table to estimate the dimensions that will produce the maximum enclosed area. x (c) Use a graphing utility to graph the area function. Use the graph to approximate the dimensions that will produce the maximum enclosed area. (d) Write the area function in standard form to find analytically the dimensions that will produce the maximum area. (e) Compare your results from parts (b), (c), and (d). 76. Geometry An indoor physical fitness room consists of a rectangular region with a semicircle on each end (see figure). The perimeter of the room is to be a 200-meter single-lane running track. x y (a) Determine the radius of the semicircular ends of the room. Determine the distance, in terms of around the inside edge of the two semicircular parts of the track. y, (b) Use the result of part (a) to write an equation, in terms for the distance traveled in one lap around x y, of and the track. Solve for y. y is the height (in feet) and where is the horizontal distance from the end of the diving board (in feet). What is the maximum height of the diver? x 78. Height of a Ball The height y (in feet) of a punted foot- ball is given by y 16 2025 x2 9 5 x 1.5 x is the horizontal distance (in feet) from the point at where which the ball is punted (see figure). y x Not drawn to scale (a) How high is the ball when it is punted? (b) What is the maximum height of the punt? (c) How long is the punt? 79. Minimum Cost A manufacturer of lighting fixtures has daily production costs of C 800 10x 0.25x 2 C is the total cost (in dollars) and where is the number of units produced. How many fixtures should be produced each day to yield a minimum cost? x 80. Minimum Cost A textile manufacturer has daily produc- tion costs of C 100,000 110x 0.045x 2 C is the total cost (in dollars) and where is the number of units produced. How many units should be produced each day to yield a minimum cost? x 81. Maximum Profit The profit P (in dollars) for a company that produces antivirus and system utilities software is P 0.0002x 2 140x 250,000
is the number of units sold. What sales level will x where yield a maximum profit? 333202_0201.qxd 12/7/05 9:10 AM Page 137 82. Maximum Profit The profit (in hundreds of dollars) that a company makes depends on the amount (in hundreds of dollars) the company spends on advertising according to the model P x P 230 20x 0.5x 2. What expenditure for advertising will yield a maximum profit? 83. Maximum Revenue The total revenue earned (in thousands of dollars) from manufacturing handheld video games is given by R p 25p2 1200p R where p is the price per unit (in dollars). (a) Find the revenue earned for each price per unit given below. $20 $25 $30 (b) Find the unit price that will yield a maximum revenue. What is the maximum revenue? Explain your results. 84. Maximum Revenue The total revenue R earned per day (in dollars) from a pet-sitting service is given by R p 12p2 150p where p is the price charged per pet (in dollars). (a) Find the revenue earned for each price per pet given below. $4 $6 $8 (b) Find the price that will yield a maximum revenue. What is the maximum revenue? Explain your results. 85. Graphical Analysis From 1960 to 2003, the per capita of cigarettes by Americans (age 18 and C consumption older) can be modeled by C 4299 1.8t 1.36t2, 0 ≤ t ≤ 43 t t 0 where is the year, with (Source: Tobacco Outlook Report) corresponding to 1960. (a) Use a graphing utility to graph the model. (b) Use the graph of the model to approximate the maximum average annual consumption. Beginning in 1966, all cigarette packages were required by law to carry a health warning. Do you think the warning had any effect? Explain. (c) In 2000, the U.S. population (age 18 and over) was 209,128,094. Of those, about 48,308,590 were smokers. What was the average annual cigarette consumption per smoker in 2000? What was the average daily cigarette consumption per smoker? Section 2.1 Quadratic Functions and Models 137 Model It 86. Data Analysis The numbers (in thousands) of hairdressers and cosmetologists in the United States for the years 1994 through 2002 are shown in the table. (Source: U.S. Bureau of Labor
Statistics) y Year 1994 1995 1996 1997 1998 1999 2000 2001 2002 Number of hairdressers and cosmetologists, y 753 750 737 748 763 784 820 854 908 (a) Use a graphing utility to create a scatter plot of the corre- represent the year, with x 4 x data. Let sponding to 1994. (b) Use the regression feature of a graphing utility to find a quadratic model for the data. (c) Use a graphing utility to graph the model in the same viewing window as the scatter plot. How well does the model fit the data? (d) Use the trace feature of the graphing utility to approximate the year in which the number of hairdressers and cosmetologists was the least. (e) Verify your answer to part (d) algebraically. (f) Use the model to predict the number of hairdressers and cosmetologists in 2008. 87. Wind Drag The number of horsepower required to overcome wind drag on an automobile is approximated by y y 0.002s 2 0.005s 0.029, 0 ≤ s ≤ 100 where s is the speed of the car (in miles per hour). (a) Use a graphing utility to graph the function. (b) Graphically estimate the maximum speed of the car if the power required to overcome wind drag is not to exceed 10 horsepower. Verify your estimate algebraically. 333202_0201.qxd 12/7/05 9:10 AM Page 138 138 Chapter 2 Polynomial and Rational Functions 88. Maximum Fuel Economy A study was done to compare the speed (in miles (in miles per hour) with the mileage per gallon) of an automobile. The results are shown in the (Source: Federal Highway Administration) table. y x Speed, x Mileage, y 15 20 25 30 35 40 45 50 55 60 65 70 75 22.3 25.5 27.5 29.0 28.8 30.0 29.9 30.2 30.4 28.8 27.4 25.3 23.3 (a) Use a graphing utility to create a scatter plot of the data. (b) Use the regression feature of a graphing utility to find a quadratic model for the data. (c) Use a graphing utility to graph the model in the same viewing window as the scatter plot. (d) Estimate the speed for which the miles per gallon is greatest.
Synthesis In Exercises 89 and 90, determine whether True or False? the statement is true or false. Justify your answer. f x 12x2 1 89. The function given by has no x -intercepts. 90. The graphs of f x 4x2 10x 7 and gx 12x2 30x 1 have the same axis of symmetry. 91. Write the quadratic function f x ax2 bx c in standard form to verify that the vertex occurs at b 2a, f b 2a. 92. Profit The profit (in millions of dollars) for a recreational vehicle retailer is modeled by a quadratic function of the form P P at 2 bt c t where represents the year. If you were president of the company, which of the models below would you prefer? Explain your reasoning. a a (a) (b) is positive and is positive and b2a ≤ t. t ≤ b2a. b2a ≤ t. t ≤ b2a. 93. Is it possible for a quadratic equation to have only one is negative and is negative and (d) (c) a a x -intercept? Explain. 94. Assume that the function given by f x ax 2 bx c, a 0 has two real zeros. Show that the vertex of the graph is the average of the zeros of Use the Quadratic Formula.) x -coordinate of the (Hint: f. Skills Review In Exercises 95–98, find the equation of the line in slope-intercept form that has the given characteristics. 4, 3 95. Passes through the points 7 2, 2 0, 3 96. Passes through the point 97. Passes through the point 4x 5y 10 98. Passes through the point 8, 4 line and 2, 1 and has a slope of 3 2 and is perpendicular to the and is parallel to the line and let gx 8x2. y 3x 2 fx 14x 3 99. 100. In Exercises 99–104, let Find the indicated value. f g3 g f 2 fg4 1.5 f g f g1 g f 0 103. 104. 101. 102. 7 105. Make a Decision To work an extended application analyzing the height of a basketball after it has been dropped, visit this text’s website at college.hmco.com. 333202_0202.qxd
12/7/05 9:11 AM Page 139 Section 2.2 Polynomial Functions of Higher Degree 139 2.2 Polynomial Functions of Higher Degree What you should learn • Use transformations to sketch graphs of polynomial functions. • Use the Leading Coefficient Test to determine the end behavior of graphs of polynomial functions. • Find and use zeros of polynomial functions as sketching aids. • Use the Intermediate Value Theorem to help locate zeros of polynomial functions. Why you should learn it You can use polynomial functions to analyze business situations such as how revenue is related to advertising expenses, as discussed in Exercise 98 on page 151. Graphs of Polynomial Functions In this section, you will study basic features of the graphs of polynomial functions. The first feature is that the graph of a polynomial function is continuous. Essentially, this means that the graph of a polynomial function has no breaks, holes, or gaps, as shown in Figure 2.10(a). The graph shown in Figure 2.10(b) is an example of a piecewise-defined function that is not continuous. y y x x (a) Polynomial functions have continuous graphs. FIGURE 2.10 (b) Functions with graphs that are not continuous are not polynomial functions. The second feature is that the graph of a polynomial function has only smooth, rounded turns, as shown in Figure 2.11. A polynomial function cannot have a sharp turn. For instance, the function given by which has a sharp turn at the point as shown in Figure 2.12, is not a polynomial function. f x x, 0, 0, y Bill Aron /PhotoEdit, Inc. f(x) = 4 −3 −2 −1 −2 3 4 1 2 (0, 0) x Polynomial functions have graphs with smooth rounded turns. FIGURE 2.11 Graphs of polynomial functions cannot have sharp turns. FIGURE 2.12 The graphs of polynomial functions of degree greater than 2 are more difficult to analyze than the graphs of polynomials of degree 0, 1, or 2. However, using the features presented in this section, coupled with your knowledge of point plotting, intercepts, and symmetry, you should be able to make reasonably accurate sketches by hand. 333202_0202.qxd 12/7/05 9:11 AM
Page 140 140 Chapter 2 Polynomial and Rational Functions if For power functions given by f x xn, n is even, then the graph of the function is symmetric with respect to the y n -axis, and if is odd, then the graph of the function is symmetric with respect to the origin. (−1, 1) −1 f x x n, The polynomial functions that have the simplest graphs are monomials of is an integer greater than zero. From Figure 2.13, f x x 2, Moreover, the the flatter the graph near the origin. Polynomial functions are often referred to as power functions. the form you can see that when n and when greater the value of of the form is even, the graph is similar to the graph of is odd, the graph is similar to the graph of n, f x xn where n f x x 31, 11, 1) −1 x 1 x 1 y xn −1 (−1, −1) y xn (b) If n is odd, the graph of x crosses the axis at the -intercept. (a) If n is even, the graph of x touches the axis at the -intercept. FIGURE 2.13 Example 1 Sketching Transformations of Monomial Functions Sketch the graph of each function. a. f x x5 b. hx x 14 Solution a. Because the degree of is odd, its graph is similar to the graph of In Figure 2.14, note that the negative coefficient has the effect of f x x 5 y x 3. reflecting the graph in the -axis. x b. The graph of hx x 14, as shown in Figure 2.15, is a left shift by one y x 4. unit of the graph of y (−1, 1) 1 f(x) = −x 5 x 1 −1 −1 (1, −1) FIGURE 2.14 Now try Exercise 9. h(x) = (x + 1) 4 y 3 2 1 (0, 1) x 1 (−2, 1) (−1, 0) −2 −1 FIGURE 2.15 333202_0202.qxd 12/7/05 9:11 AM Page 141 Exploration For each function, identify the degree of the function and whether the degree of the function is even or odd. Identify the leading coefficient and whether the leading coefficient is positive or negative. Use a graphing utility to graph each
function. Describe the relationship between the degree and the sign of the leading coefficient of the function and the right-hand and left-hand behavior of the graph of the function. a. b. c. d. e. f. g. f x x3 2x2 x 1 f x 2x5 2x2 5x 1 f x 2x5 x2 5x 3 f x x3 5x 2 f x 2x2 3x 4 f x x 4 3x2 2x 1 f x x2 3x 2 as f x → ” indicates that the The notation “ x → graph falls to the left. The f x → x → ” notation “ indicates that the graph rises to the right. as Section 2.2 Polynomial Functions of Higher Degree 141 The Leading Coefficient Test x In Example 1, note that both graphs eventually rise or fall without bound as moves to the right. Whether the graph of a polynomial function eventually rises or falls can be determined by the function’s degree (even or odd) and by its leading coefficient, as indicated in the Leading Coefficient Test. x Leading Coefficient Test As moves without bound to the left or to the right, the graph of the polynomial function falls in the following manner. f x anxn... a1x a0 eventually rises or 1. When n is odd: y y f(x) → ∞ as x → ∞ f(x) → ∞ as x → −∞ f(x) → −∞ as x → −∞ x f(x) → − ∞ as x → ∞ x If the leading coefficient is an > 0, positive the graph falls to the left and rises to the right. If the leading coefficient is an < 0, negative to the left and falls to the right. the graph rises 2. When n is even: y y f(x) → ∞ as x → −∞ f(x) → ∞ as x → ∞ f(x) → −∞ as x → −∞ f(x) → −∞ as x → ∞ x x If the leading coefficient is an > 0, positive the graph rises to the left and right. If the leading coefficient is an < 0, negative falls to the left and right. the graph The dashed portions of the graphs indicate that the test determines only the right-hand and left-hand behavior of
the graph. 333202_0202.qxd 12/7/05 9:11 AM Page 142 142 Chapter 2 Polynomial and Rational Functions Example 2 Applying the Leading Coefficient Test A polynomial function is written in standard form if its terms are written in descending order of exponents from left to right. Before applying the Leading Coefficient Test to a polynomial function, it is a good idea to check that the polynomial function is written in standard form. Exploration For each of the graphs in Example 2, count the number of zeros of the polynomial function and the number of relative minima and relative maxima. Compare these numbers with the degree of the polynomial. What do you observe? Remember that the zeros of a function of are the -values for which the function is zero. x x Describe the right-hand and left-hand behavior of the graph of each function. f x x 4 5x 2 4 f x x3 4x f x x 5 x b. a. c. Solution a. Because the degree is odd and the leading coefficient is negative, the graph rises to the left and falls to the right, as shown in Figure 2.16. b. Because the degree is even and the leading coefficient is positive, the graph rises to the left and right, as shown in Figure 2.17. c. Because the degree is odd and the leading coefficient is positive, the graph falls to the left and rises to the right, as shown in Figure 2.18. f(x) = − x3 + 4x y 3 2 1 −3 −1 1 3 x f(x) = x4 − 5x2 + 4 y 6 4 −4 −2 x 4 f(x) = x5 − x y x 2 2 1 −1 −2 FIGURE 2.16 FIGURE 2.17 FIGURE 2.18 Now try Exercise 15. In Example 2, note that the Leading Coefficient Test tells you only whether the graph eventually rises or falls to the right or left. Other characteristics of the graph, such as intercepts and minimum and maximum points, must be determined by other tests. Zeros of Polynomial Functions It can be shown that for a polynomial function statements are true. f of degree n, the following 1. The function has, at most, f n real zeros. (You will study this result in detail in the discussion of the Fundamental Theorem of Algebra in
Section 2.5.) f 2. The graph of has, at most, turning points. (Turning points, also called relative minima or relative maxima, are points at which the graph changes from increasing to decreasing or vice versa.) n 1 Finding the zeros of polynomial functions is one of the most important problems in algebra. There is a strong interplay between graphical and algebraic approaches to this problem. Sometimes you can use information about the graph of a function to help find its zeros, and in other cases you can use information about the zeros of a function to help sketch its graph. Finding zeros of polynomial functions is closely related to factoring and finding -intercepts. x 333202_0202.qxd 12/7/05 9:11 AM Page 143 Section 2.2 Polynomial Functions of Higher Degree 143 Real Zeros of Polynomial Functions If ments are equivalent. is a polynomial function and a f is a real number, the following state- 1. 2. 3. 4. x a x a x a a, 0 is a zero of the function f. is a solution of the polynomial equation f x 0. is a factor of the polynomial f x. is an -intercept of the graph of f. x Example 3 Finding the Zeros of a Polynomial Function Find all real zeros of f (x) 2x4 2x 2. Then determine the number of turning points of the graph of the function. Algebraic Solution To find the real zeros of the function, set x. zero and solve for f x equal to 2x4 2x2 0 2x2x2 1 0 2x2x 1x 1 0 f x equal to 0. Set Remove common monomial factor. Factor completely. x 0, x 1. So, the real zeros are Because the function is a fourth-degree polynomial, the turning points. graph of can have at most 4 1 3 x 1, and f y 2x4 2x2. 0, 0, Graphical Solution In Figure Use a graphing utility to graph 1, 0, 2.19, the graph appears to have zeros at and 1, 0. Use the zero or root feature, or the zoom and trace features, of the graphing utility to verify these zeros. So, the real zeros are From the figure, you can see that the graph has three turning points
. This is consistent with the fact that a fourth-degree polynomial can have at most three turning points. x 1. x 1, x 0, and 2 y = − 2x 4 + 2x 2 −3 3 Now try Exercise 27. −2 FIGURE 2.19 k In Example 3, note that because x 0. x The graph touches the -axis at is even, the factor x 0, zero 2x2 yields the repeated as shown in Figure 2.19. Repeated Zeros A factor x ak, k > 1, yields a repeated zero x a of multiplicity k. 1. If k is odd, the graph crosses the -axis at x x a. is even, the graph touches the -axis (but does not cross the -axis) x x 2. If k at x a. 333202_0202.qxd 12/7/05 9:12 AM Page 144 144 Chapter 2 Polynomial and Rational Functions Te c h n o l o g y Example 4 uses an algebraic approach to describe the graph of the function. A graphing utility is a complement to this approach. Remember that an important aspect of using a graphing utility is to find a viewing window that shows all significant features of the graph. For instance, the viewing window in part (a) illustrates all of the significant features of the function in Example 4. a. 3 − 4 b. −2 − 3 0.5 −0.5 5 2 If you are unsure of the shape of a portion of the graph of a polynomial function, plot some additional points, such as the 0.5, 0.3125 point in Figure 2.21. as shown To graph polynomial functions, you can use the fact that a polynomial function can change signs only at its zeros. Between two consecutive zeros, a polynomial must be entirely positive or entirely negative. This means that when the real zeros of a polynomial function are put in order, they divide the real number line into intervals in which the function has no sign changes. These resulting intervals are test intervals in which a representative -value in the interval is chosen to determine if the value of the polynomial function is positive (the graph lies above the -axis) or negative (the graph lies below the -axis). x x x Example 4 Sketching the Graph of a Polynomial Function f x 3x 4 4x 3. Sketch the graph of Solution 1
. Apply the Leading Coefficient Test. Because the leading coefficient is positive and the degree is even, you know that the graph eventually rises to the left and to the right (see Figure 2.20). 2. Find the Zeros of the Polynomial. By factoring you can see that the zeros of are 0, 0 f x x 33x 4, of odd multiplicity). So, the -intercepts occur at points to your graph, as shown in Figure 2.20. x f f x 3x 4 4x 3 x 0 and 3, 0. 4 and x 4 3 as (both Add these 3. Plot a Few Additional Points. Use the zeros of the polynomial to find the -value and test intervals. In each test interval, choose a representative evaluate the polynomial function, as shown in the table. x Test interval Representative Value of f Sign x-value, 0 1 f 1 7 Positive Point on graph 1, 7 0, 4 3 3, 4 1 1.5 f 1 1 Negative 1, 1 f 1.5 1.6875 Positive 1.5, 1.6875 4. Draw the Graph. Draw a continuous curve through the points, as shown in Figure 2.21. Because both zeros are of odd multiplicity, you know that the graph should cross the -axis at x 0 and x x 4 3. y 7 6 5 4 3 2 Up to left Up to right f(x) = 3x4 − 4x3 y 7 6 5 4 3 ) ), 04 3 (0, 0) −4 −3 −2 −1 −1 1 2 3 4 FIGURE 2.20 x −4 −3 −2 −1 −1 FIGURE 2.21 Now try Exercise 67. x 2 3 4 333202_0202.qxd 12/7/05 9:12 AM Page 145 Section 2.2 Polynomial Functions of Higher Degree 145 Example 5 Sketching the Graph of a Polynomial Function Sketch the graph of f x 2x 3 6x2 9 2x. Solution 1. Apply the Leading Coefficient Test. Because the leading coefficient is negative and the degree is odd, you know that the graph eventually rises to the left and falls to the right (see Figure 2.22). 2. Find the Zeros of the Polynomial. By factoring f x 2x3 6x2 9 2 x 1 1 2x4x2 12x 9 2
x2x 32 x 0 you can see that the zeros of x multiplicity). So, the -intercepts occur at to your graph, as shown in Figure 2.22. are f (odd multiplicity) and 2, 0. 3 0, 0 x 3 (even 2 Add these points and 3. Plot a Few Additional Points. Use the zeros of the polynomial to find the test intervals. In each test interval, choose a representative -value and evaluate the polynomial function, as shown in the table. x f x is positive to the Observe in Example 5 that the sign of left of and negative to the right x 0. of the zero Similarly, the f x sign of is negative to the left and to the right of the zero x 3 2. This suggests that if the zero of a polynomial function is of odd multiplicity, then the sign changes from one side of of the zero to the other side. If the zero is of even multiplicity, then the sign of change from one side of the zero to the other side. does not f x f x Test interval Representative Value of f Sign, 0 0, 3 2 2, 3 x-value 0.5 0.5 2 Point on graph 0.5, 4 f 0.5 4 Positive f 0.5 1 Negative 0.5, 1 f 2 1 Negative 2, 1 4. Draw the Graph. Draw a continuous curve through the points, as shown in Figure 2.23. As indicated by the multiplicities of the zeros, the graph crosses the -axis at but does not cross the -axis at 0, 0 x x 2, 0. 3 y y 6 5 4 3 2 Up to left Down to right − 0, 0) ) 2(, 03 1 2 3 4 −4 −3 −2 −1 −1 −2 FIGURE 2.22 x −4 −3 1 −2 −1 −1 −2 FIGURE 2.23 Now try Exercise 69. x 3 4 333202_0202.qxd 12/7/05 9:12 AM Page 146 146 Chapter 2 Polynomial and Rational Functions The Intermediate Value Theorem The next theorem, called the Intermediate Value Theorem, illustrates the existence of real zeros of polynomial functions. This theorem implies that if a, f a are two points on the graph of a polynomial function such f a that there must be a number between and then for any number between a
f b and (See Figure 2.24.) and f a f b, f c d. b, f b such that b d c y f b( ) FIGURE 2.24 a c b x Intermediate Value Theorem a Let and be real numbers such that such that between b f a f b, and f b. f a then, in the interval a < b. f If a, b, is a polynomial function f takes on every value The Intermediate Value Theorem helps you locate the real zeros of a at which polynomial function in the following way. If you can find a value at which it is negaa polynomial function is positive, and another value tive, you can conclude that the function has at least one real zero between these is negative two values. For example, the function given by when it follows from the Intermediate Value Theorem that must have a real zero somewhere between 2 f x x 3 x 2 1 Therefore, and positive when f as shown in Figure 2.25. x 1. x 2 x a x b 1, and ( 1) = 1 x 1 2 f has a zero between − − 2 and 1. − − f( 2) = 3 − ( 1, 1) −2 −1 −2 −3 − − ( 2, 3) FIGURE 2.25 By continuing this line of reasoning, you can approximate any real zeros of a polynomial function to any desired accuracy. This concept is further demonstrated in Example 6. 333202_0202.qxd 12/7/05 9:12 AM Page 147 Section 2.2 Polynomial Functions of Higher Degree 147 Example 6 Approximating a Zero of a Polynomial Function Use the Intermediate Value Theorem to approximate the real zero of f x x 3 x 2 1. Solution Begin by computing a few function values, as follows. x 2 1 0 1 f x 11 1 1 1 f 1 is negative and f 0 Because Value Theorem to conclude that the function has a zero between pinpoint this zero more closely, divide the interval evaluate the function at each point. When you do this, you will find that is positive, you can apply the Intermediate and 0. To into tenths and 1, 0 1 f 0.8 0.152 and f 0.7 0.167. 0.7, and f So, must have a zero between more accurate approximation, compute function values between f 0.7 process, you can approximate this zero to
any desired accuracy. as shown in Figure 2.26. For a and and apply the Intermediate Value Theorem again. By continuing this f 0.8 0.8 f x ( ) = 3 x 2− x + 1 y 2 (0, 1) (1, 1) x 2 1 f has a zero between 0.8− − and 0.7. −1 −1 − − ( 1, 1) FIGURE 2.26 Now try Exercise 85. Te c h n o l o g y You can use the table feature of a graphing utility to approximate the zeros of a polynomial function. For instance, for the function given by fx 2x3 3x2 3, as 20 ≤ x ≤ 20 and f1 f0 create a table that shows the function values for shown in the first table at the right. Scroll through the table looking for consecutive function values that differ in sign. From the table, you can see that differ in sign. So, you can conclude from the Intermediate Value Theorem that the function has a zero between 0 and 1. You can adjust your table to show function values for using increments of 0.1, as shown in the second table at the right. By scrolling through the table you can see that and and 0.9. If you repeat this process several times, you should obtain x 0.806 as the zero of the function. Use the zero or root feature of a graphing utility to confirm this result. f0.8 differ in sign. So, the function has a zero between 0.8 0 ≤ x ≤ 1 f0.9 333202_0202.qxd 12/7/05 2:50 PM Page 148 148 Chapter 2 Polynomial and Rational Functions 2.2 Exercises VOCABULARY CHECK: Fill in the blanks. 1. The graphs of all polynomial functions are ________, which means that the graphs have no breaks, holes, or gaps. 2. The ________ ________ ________ is used to determine the left-hand and right-hand behavior of the graph of a polynomial function. 3. A polynomial function of degree has at most ________ real zeros and at most ________ turning points. n 4. If x a is a zero of a polynomial function f, then the following three statements are true. (a) x a is a ________ of the polynomial equation f x 0. (b
) ________ is a factor of the polynomial f x. (c) a, 0 is an ________ of the graph f. 5. If a real zero of a polynomial function is of even multiplicity, then the graph of ________ the -axis at x f x a, and if it is of odd multiplicity then the graph of ________ the -axis at x f x a. 6. A polynomial function is written in ________ form if its terms are written in descending order of exponents from left to right. 7. The ________ ________ Theorem states that if a, b, interval f b. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. takes on every value between f f f a is a polynomial function such that and f a f b, then in the In Exercises 1– 8, match the polynomial function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), (f), (g), and (h).] (a) y (b) y 8 −8 x 8 −8 −4 4 8 x (c) (e) −4 −8 y 8 4 x x −8 −4 4 8 −4 −8 y 8 −8 −4 4 8 −4 −8 (d) −4 (f) y 6 4 2 −2 y 4 (g) y (h) y 4 −2 2 x 6 −4 x 2 −2 −4 −2 −4 1. 3. 5. 7. f x 2x 3 f x 2x 2 5x f x 1 4x 4 3x 2 f x x 4 2x 3 2. 4. 6. 8. f x x 2 4x f x 2x3 3x 1 f x 1 3x3 x 2 4 3 f x 1 5x 5 2x 3 9 5x In Exercises 9–12, sketch the graph of transformation. y x n and each x 2 4 9. 10. 11. f x x 23 f x 1 2x 3 f x x 15 f x 1 1 2x 5 y x 3 (a) (c) y x 5 (a) (c) y x 4 (a) f x x 34 f x 4 x 4 fx 2x4 1 (c) (e)
(b) (d) f x x 3 2 f x x 23 2 (b) (d 15 (bd) 2 (f) fx 1 x 14 2 x4 2 −4 −2 2 4 x −4 333202_0202.qxd 12/7/05 9:12 AM Page 149 Section 2.2 Polynomial Functions of Higher Degree 149 12. y x 6 (a) 8x 6 f x 1 f x x 6 4 fx 1 4 x6 2 (c) (e) (b) (d) (f) f x x 26 4 f x 1 4x 6 1 fx 2x6 1 In Exercises 13–22, describe the right-hand and left-hand behavior of the graph of the polynomial function. 13. 15. 17. 18. 19. 20. 21. 22. 14. 16. f x 2x 2 3x 1 hx 1 x 6 2x 3x 2 3x 3 5x f x 1 gx 5 7 f x 2.1x 5 4x 3 2 f x 2x 5 5x 7.5 f x 6 2x 4x 2 5x 3 f x 3x 4 2x 5 4 ht 2 3 fs 7 8 t 2 5t 3 s3 5s 2 7s 1 Graphical Analysis In Exercises 23–26, use a graphing f utility to graph the functions and in the same viewing window. Zoom out sufficiently far to show that the right-hand and left-hand behaviors of appear identical. and g g f gx 3x 3 23. 24. 25. 26. x3 3x 2, f x 3x 3 9x 1, fx 1 3 f x x 4 4x 3 16x, fx 3x 4 6x 2, gx 3x 4 gx 1 3x 3 gx x 4 In Exercises 27– 42, (a) find all the real zeros of the polynomial function, (b) determine the multiplicity of each zero and the number of turning points of the graph of the function, and (c) use a graphing utility to graph the function and verify your answers. Graphical Analysis In Exercises 43–46, (a) use a graphing utility to graph the function, (b) use the graph to approximate any -intercepts of the graph, (c) set and solve the resulting equation,
and (d) compare the results of part (c) with any -intercepts of the graph. y 0 x x 43. 44. 45. 46. y 4x3 20x 2 25x y 4x 3 4x 2 8x 8 y x 5 5x 3 4x 4x 3x 2 9 y 1 In Exercises 47–56, find a polynomial function that has the given zeros. (There are many correct answers.) 47. 49. 51. 53. 55. 0, 10 2, 6 0, 2, 3 4, 3, 3, 0 1 3, 1 3 48. 50. 52. 54. 56. 0, 3 4, 5 0, 2, 5 2, 1, 0, 1, 2 2, 4 5, 4 5 In Exercises 57–66, find a polynomial of degree the given zero(s). (There are many correct answers.) n that has Zero(s) x 2 x 8, 4 x 3, 0, 1 x 2, 4, 7 x 0, 3, 3 x 9 x 5, 1, 2 x 4, 1, 3, 6 x 0, 4 x 3, 1, 5, 6 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. Degree 27. 29. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 28. 30. 3 2 3 x 2 2x 3 fx x 2 25 ht t 2 6t 2x 2 5 fx 3x3 12x2 3x gx 5xx 2 2x 1 f t t 3 4t 2 4t fx x 4 x 3 20x 2 gt t 5 6t 3 9t fx x 5 x 3 6x fx 5x 4 15x 2 10 fx 2x 4 2x 2 40 gx x3 3x 2 4x 12 fx x 3 4x 2 25x 100 fx 49 x 2 fx x 2 10x 25 In Exercises 67– 80, sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points. 73. 69. 71. 67. 70. f x x 3 9x f t 1 t
2 2t 15 4 gx x 2 10x 16 f x x 3 3x 2 f x 3x3 15x2 18x f x 4x 3 4x2 15x f x 5x2 x3 f x x 2x 4 gt 1 t 22t 22 79. 4 x 12x 33 80. gx 1 10 77. 75. 74. 68. gx x 4 4x2 72. f x 1 x 3 76. 78. f x 48x2 3x4 hx 1 3x 3x 42 333202_0202.qxd 12/7/05 9:12 AM Page 150 150 Chapter 2 Polynomial and Rational Functions In Exercises 81–84, use a graphing utility to graph the function. Use the zero or root feature to approximate the real zeros of the function. Then determine the multiplicity of each zero. 81. 82. 83. 84. f x x 3 4x f x 1 gx 1 5 hx 1 5 4x 4 2x 2 x 12x 32x 9 x 223x 52 In Exercises 85– 88, use the Intermediate Value Theorem and the table feature of a graphing utility to find intervals one unit in length in which the polynomial function is guaranteed to have a zero. Adjust the table to approximate the zeros of the function. Use the zero or root feature of a graphing utility to verify your results. 85. 86. 87. 88. f x x 3 3x 2 3 f x 0.11x 3 2.07x 2 9.81x 6.88 gx 3x 4 4x 3 3 hx x 4 10x 2 3 90. Maximum Volume An open box with locking tabs is to be made from a square piece of material 24 inches on a side. This is to be done by cutting equal squares from the corners and folding along the dashed lines shown in the figure. x x x 24 ina) Verify that the volume of the box is given by the function Vx 8x6 x12 x. (b) Determine the domain of the function V. (c) Sketch a graph of the function and estimate the value of x for which Vx is maximum. 89. Numerical and Graphical Analysis An open box is to be made from a square piece of material, 36 inches on a side, by cutting equal squares with sides of length from the corners and turning up the sides (see figure). x
91. Construction A roofing contractor is fabricating gutters from 12-inch aluminum sheeting. The contractor plans to use an aluminum siding folding press to create the gutter by creasing equal lengths for the sidewalls (see figure). x x 12 − 2x x x 36 2− x x (a) Let x (a) Verify that the volume of the box is given by the function Vx x36 2x2. (b) Determine the domain of the function. (c) Use a graphing utility to create a table that shows the box height and the corresponding volumes Use the table to estimate the dimensions that will produce a maximum volume. V. x (d) Use a graphing utility to graph V Vx estimate the value of Compare your result with that of part (c). for which x and use the graph to is maximum. represent the height of the sidewall of the gutter. that represents the cross-sectional A Write a function area of the gutter. (b) The length of the aluminum sheeting is 16 feet. Write that represents the volume of one run of V a function gutter in terms of x. (c) Determine the domain of the function in part (b). (d) Use a graphing utility to create a table that shows the V. sidewall height Use the table to estimate the dimensions that will produce a maximum volume. and the corresponding volumes x (e) Use a graphing utility to graph Use the graph to is a maximum. estimate the value of Compare your result with that of part (d). for which V. Vx x (f) Would the value of change if the aluminum sheeting x were of different lengths? Explain. 333202_0202.qxd 12/7/05 9:12 AM Page 151 Section 2.2 Polynomial Functions of Higher Degree 151 92. Construction An industrial propane tank is formed by adjoining two hemispheres to the ends of a right circular cylinder. The length of the cylindrical portion of the tank is four times the radius of the hemispherical components (see figure). 4r r 96. Use the graphs of the models in Exercises 93 and 94 to write a short paragraph about the relationship between the median prices of homes in the two regions. Model It 97. Tree Growth The growth of a red oak tree is approx- imated by the function G 0.003t 3 0.137t 2 0.458t 0.839
(a) Write a function that represents the total volume V of the tank in terms of r. (b) Find the domain of the function. (c) Use a graphing utility to graph the function. (d) The total volume of the tank is to be 120 cubic feet. Use the graph from part (c) to estimate the radius and length of the cylindrical portion of the tank. Data Analysis: Home Prices In Exercise 93–96, use the table, which shows the median prices (in thousands of dollars) of new privately owned U.S. homes in the Midwest for the years 1997 through 2003.The y1 data can be approximated by the following models. and in the South y2 0.139t3 4.42t2 51.1t 39 0.056t3 1.73t2 23.8t 29 y1 y2 In the models, t ding to 1997. Department of Housing and Urban Development) represents the year, with correspon(Source: U.S. Census Bureau; U.S. t 7 Year, t 7 8 9 10 11 12 13 y1 150 158 164 170 173 178 184 y2 130 136 146 148 155 163 168 93. Use a graphing utility to plot the data and graph the model in the same viewing window. How closely does the y1 for model represent the data? 94. Use a graphing utility to plot the data and graph the model in the same viewing window. How closely does the y2 for model represent the data? 95. Use the models to predict the median prices of a new privately owned home in both regions in 2008. Do your answers seem reasonable? Explain. G is the height of the tree (in feet) and where 2 ≤ t ≤ 34 (a) Use a graphing utility to graph the function. (Hint: 10 ≤ x ≤ 45 is its age (in years). Use a viewing window in which and 5 ≤ y ≤ 60.) t (b) Estimate the age of the tree when it is growing most rapidly. This point is called the point of diminishing returns because the increase in size will be less with each additional year. (c) Using calculus, the point of diminishing returns can also be found by finding the vertex of the parabola given by y 0.009t2 0.274t 0.458. Find the vertex of this parabola. (d) Compare your results from parts (b) and (c). 98
. Revenue The total revenue R (in millions of dollars) for a company is related to its advertising expense by the function R 1 100,000 x3 600x 2, 0 ≤ x ≤ 400 x is the amount spent on advertising (in tens of thouwhere sands of dollars). Use the graph of this function, shown in the figure, to estimate the point on the graph at which the function is increasing most rapidly. This point is called the point of diminishing returns because any expense above this amount will yield less return per dollar invested in advertising ( 350 300 250 200 150 100 50 100 200 300 400 Advertising expense (in tens of thousands of dollars) x 333202_0202.qxd 12/7/05 9:12 AM Page 152 152 Chapter 2 Polynomial and Rational Functions Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 99–101, determine whether 99. A fifth-degree polynomial can have five turning points in its graph. 100. It is possible for a sixth-degree polynomial to have only one solution. 101. The graph of the function given by fx 2 x x2 x3 x 4 x5 x6 x7 rises to the left and falls to the right. 102. Graphical Analysis For each graph, describe a polynomial function that could represent the graph. (Indicate the degree of the function and the sign of its leading coefficient.) (a) y (b) y (c) y (d) y x x x x 103. Graphical Reasoning Sketch a graph of the function f x x 4. Explain how the graph of each differs (if it does) from the graph of each is odd, even, or neither. given by g function f. function Determine whether g (a) (b) (c) (d) (e) (f ) (g) (h) gx f x 2 gx f x 2 gx f x gx fx gx f 1 2x 2 f x gx 1 gx f x34 gx f f x 104. Exploration Explore the transformations of the form gx ax h5 k. (a) Use a graphing utility to graph the functions given by y1 1 x 25 1 3 and y2 3 x 25 3. 5 Determine whether the graphs are increasing or decreasing. Explain. (b) Will the graph of always be increasing or decreask? ing? If
so, is this behavior determined by Explain. a, h, or g (c) Use a graphing utility to graph the function given by Hx x 5 3x 3 2x 1. Use the graph and the result of part (b) to determine whether form can be written Hx ax h5 k. Explain. the in H Skills Review In Exercises 105–108, factor the expression completely. 105. 107. 5x2 7x 24 4x 4 7x3 15x2 106. 108. 6x3 61x2 10x y3 216 In Exercises 109 –112, solve the equation by factoring. 109. 110. 111. 112. 2x2 x 28 0 3x2 22x 16 0 12x2 11x 5 0 x2 24x 144 0 In Exercises 113–116, solve the equation by completing the square. 113. 115. x2 2x 21 0 2x2 5x 20 0 114. 116. x2 8x 2 0 3x2 4x 9 0 In Exercises 117–122, describe the transformation from a Then sketch its graph. common function that occurs in f x. 117. 118. 119. 120. 121. 122. f x x 42 f x 3 x2 2x 9 f x 10 1 3 x 3 333202_0203.qxd 12/7/05 9:23 AM Page 153 2.3 Polynomial and Synthetic Division Section 2.3 Polynomial and Synthetic Division 153 What you should learn • Use long division to divide polynomials by other polynomials. • Use synthetic division to divide polynomials by binomials of x k the form. • Use the Remainder Theorem and the Factor Theorem. Why you should learn it Synthetic division can help you evaluate polynomial functions. For instance, in Exercise 73 on page 160, you will use synthetic division to determine the number of U.S. military personnel in 2008. © Kevin Fleming/Corbis Long Division of Polynomials In this section, you will study two procedures for dividing polynomials. These procedures are especially valuable in factoring and finding the zeros of polynomial functions. To begin, suppose you are given the graph of fx 6x3 19x 2 16x 4. f, f you know that Notice that a zero of occurs at is a zero of exists a second-degree polynomial f
x x 2 qx. x 2, x 2 qx as shown in Figure 2.27. Because x 2 This means that there fx. is a factor of such that To find qx, you can use long division, as illustrated in Example 1. Example 1 Long Division of Polynomials 6x3 19x 2 16x 4 Divide mial completely. Solution by x 2, and use the result to factor the polyno- 6x2. Think Think Think 6x3 x 7x2 x 2x x 2. 7x. 6x2 7x 2 x 2 ) 6x3 19x2 16x 4 6x3 12x2 7x2 16x 7x2 14x 2x 4 2x 4 0 Multiply: 6x2x 2. Subtract. Multiply: 7xx 2. Subtract. Multiply: 2x 2. Subtract. y 1 ( 1 2 ), 0 (, 02 From this division, you can conclude that 6x3 19x 2 16x 4 x 26x2 7x 2 and by factoring the quadratic 6x2 7x 2, you have 6x3 19x 2 16x 4 x 22x 13x 2. f(x) = 6x3 − 19x2 + 16x − 4 Note that this factorization agrees with the graph shown in Figure 2.27 in that the x 2, x 1 2, three -intercepts occur at x 2 3. and x FIGURE 2.27 Now try Exercise 5. 333202_0203.qxd 12/7/05 9:23 AM Page 154 154 Chapter 2 Polynomial and Rational Functions x 2 In Example 1, 6x3 19x 2 16x 4, is a factor of the polynomial and the long division process produces a remainder of zero. Often, long division will produce a nonzero remainder. For instance, if you divide by x 1, you obtain the following. x2 3x 5 Divisor x 2 x 1 ) x2 3x 5 x2 x Quotient Dividend 2x 5 2x 2 3 Remainder In fractional form, you can write this result as follows. Remainder Dividend Quotient x 2 3x 5 x 1 x 2 3 x 1 Divisor Divisor This implies that x 2 3x 5 x 1(x 2 3 Multiply each side by x
1. which illustrates the following theorem, called the Division Algorithm. and f x dx The Division Algorithm If less than or equal to the degree of and rx f x dxqx rx such that are polynomials such that dx 0, and the degree of there exist unique polynomials dx is qx f x, Dividend Quotient Divisor Remainder where remainder rx 0 rx or the degree of dx is zero, rx divides evenly into f x. is less than the degree of dx. If the The Division Algorithm can also be written as f x dx qx rx dx. In the Division Algorithm, the rational expression the degree of hand, the rational expression than the degree of dx. rxdx f x is greater than or equal to the degree of f xdx is proper because the degree of is improper because On the other is less dx. rx 333202_0203.qxd 12/7/05 9:23 AM Page 155 Section 2.3 Polynomial and Synthetic Division 155 Before you apply the Division Algorithm, follow these steps. 1. Write the dividend and divisor in descending powers of the variable. 2. Insert placeholders with zero coefficients for missing powers of the variable. Example 2 Long Division of Polynomials Divide x3 1 by x 1. Solution Because there is no -term or -term in the dividend, you need to line up the subtraction by using zero coefficients (or leaving spaces) for the missing terms. x2 x x2 x 1 x 1 ) x 3 0x2 0x 1 x 3 x2 x2 0x x2 x x 1 x 1 0 x3 1, So, divides evenly into and you can write x 1 x3 1 x 1 x2 x 1, x 1. Now try Exercise 13. You can check the result of Example 2 by multiplying. x 1x2 x 1 x3 x2 x x2 x 1 x3 1 Example 3 Long Division of Polynomials Divide 2x4 4x3 5x 2 3x 2 by x 2 2x 3. Solution 1 x2 2x 3 ) 2x 4 4x 3 5x2 3x 2 2x2 2x 4 4x 3 6x2 x2 3x 2 x2 2x 3 x 1 Note that the first subtraction eliminated two terms from the dividend. When this happens, the quotient skips a term. You
can write the result as 2x4 4x3 5x 2 3x 2 x 2 2x 3 2x 2 1 x 1 x 2 2x 3. Now try Exercise 15. 333202_0203.qxd 12/7/05 9:23 AM Page 156 156 Chapter 2 Polynomial and Rational Functions Synthetic Division There is a nice shortcut for long division of polynomials when dividing by This shortcut is called synthetic division. The pattern divisors of the form for synthetic division of a cubic polynomial is summarized as follows. (The pattern for higher-degree polynomials is similar.) x k. Synthetic Division (for a Cubic Polynomial) To divide ax3 bx2 cx d x k, by use the following pattern. k a b ka c d Coefficients of dividend a r Remainder Coefficients of quotient Vertical pattern: Add terms. Diagonal pattern: Multiply by k. Synthetic division works only for divisors of the form x k x k. [Remember ] You cannot use synthetic division to divide a polynomial x k. that by a quadratic such as x 2 3. Example 4 Using Synthetic Division Use synthetic division to divide x 4 10x 2 2x 4 by x 3. Solution You should set up the array as follows. Note that a zero is included for the missing x3 -term in the dividend. 3 1 0 10 2 4 Then, use the synthetic division pattern by adding terms in columns and multiplying the results by 3. Divisor: x 3 Dividend: x 4 10x2 2x 4 3 1 1 0 3 3 10 9 1 2 3 1 4 3 1 Quotient: x3 3x2 x 1 Remainder: 1 So, you have x4 10x2 2x 4 x 3 x3 3x2 x 1 1 x 3. Now try Exercise 19. 333202_0203.qxd 12/7/05 9:23 AM Page 157 Section 2.3 Polynomial and Synthetic Division 157 The Remainder and Factor Theorems The remainder obtained in the synthetic division process has an important interpretation, as described in the Remainder Theorem. The Remainder Theorem If a polynomial r f k. f x is divided by x k, the remainder is For a proof of the Remainder Theorem, see Proofs in Mathematics on page 213. The Remainder Theorem tells you that synthetic division
can be used to f x as illustrated in evaluate a polynomial function. That is, to evaluate a polynomial function x k, when Example 5. The remainder will be x k. divide f k, f x by Example 5 Using the Remainder Theorem Use the Remainder Theorem to evaluate the following function at x 2. f x 3x3 8x2 5x 7 Solution Using synthetic division, you obtain the following. 2 3 3 8 6 2 Because the remainder is f 2 9. This means that substituting x 2 2, r f k you can conclude that is a point on the graph of You can check this by f. in the original function. Check f 2 323 822 52 7 38 84 10 7 9 Now try Exercise 45. Another important theorem is the Factor Theorem, stated below. This theoas a factor rem states that you can test to see whether a polynomial has by evaluating the polynomial at If the result is 0, is a factor. x k x k x k. The Factor Theorem A polynomial f x has a factor x k if and only if f k 0. For a proof of the Factor Theorem, see Proofs in Mathematics on page 213. 333202_0203.qxd 12/7/05 9:23 AM Page 158 158 Chapter 2 Polynomial and Rational Functions Example 6 Factoring a Polynomial: Repeated Division Show that x 2 and x 3 are factors of fx 2x4 7x3 4x2 27x 18. Then find the remaining factors of f x. Solution Using synthetic division with the factor 18 18 27 36 4 22 7 4 2 2 x 2, you obtain the following. 2 11 18 9 0 0 remainder, so x 2 is a factor. f 2 0 and Take the result of this division and perform synthetic division again using the factor x 3. f(x) = 2x 4 + 7x3 − 4x 2 − 27x − 18 y 3 2 2 11 6 5 18 15 9 9 3 0 0 remainder, so x 3 and is a factor. f 3 0 40 30 20 10 2( 3 −, 0 ( −4 −1 (−1, 0) (−3, 0) −20 −30 −40 Because the resulting quadratic expression factors as (2, 0) 1 3 4 2x 2 5x 3 2x 3x 1 x the complete factorization of fx is fx x 2x
32x 3x 1. Note that this factorization implies that has four real zeros: x 2, x 3, x 3 2, and f x 1. FIGURE 2.28 This is confirmed by the graph of which is shown in Figure 2.28. f, Now try Exercise 57. Uses of the Remainder in Synthetic Division f x The remainder obtained in the synthetic division of provides the following information. r, by x k, 1. The remainder gives the value of at f x k. That is, r f k. r r 0, x k r 0, k, 0 2. If 3. If is a factor of f x. is an -intercept of the graph of f. x Throughout this text, the importance of developing several problem-solving strategies is emphasized. In the exercises for this section, try using more than one x k strategy to solve several of the exercises. For instance, if you find that f. (with no remainder), try sketching the graph of You divides evenly into should find that is an -intercept of the graph. k, 0 f x x 333202_0203.qxd 12/7/05 9:23 AM Page 159 2.3 Exercises Section 2.3 Polynomial and Synthetic Division 159 VOCABULARY CHECK: 1. Two forms of the Division Algorithm are shown below. Identify and label each term or function. qx rx dx f x dxqx rx f x dx In Exercises 2–5, fill in the blanks. 2. The rational expression pxqx is called ________ if the degree of the numerator is greater than or equal to that of the denominator, and is called ________ if the degree of the numerator is less than that of the denominator. 3. An alternative method to long division of polynomials is called ________ ________, in which the divisor must be of the form x k. 4. The ________ Theorem states that a polynomial f x has a factor x k if and only if f k 0. 5. The ________ Theorem states that if a polynomial f x is divided by x k, the remainder is r f k. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. Analytical Analysis to verify that y1 y2. In
Exercises 1 and 2, use long division 1. y1 x2, x 2 y2 x 2 4 2. y1 x4 3x 2 1, x2 5 y2 x 2 x 2 8 39 x2 5 Graphical Analysis In Exercises 3 and 4, (a) use a graphing utility to graph the two equations in the same viewing window, (b) use the graphs to verify that the expressions are equivalent, and (c) use long division to verify the results algebraically. 3. y1 4. y1, y2 x5 3x3 x2 1 x3 2x2 5, x2 x 1 x3 4x 4x x2 1 x 3 2x 4 x2 x 1 y2 In Exercises 5 –18, use long division to divide. 5. 6. 7. 8. 9. 10. 11. 13. 14. 15. 16. 2x2 10x 12 x 3 5x2 17x 12 x 4 4x3 7x 2 11x 5 4x 5 6x3 16x 2 17x 6 3x 2 x4 5x3 6x2 x 2 x 2 x3 4x 2 3x 12 x 3 7x 3 x 2 12. 6x3 10x2 x 8 2x2 1 x3 9 x2 1 x4 3x2 1 x2 2x 3 x5 7 x3 1 8x 5 2x 1 17. x 4 x 13 18. 2x3 4x2 15x 5 x 12 In Exercises 19 –36, use synthetic division to divide. 19. 20. 21. 22. 23. 24. 25. 26. 27. 29. 31. 33. 35. 3x3 17x2 15x 25 x 5 5x3 18x2 7x 6 x 3 4x3 9x 8x2 18 x 2 9x3 16x 18x2 32 x 2 x3 75x 250 x 10 3x3 16x2 72 x 6 5x3 6x2 8 x 4 5x3 6x 8 x 2 10x4 50x3 800 x 6 28. x5 13x4 120x 80 x 3 x3 512 x 8 3x4 x 2 180x x4 x 6 4x3 16x2 23x 15 x 1 2 30. 32. 34. 36. x 3 729 x 9 3x 4 x 2 5 3x 2
x2 x3 x 1 3x3 4x2 5 x 3 2 In Exercises 37– 44, write the function in the form f x x kqx r for the given value of and demonstrate that f k r. k, Function fx x3 x2 14x 11 fx x3 5x2 11x 8 37. 38. Value of k k 4 k 2 333202_0203.qxd 12/7/05 9:23 AM Page 160 160 Chapter 2 Polynomial and Rational Functions Function fx 15x 4 10x3 6x2 14 fx 10x3 22x2 3x 4 fx x3 3x2 2x 14 fx x 3 2x2 5x 4 fx 4x3 6x2 12x 4 fx 3x3 8x2 10x 8 39. 40. 41. 42. 43. 44. Value of In Exercises 45–48, use synthetic division to find each function value. Verify your answers using another method. 45. f x 4x3 13x 10 f 2 f 1 (b) (a) 46. gx x6 4x4 3x2 2 (c) f 1 2 (d) f 8 47. g2 g4 (a) (b) hx 3x3 5x2 10x 1 (b) h3 h 1 (a) 3 (c) (c) g3 (d) g1 h2 (d) h5 Function f x 6x3 41x2 9x 14 f x 10x3 11x2 72x 45 f x 2x3 x2 10x 5 f x x3 3x2 48x 144 61. 62. 63. 64. Factors 2x 1, 2x 5, 2x 1, x 43, 3x 2 5x 3 x5 x 3 Graphical Analysis In Exercises 65–68, (a) use the zero or root feature of a graphing utility to approximate the zeros of the function accurate to three decimal places, (b) determine one of the exact zeros, and (c) use synthetic division to verify your result from part (b), and then factor the polynomial completely. 65. 66. 67. 68. f x x3 2x2 5x 10 gx x3 4x2 2x 8 ht t3 2t 2 7t 2 f s s3 12s2 40s
24 48. f x 0.4x4 1.6x3 0.7x2 2 (a) f 1 (b) f 2 (c) f 5 (d) f 10 In Exercises 49–56, use synthetic division to show that is a solution of the third-degree polynomial equation, and use the result to factor the polynomial completely. List all real solutions of the equation. x In Exercises 69–72, simplify the rational expression by using long division or synthetic division. 69. 71. 4x3 8x2 x 3 2x 3 x4 6x3 11x 2 6x x2 3x 2 70. 72. x3 x2 64x 64 x 8 x4 9x3 5x 2 36x 4 x2 4 Polynomial Equation 49. 50. 51. 52. 53. 54. 55. 56. x3 7x 6 0 x3 28x 48 0 2x3 15x 2 27x 10 0 48x3 80x 2 41x 6 0 x3 2x 2 3x 6 0 x3 2x 2 2x 4 0 x3 3x 2 2 0 x3 x 2 13x 3 0 Value of Model It 73. Data Analysis: Military Personnel The numbers M (in thousands) of United States military personnel on active duty for the years 1993 through 2003 are shown t 3 in the table, where corresponding to 1993. (Source: U.S. Department of Defense) represents the year, with t f, In Exercises 57– 64, (a) verify the given factors of the funcf, (c) use your results (b) find the remaining factors of tion f, to write the complete factorization of (d) list all real zeros of and (e) confirm your results by using a graphing utility to graph the function. f, 57. 58. 59. 60. Function f x 2x3 x2 5x 2 f x 3x3 2x2 19x 6 f x x 4 4x3 15x2 58x 40 f x 8x 4 14x3 71x2 10x 24 Factors x 2, x 3, x 5, x 1 x 2 x 4 x 2, x 4 Year, t Military personnel, M 3 4 5 6 7 8 9 10 11 12 13 1705 1611 1518 1472 1439 1407 1386 1384 1385 1412 1434 333202_0
203.qxd 12/7/05 9:23 AM Page 161 Model It (co n t i n u e d ) (a) Use a graphing utility to create a scatter plot of the data. (b) Use the regression feature of the graphing utility to find a cubic model for the data. Graph the model in the same viewing window as the scatter plot. (c) Use the model to create a table of estimated values of M. Compare the model with the original data. (d) Use synthetic division to evaluate the model for the year 2008. Even though the model is relatively accurate for estimating the given data, would you use this model to predict the number of military personnel in the future? Explain. R 74. Data Analysis: Cable Television The average monthly (in dollars) for cable television in the United basic rates States for the years 1992 through 2002 are shown in the table, where represents the year, with corresponding to 1992. t 2 (Source: Kagan Research LLC) t Year, t Basic rate 10 11 12 19.08 19.39 21.62 23.07 24.41 26.48 27.81 28.92 30.37 32.87 34.71 (a) Use a graphing utility to create a scatter plot of the data. (b) Use the regression feature of the graphing utility to find a cubic model for the data. Then graph the model in the same viewing window as the scatter plot. Compare the model with the data. (c) Use synthetic division to evaluate the model for the year 2008. Synthesis True or False? statement is true or false. Justify your answer. In Exercises 75–77, determine whether the 75. If 7x 4 is a zero of f. 4 7 is a factor of some polynomial function f, then Section 2.3 Polynomial and Synthetic Division 161 76. 2x 1 is a factor of the polynomial 6x 6 x 5 92x 4 45x 3 184x2 4x 48. 77. The rational expression x3 2x2 13x 10 x2 4x 12 is improper. 78. Exploration Use the form f x x kqx r to create a cubic function that (a) passes through the point 2, 5 and rises to the right, and (b) passes through the point 3, 1 and falls to the right. (There are many correct answers.) Think About It division by assuming that n is a positive integer. In
Exercises 79 and 80, perform the 79. x3n 9x2n 27xn 27 xn 3 80. x3n 3x2n 5xn 6 x n 2 81. Writing Briefly explain what it means for a divisor to divide evenly into a dividend. 82. Writing Briefly explain how to check polynomial divi- sion, and justify your reasoning. Give an example. Exploration In Exercises 83 and 84, find the constant c such that the denominator will divide evenly into the numerator. 83. x3 4x2 3x c x 5 84. x5 2x2 x c x 2 Think About It questions about the division f x x 32x 3x 13. In Exercises 85 and 86, answer the where f x x k, 85. What is the remainder when k 3? Explain. 86. If it is necessary to find f2, function directly or to use synthetic division? Explain. is it easier to evaluate the Skills Review In Exercises 87–92, use any method to solve the quadratic equation. 87. 89. 91. 9x2 25 0 5x2 3x 14 0 2x2 6x 3 0 88. 90. 92. 16x2 21 0 8x2 22x 15 0 x2 3x 3 0 In Exercises 93– 96, find a polynomial function that has the given zeros. (There are many correct answers.) 93. 95. 0, 3, 4 3, 1 2, 1 2 94. 96. 6, 1 1, 2, 2 3, 2 3 333202_0204.qxd 12/7/05 9:30 AM Page 162 162 Chapter 2 Polynomial and Rational Functions 2.4 Complex Numbers What you should learn • Use the imaginary unit i to write complex numbers. • Add, subtract, and multiply complex numbers. • Use complex conjugates to write the quotient of two complex numbers in standard form. • Find complex solutions of quadratic equations. Why you should learn it You can use complex numbers to model and solve real-life problems in electronics. For instance, in Exercise 83 on page 168, you will learn how to use complex numbers to find the impedance of an electrical circuit. The Imaginary Unit i x 2 1 0 You have learned that some quadratic equations have no real solutions. For instance, the quadratic equation has no real solution because there
is x To overcome this deficienno real number cy, mathematicians created an expanded system of numbers using the imaginary i, unit defined as i 1 that can be squared to produce Imaginary unit 1. i 2 1. where By adding real numbers to real multiples of this imaginary unit, the set of complex numbers is obtained. Each complex number can be written in the standard form For instance, the standard form of the complex num5 9 is ber 5 9 5 321 5 31 5 3i. a bi. 5 3i because In the standard form a bi, bi complex number the imaginary part of the complex number. and the number the real number a bi, a (where is called the real part of the is a real number) is called b a b Definition of a Complex Number If and are real numbers, the number is said to be written in standard form. If b 0, a bi a real number. If bi, A number of the form where the number b 0, a bi b 0, is a complex number, and it the number is is called an imaginary number. is called a pure imaginary number. a bi a The set of real numbers is a subset of the set of complex numbers, as shown in Figure 2.29. This is true because every real number can be written as a complex a a 0i. number using That is, for every real number you can write b 0. a a, © Richard Megna/Fundamental Photographs Real numbers Imaginary numbers FIGURE 2.29 Complex numbers Equality of Complex Numbers Two complex numbers equal to each other a bi and c di, written in standard form, are a bi c di Equality of two complex numbers if and only if a c and b d. 333202_0204.qxd 12/7/05 9:30 AM Page 163 Section 2.4 Complex Numbers 163 Operations with Complex Numbers To add (or subtract) two complex numbers, you add (or subtract) the real and imaginary parts of the numbers separately. Addition and Subtraction of Complex Numbers If their sum and difference are defined as follows. are two complex numbers written in standard form, a bi c di and a bi c di a c b di Sum: Difference: a bi c di a c b di The additive identity in the complex number system is zero (the same as in the real number system). Furthermore, the additive inverse of the complex number a bi is (a bi) a bi. Additive inverse So, you have a
bi a bi 0 0i 0. Example 1 Adding and Subtracting Complex Numbers a. 4 7i 1 6i 4 7i 1 6i (4 1) (7i 6i) 5 i b. (1 2i) 4 2i 1 2i 4 2i 1 4 2i 2i 3 0 3 Remove parentheses. Group like terms. Write in standard form. Remove parentheses. Group like terms. Simplify. Write in standard form. c. 3i 2 3i 2 5i 3i 2 3i 2 5i 2 2 3i 3i 5i 0 5i 5i d. 3 2i 4 i 7 i 3 2i 4 i 7 i 3 4 7 2i i i 0 0i 0 Now try Exercise 17. Note in Examples 1(b) and 1(d) that the sum of two complex numbers can be a real number. 333202_0204.qxd 12/7/05 9:30 AM Page 164 164 Chapter 2 Polynomial and Rational Functions Exploration Complete the following. i1 i i 2 1 i3 i10 i11 i12 What pattern do you see? Write a brief description of how you i would find raised to any positive integer power. Many of the properties of real numbers are valid for complex numbers as well. Here are some examples. Associative Properties of Addition and Multiplication Commutative Properties of Addition and Multiplication Distributive Property of Multiplication Over Addition Notice below how these properties are used when two complex numbers are multiplied. a bic di ac di bic di ac adi bci bdi 2 ac adi bci bd1 ac bd adi bci ac bd ad bci Distributive Property Distributive Property i2 1 Commutative Property Associative Property Rather than trying to memorize this multiplication rule, you should simply remember how the Distributive Property is used to multiply two complex numbers. The procedure described above is similar to multiplying two polynomials and combining like terms, as in the FOIL Method shown in Appendix A.3. For instance, you can use the FOIL Method to multiply the two complex numbers from Example 2(b). F L 2 i4 3i 8 6i 4i 3i2 O I Example 2 Multiplying Complex Numbers a. 42 3i 42 43i 8 12i b. 2 i4 3i 24 3i i4 3i 8 6i 4i 3i 2 8
6i 4i 31 8 3 6i 4i 11 2i (3 2i)(3 2i) 33 2i 2i3 2i c. 9 6i 6i 4i 2 9 6i 6i 41 9 4 13 d. 3 2i2 3 2i3 2i 33 2i 2i3 2i 9 6i 6i 4i 2 9 6i 6i 41 9 12i 4 5 12i Now try Exercise 27. Distributive Property Simplify. Distributive Property Distributive Property i2 1 Group like terms. Write in standard form. Distributive Property Distributive Property i2 1 Simplify. Write in standard form. Square of a binomial Distributive Property Distributive Property i2 1 Simplify. Write in standard form. 333202_0204.qxd 12/7/05 9:30 AM Page 165 Section 2.4 Complex Numbers 165 Complex Conjugates Notice in Example 2(c) that the product of two complex numbers can be a real number. This occurs with pairs of complex numbers of the form and a bi, called complex conjugates. a bi a bia bi a2 abi abi b2i 2 a2 b21 a2 b2 Example 3 Multiplying Conjugates Multiply each complex number by its complex conjugate. a. 1 i b. 4 3i Solution a. The complex conjugate of 1 i1 i 12 i 2 b. The complex conjugate of is 4 3i4 3i 42 3i2 4 3i 16 91 25 1 i is 1 i. 1 1 4 3i. 16 9i 2 2 Note that when you multiply the numerator and denominator of a quotient of complex numbers by c di c di you are actually multiplying the quotient by a form of 1. You are not changing the original expression, you are only creating an expression that is equivalent to the original expression. Now try Exercise 37. To write the quotient of d are not both zero, multiply the numerator and denominator by the complex conjugate of the denominator to obtain in standard form, where and and c a bi c di a bi c di c di a bi c di c di ac bd bc adi c 2 d 2. Standard form Example 4 Writing a Quotient of Complex Numbers in Standard Form 2 3i 4 2i 2 3i 4 2i 4 2i 4 2i Multiply numerator and denominator
by complex conjugate of denominator. 8 4i 12i 6i 2 16 4i 2 8 6 16i 16 4 2 16i 20 4 5 1 10 i Now try Exercise 49. Expand. i2 1 Simplify. Write in standard form. 333202_0204.qxd 12/7/05 9:30 AM Page 166 166 Chapter 2 Polynomial and Rational Functions Complex Solutions of Quadratic Equations When using the Quadratic Formula to solve a quadratic equation, you often obtain a result such as which you know is not a real number. By factoring out you can write this number in standard form. 3, i 1, 3 31 31 3i The number 3i is called the principal square root of 3. Principal Square Root of a Negative Number a If a is a positive number, the principal square root of the negative number is defined as a ai. The definition of principal square root uses the rule ab ab and a > 0 b < 0. a This rule b and are for is not valid if both negative. For example, 55 5151 5i5i 25i 2 5i 2 5 whereas 55 25 5. To avoid problems with square roots of negative numbers, be sure to convert complex numbers to standard form before multiplying. Example 5 Writing Complex Numbers in Standard Form a. b. c. 312 3 i12 i 36 i 2 61 6 48 27 48i 27 i 43i 33i 3 i 1 32 1 3i2 12 23i 32i 2 1 23i 31 2 23i Now try Exercise 59. Example 6 Complex Solutions of a Quadratic Equation Solve (a) x2 4 0 and (b) 3x 2 2x 5 0. Solution a. x2 4 0 x2 4 x ±2i 3x2 2x 5 0 b. x 2 ± 22 435 23 2 ± 56 6 2 ± 214i 6 1 3 ± 14 3 i Now try Exercise 65. Write original equation. Subtract 4 from each side. Extract square roots. Write original equation. Quadratic Formula Simplify. Write 56 in standard form. Write in standard form. 333202_0204.qxd 12/7/05 9:30 AM Page 167 Section 2.4 Complex Numbers 167 2.4 Exercises VOCABULARY CHECK: 1. Match the type of complex number with its definition. (a) Real Number (b) Imaginary number (c) Pure imaginary
number (i) (ii) (iii) a bi, a bi, a bi, a 0, a 0, b 0 b 0 b 0 In Exercises 2–5, fill in the blanks. 2. The imaginary unit i is defined as i ________, where i2 ________. 3. If a is a positive number, the ________ ________ root of the negative number a is defined as 4. The numbers a bi and a bi are called ________ ________, and their product is a real number a a i. a2 b2. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1– 4, find real numbers and equation is true. a b such that the 1. 3. 4. a bi 10 6i a 1 b 3i 5 8i a 6 2bi 6 5i 2. a bi 13 4i In Exercises 5–16, write the complex number in standard form. 5. 7. 9. 4 9 2 27 75 11. 8 13. 15. 6i i 2 0.09 6. 8. 10. 3 16 1 8 4 12. 45 14. 16. 4i 2 2i 0.0004 In Exercises 17–26, perform the addition or subtraction and write the result in standard form. 17. 19. 21. 22. 23. 25. 26. 18. 5 i 6 2i 8 i 4 i 20. 2 8 5 50 8 18 4 32 i 13i 14 7i 3 2i 5 5 3 1.6 3.2i 5.8 4.3i 3 i 11 24. 2 13 2i 5 6i 3 2i 6 13i 22 5 8i 10i 27. In Exercises 27–36, perform the operation and write the result in standard form. 1 i3 2i 6i5 2i 14 10 i14 10 i 6 2i2 3i 8i9 4i 31. 29. 28. 30. 32. 33. 35. 3 15 i3 15 i 4 5i2 2 3i2 2 3i2 34. 36. 2 3i2 1 2i2 1 2i2 In Exercises 37– 44, write the complex conjugate of the complex number.Then multiply the number by its complex conjugate. 37. 39. 41. 43.
6 3i 1 5 i 20 8 38. 40. 42. 44. 7 12i 3 2 i 15 1 8 In Exercises 45–54, write the quotient in standard form. 45. 47. 49. 51. 53. 5 i 2 4 5i 3 i 3 i 6 5i i 3i 4 5i 2 46. 48. 50. 52. 54. 14 2i 5 1 i 6 7i 1 2i 8 16i 2i 5i 2 3i2 In Exercises 55–58, perform the operation and write the result in standard form. 55. 57. 3 1 i 2i 2 1 i i 3 2i 3 8i 56. 58. 2i 333202_0204.qxd 12/7/05 2:51 PM Page 168 168 Chapter 2 Polynomial and Rational Functions In Exercises 59–64, write the complex number in standard form. 59. 61. 63. 6 2 102 3 57 10 60. 62. 64. 5 10 752 2 62 84. Cube each complex number. (a) 2 (b) 1 3 i (c) 1 3 i 85. Raise each complex number to the fourth power. (a) 2 (b) 2 (c) 2i 86. Write each of the powers of as 2i (d) i, i, 1, (d) i 67 or 1. i i 50 In Exercises 65–74, use the Quadratic Formula to solve the quadratic equation. Synthesis (a) i 40 (b) i 25 (c) 65. 67. 69. 71. 73. x 2 2x 2 0 4x 2 16x 17 0 4x 2 16x 15 0 3 2 x2 6x 9 0 1.4x2 2x 10 0 66. 68. 70. 72. 74. x 2 6x 10 0 9x 2 6x 37 0 16t 2 4t 3 0 7 8 x2 3 4x 5 0 4.5x2 3x 12 0 16 In Exercises 75–82, simplify the complex number and write it in standard form. 75. 77. 79. 81. 6i 3 i 2 5i 5 753 1 i 3 76. 78. 80. 82. 4i 2 2i 3 i 3 26 1 2i 3 Model It 83. Impedance The opposition to current in an electrical circuit is called its impedance. The impedance in
a parallel circuit with two pathways satisfies the equation z 1 z 1 z1 1 z 2 z1 is the impedance of pathway 2. is the impedance (in ohms) of pathway 1 and where z2 (a) The impedance of each pathway in a parallel circuit is found by adding the impedances of all compoz2. nents in the pathway. Use the table to find and z1 (b) Find the impedance z. Resistor Inductor Capacitor Symbol Impedance aΩ a bΩ bi cΩ ci 1 16 Ω 2 20 Ω 9 Ω 10 Ω True or False? statement is true or false. Justify your answer. In Exercises 87– 89, determine whether the 87. There is no complex number that is equal to its complex 88. conjugate. i6 i 44 i 150 i 74 i 109 i61 1 90. Error Analysis Describe the error. is a solution of 89. x 4 x2 14 56. 66 66 36 6 91. Proof Prove that the complex conjugate of the product is the b1i of two complex numbers product of their complex conjugates. b2i and a2 a1 92. Proof Prove that the complex conjugate of the sum of b1i is the sum of b2i and a2 a1 two complex numbers their complex conjugates. Skills Review In Exercises 93–96, perform the operation and write the result in standard form. 93. 94. 95. 4 3x 8 6x x 2 x3 3x2 6 2x 4x 2 3x 1 x 4 2 96. 2x 52 In Exercises 97–100, solve the equation and check your solution. 97. 99. 100. x 12 19 45x 6 36x 1 0 5x 3x 11 20x 15 98. 8 3x 34 101. Volume of an Oblate Spheroid a Solve for : V 4 3 102. Newton’s Law of Universal Gravitation a2b r Solve for : F m1m2 r 2 103. Mixture Problem A five-liter container contains a mixture with a concentration of 50%. How much of this mixture must be withdrawn and replaced by 100% concentrate to bring the mixture up to 60% concentration? 333202_0205.qxd 12/7/05 9:36 AM Page 169 2.5 Z
eros of Polynomial Functions Section 2.5 Zeros of Polynomial Functions 169 What you should learn • Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions. • Find rational zeros of polyno- mial functions. • Find conjugate pairs of com- plex zeros. • Find zeros of polynomials by factoring. • Use Descartes’s Rule of Signs and the Upper and Lower Bound Rules to find zeros of polynomials. Why you should learn it Finding zeros of polynomial functions is an important part of solving real-life problems. For instance, in Exercise 112 on page 182, the zeros of a polynomial function can help you analyze the attendance at women’s college basketball games. set f x, Recall that in order to find the f x zeros of a function equal to 0 and solve the resulting equation for For instance, the function in Example 1(a) has a zero at because x. x 2 x 2 0 x 2. The Fundamental Theorem of Algebra n th-degree polynomial can have at most You know that an real zeros. In the complex number system, this statement can be improved. That is, in the complex number system, every th-degree polynomial function has precisely zeros. This important result is derived from the Fundamental Theorem of Algebra, first proved by the German mathematician Carl Friedrich Gauss (1777–1855). n n n The Fundamental Theorem of Algebra n, n > 0, If is a polynomial of degree where zero in the complex number system. f x then has at least one f Using the Fundamental Theorem of Algebra and the equivalence of zeros and factors, you obtain the Linear Factorization Theorem. f x is a polynomial of degree where Linear Factorization Theorem If linear factors f x an n, x c1 c1, c2,..., cn... x cn x c2 are complex numbers. where n > 0, then has precisely f n For a proof of the Linear Factorization Theorem, see Proofs in Mathematics on page 214. Note that the Fundamental Theorem of Algebra and the Linear Factorization Theorem tell you only that the zeros or factors of a polynomial exist, not how to find them. Such theorems are called existence theorems.
Example 1 Zeros of Polynomial Functions a. The first-degree polynomial b. Counting multiplicity, the second-degree polynomial function has exactly one zero: f x x 2 x 2. f x x 2 6x 9 x 3x 3 x 3 x 3. has exactly two zeros: and (This is called a repeated zero.) c. The third-degree polynomial function f x x3 4x xx 2 4 xx 2ix 2i x 0, x 2i, d. The fourth-degree polynomial function has exactly three zeros: x 2i. and f x x4 1 x 1x 1x ix i has exactly four zeros: x 1, x 1, x i, and x i. Now try Exercise 1. 333202_0205.qxd 12/7/05 9:36 AM Page 170 170 Chapter 2 Polynomial and Rational Functions The Rational Zero Test The Rational Zero Test relates the possible rational zeros of a polynomial (having integer coefficients) to the leading coefficient and to the constant term of the polynomial Historical Note Although they were not contemporaries, Jean Le Rond d’Alembert (1717–1783) worked independently of Carl Gauss in trying to prove the Fundamental Theorem of Algebra. His efforts were such that, in France, the Fundamental Theorem of Algebra is frequently known as the Theorem of d’Alembert. The Rational Zero Test If the polynomial has integer coefficients, every rational zero of has the form f x anx n an1x n1... a 2 x 2 f a1x a0 Rational zero p q where and have no common factors other than 1, and q p p a factor of the constant term a0 q a factor of the leading coefficient an. To use the Rational Zero Test, you should first list all rational numbers whose numerators are factors of the constant term and whose denominators are factors of the leading coefficient. Possible rational zeros factors of constant term factors of leading coefficient Having formed this list of possible rational zeros, use a trial-and-error method to determine which, if any, are actual zeros of the polynomial. Note that when the leading coefficient is 1, the possible rational zeros are simply the factors of the constant term. Example 2 Rational Zero Test with Leading Coefficient of 1 Find the rational zeros of f x x3 x 1. f
(x) = x3 + x + 1 y Solution Because the leading coefficient is 1, the possible rational zeros are the factors of the constant term. By testing these possible zeros, you can see that neither works. ±1, f 1 13 1 1 3 f 1 13 1 1 1 2 3 x 1 So, you can conclude that the given polynomial has no rational zeros. Note from the graph of and 0. However, by the Rational Zero Test, you know that this real zero is not a rational number. in Figure 2.30 that does have one real zero between 1 f f −3 −2 3 2 1 −1 −2 −3 FIGURE 2.30 Now try Exercise 7. 333202_0205.qxd 12/7/05 9:36 AM Page 171 When the list of possible rational zeros is small, as in Example 2, it may be quicker to test the zeros by evaluating the function. When the list of possible rational zeros is large, as in Example 3, it may be quicker to use a different approach to test the zeros, such as using synthetic division or sketching a graph. Section 2.5 Zeros of Polynomial Functions 171 Example 3 Rational Zero Test with Leading Coefficient of 1 Find the rational zeros of f x x4 x3 x 2 3x 6. Solution Because the leading coefficient is 1, the possible rational zeros are the factors of the constant term. Possible rational zeros: ±1, ±2, ±3, ±6 By applying synthetic division successively, you can determine that x 2 1 are the only two rational zeros and 1 2 3 6 0 0 remainder, so x 1 is a zero So, factors as f x f x x 1x 2x 2 3. 0 remainder, so x 2 is a zero. Because the factor x 1 x 2 and x 2 3 produces no real zeros, you can conclude that are the only real zeros of which is verified in Figure 2.31. f, 0) (2, 0) −8 −6 −4 −2 4 6 8 x −6 −8 FIGURE 2.31 Now try Exercise 11. If the leading coefficient of a polynomial is not 1, the list of possible rational zeros can increase dramatically. In such cases, the search can be shortened in several ways: (1) a programmable calculator can be used to speed up the calculations; (2) a graph, drawn
either by hand or with a graphing utility, can give a good estimate of the locations of the zeros; (3) the Intermediate Value Theorem along with a table generated by a graphing utility can give approximations of zeros; and (4) synthetic division can be used to test the possible rational zeros. Finding the first zero is often the most difficult part. After that, the search is simplified by working with the lower-degree polynomial obtained in synthetic division, as shown in Example 3. 333202_0205.qxd 12/7/05 9:36 AM Page 172 172 Chapter 2 Polynomial and Rational Functions Remember that when you try to find the rational zeros of a polynomial function with many possible rational zeros, as in Example 4, you must use trial and error. There is no quick algebraic method to determine which of the possibilities is an actual zero; however, sketching a graph may be helpful. y 15 10 5 −5 −10 3 f x ( ) = 10 + 15 + 16 x x 2 − − x 12 FIGURE 2.32 Example 4 Using the Rational Zero Test Find the rational zeros of f x 2x3 3x 2 8x 3. Solution The leading coefficient is 2 and the constant term is 3. Possible rational zeros: Factors of 3 Factors of 2 By synthetic division, you can determine that ±1, ±3 ±1, ±2 x 1 ±1, ±3, ± 1 2, ± 3 2 is a rational zero So, factors as f x f x x 12x 2 5x 3 x 12x 1x 3 and you can conclude that the rational zeros of are f x 1, x 1 2, and x 3. Now try Exercise 17. Recall from Section 2.2 that if x a is a solution of the polynomial equation x a then f x 0. is a zero of the polynomial function f, Solution The leading coefficient is 10 Possible rational solutions: 12. and the constant term is Factors of 12 Factors of 10 ±1, ±2, ±3, ±4, ±6, ±12 ±1, ±2, ±5, ±10 With so many possibilities (32, in fact), it is worth your time to stop and sketch a graph. From Figure 2.32, it looks like three reasonable solutions would be x 6 x 2 5, is the only rational solution. So, you have x 210x2 5x 6
0. Testing these by synthetic division shows that x 1 2, x 2. and Using the Quadratic Formula for the second factor, you find that the two additional solutions are irrational numbers. x 5 265 20 1.0639 and x 5 265 20 0.5639 Now try Exercise 23. Example 5 Solving a Polynomial Equation 1 x Find all the real solutions of 10x3 15x2 16x 12 0. 333202_0205.qxd 12/7/05 9:36 AM Page 173 Section 2.5 Zeros of Polynomial Functions 173 Conjugate Pairs In Example 1(c) and (d), note that the pairs of complex zeros are conjugates. That is, they are of the form a bi. a bi and Complex Zeros Occur in Conjugate Pairs f x Let b 0, function. is a zero of the function, the conjugate a bi be a polynomial function that has real coefficients. If a bi, where is also a zero of the Be sure you see that this result is true only if the polynomial function has real f x coefficients. For instance, the result applies to the function given by x2 1 but not to the function given by gx x i. Example 6 Finding a Polynomial with Given Zeros Find a fourth-degree polynomial function with real coefficients that has and as zeros. 3i 1, 1, Solution 3i Because know that the conjugate f x Theorem, can be written as is a zero and the polynomial is stated to have real coefficients, you must also be a zero. So, from the Linear Factorization 3i f x ax 1x 1x 3ix 3i. For simplicity, let to obtain f x x2 2x 1x2 9 a 1 x4 2x3 10x 2 18x 9. Now try Exercise 37. Factoring a Polynomial The Linear Factorization Theorem shows that you can write any n linear factors. polynomial as the product of x c3 x c2... x cn f x an x c1 n th-degree However, this result includes the possibility that some of the values of are complex. The following theorem says that even if you do not want to get involved with “complex factors,” you can still write as the product of linear and/or quadratic factors. For a proof of this theorem,
see Proofs in Mathematics on page 214. f x ci Factors of a Polynomial with real coefficients can be written as Every polynomial of degree the product of linear and quadratic factors with real coefficients, where the quadratic factors have no real zeros. n > 0 333202_0205.qxd 12/7/05 9:36 AM Page 174 174 Chapter 2 Polynomial and Rational Functions A quadratic factor with no real zeros is said to be prime or irreducible over the reals. Be sure you see that this is not the same as being irreducible over the rationals. For example, the quadratic is irreducible over the reals (and therefore over the rationals). On the other hand, the quadratic x2 2 x 2 x 2 is irreducible over the rationals but reducible over the reals. x2 1 x ix i Example 7 Finding the Zeros of a Polynomial Function Find all the zeros of zero of f. f x x4 3x3 6x2 2x 60 given that 1 3i is a Algebraic Solution Because complex zeros occur in conjugate pairs, you know that 1 3i is also a zero of This means that both f. x 1 3i and x 1 3i are factors of Multiplying these two factors produces f. x 1 3ix 1 3i x 1 3ix 1 3i Graphical Solution Because complex zeros always occur in conjugate pairs, you know that is also a zero of f. Because the polynomial is a fourth-degree polynomial, you know that there are at most two other zeros of the function. Use a graphing utility to graph 1 3i y x4 3x3 6x2 2x 60 as shown in Figure 2.33. x 12 9i 2 x 2 2x 10. x 2 2x 10 into Using long division, you can divide the following. x2 x 6 x2 2x 10 ) x 4 3x3 6x2 2x 60 x 4 2x3 10x2 x3 4x2 2x x3 2x2 10x 6x2 12x 60 6x2 12x 60 0 So, you have f x x2 2x 10x2 x 6 x2 2x 10x 3x 2 and you can conclude that the zeros of x
1 3i, and x 2. x 3, f are x 1 3i, Now try Exercise 47. f to obtain y = x4 − 3x3 + 6x2 + 2x − 60 80 −80 −4 FIGURE 2.33 5 2 You can see that and 3 appear to be zeros of the graph of the function. Use the zero or root feature or the zoom and trace features of the and graphing utility to confirm that x 3 are zeros of the graph. So, you can x 1 3i, conclude that the zeros of x 1 3i, and x 2. x 2 x 3, are f In Example 7, if you were not told that is a zero of you could still 2 find all zeros of the function by using synthetic division to find the real zeros x 2x 3x2 2x 10. and 3. Then you could factor the polynomial as Finally, by using the Quadratic Formula, you could determine that the zeros are x 2, and x 1 3i. x 1 3i, x 3, f, 1 3i 333202_0205.qxd 12/7/05 9:36 AM Page 175 In Example 8, the fifth-degree polynomial function has three real zeros. In such cases, you can use the zoom and trace features or the zero or root feature of a graphing utility to approximate the real zeros. You can then use these real zeros to determine the complex zeros algebraically. Section 2.5 Zeros of Polynomial Functions 175 Example 8 shows how to find all the zeros of a polynomial function, including complex zeros. Example 8 Finding the Zeros of a Polynomial Function f x x5 x3 2x2 12x 8 Write all of its zeros. as the product of linear factors, and list Solution The possible rational zeros are the following. ±1, ±2, ±4, and ±8. Synthetic division produces 12 is a zero. 2 is a zero. f(x) = x5 + x3 + 2x2 −12x + 8 f x x5 x3 2x2 12x 8 So, you have y 10 5 (−2, 0) −4 FIGURE 2.34 x 1x 2x3 x2 4x 4. You can factor x3 x2 4x 4 as x 1x2 4, and by factoring x 2
4 as x 2 4 x 4x 4 x 2ix 2i you obtain f x x 1x 1x 2x 2ix 2i (1, 0) 2 4 x which gives the following five zeros of f. x 1, x 1, x 2, x 2i, and x 2i From the graph of only ones that appear as -intercepts. Note that shown in Figure 2.34, you can see that the real zeros are the x 1 is a repeated zero. x f Now try Exercise 63. Te c h n o l o g y You can use the table feature of a graphing utility to help you determine which of the possible rational zeros are zeros of the polynomial in Example 8. The table should be set to ask mode. Then enter each of the possible rational zeros in the table. When you do this, you will see that there are two rational zeros, right. and 1, as shown at the 2 333202_0205.qxd 12/7/05 9:36 AM Page 176 176 Chapter 2 Polynomial and Rational Functions Other Tests for Zeros of Polynomials n You know that an th-degree polynomial function can have at most real zeros. n th-degree polynomials do not have that many real zeros. For Of course, many f x x2 1 has only one real instance, zero. The following theorem, called Descartes’s Rule of Signs, sheds more light on the number of real zeros of a polynomial. has no real zeros, and f x x3 1 n Descartes’s Rule of Signs Let real coefficients and 0. a0 f (x) anxn an1xn1... a2x2 a1x a0 be a polynomial with 1. The number of positive real zeros of f is either equal to the number of variations in sign of or less than that number by an even integer. f x 2. The number of negative real zeros of variations in sign of f x is either equal to the number of or less than that number by an even integer. f A variation in sign means that two consecutive coefficients have opposite signs. When using Descartes’s Rule of Signs, a zero of multiplicity counted as zeros. For instance, the polynomial in sign, and so has either two positive or no positive real zeros. Because
k x 3 3x 2 k should be has two variations x3 3x 2 x 1x 1x 2 you can see that the two positive real zeros are x 1 of multiplicity 2. Example 9 Using Descartes’s Rule of Signs Describe the possible real zeros of f x 3x3 5x2 6x 4. Solution The original polynomial has three variations in sign. to to f x 3x3 5x2 6x 4 to The polynomial f(x) = 3x3 − 5x2 + 6x − 4 y 3 2 1 −1 −2 −3 −3 −2 −1 f x 3x3 5x2 6x 4 3x3 5x 2 6x 4 x 2 3 has no variations in sign. So, from Descartes’s Rule of Signs, the polynomial f x 3x3 5x 2 6x 4 has either three positive real zeros or one positive real zero, and has no negative real zeros. From the graph in Figure 2.35, you can x 1 ). see that the function has only one real zero (it is a positive number, near FIGURE 2.35 Now try Exercise 79. 333202_0205.qxd 12/7/05 9:36 AM Page 177 Section 2.5 Zeros of Polynomial Functions 177 Another test for zeros of a polynomial function is related to the sign pattern in the last row of the synthetic division array. This test can give you an upper or is an upper bound for the lower bound of the real zeros of A real number is a lower bound if no f real zeros of f real zeros of are less than f. if no zeros are greater than Similarly, b. b. b b Upper and Lower Bound Rules f x Let cient. Suppose is divided by x c, fx be a polynomial with real coefficients and a positive leading coeffi- using synthetic division. 1. If c > 0 and each number in the last row is either positive or zero, c is an upper bound for the real zeros of f. 2. If c < 0 and the numbers in the last row are alternately positive and negative (zero entries count as positive or negative), for the real zeros of f. c is a lower bound Example 10 Finding the Zeros of a Polynomial Function Find the real zeros of f x 6x 3 4
x 2 3x 2. Solution The possible real zeros are as follows. Factors of 2 Factors of 6 ±1, ±2 ±1, ±2, ±3, ±6 f x ±12 The original polynomial has three variations in sign. The polynomial f x 6x3 4x2 3x 2 6x3 4x2 3x 2 has no variations in sign. As a result of these two findings, you can apply Descartes’s Rule of Signs to conclude that there are three positive real zeros or one positive real zero, and no negative zeros. Trying produces the following So, that zeros between 0 and 1. By trial and error, you can determine that So, is not a zero, but because the last row has all positive entries, you know is an upper bound for the real zeros. So, you can restrict the search to is a zero. x 2 3 f x x 2 3 6x2 3. Because 6x 2 3 has no real zeros, it follows that x 2 3 is the only real zero. Now try Exercise 87. 333202_0205.qxd 12/7/05 9:36 AM Page 178 178 Chapter 2 Polynomial and Rational Functions Before concluding this section, here are two additional hints that can help you find the real zeros of a polynomial. f x 1. If the terms of have a common monomial factor, it should be factored out before applying the tests in this section. For instance, by writing f x x4 5x3 3x 2 x xx3 5x 2 3x 1 x 0 f you can see that obtained by analyzing the cubic factor. is a zero of and that the remaining zeros can be 2. If you are able to find all but two zeros of you can always use the Quadratic Formula on the remaining quadratic factor. For instance, if you succeeded in writing f x, f x x4 5x3 3x 2 x xx 1x 2 4x 1 you can apply the Quadratic Formula to remaining zeros are x 2 5 and x2 4x 1 x 2 5. to conclude that the two Example 11 Using a Polynomial Model You are designing candle-making kits. Each kit contains 25 cubic inches of candle wax and a mold for making a pyramid-shaped candle. You want the height of the candle to be 2 inches less than the length of each side of the candle
’s square base. What should the dimensions of your candle mold be? V 1 x2 B 3 Bh, where and the height is is the x 2. So, the volume of the Substituting 25 for the volume yields the following. h is the area of the base and 3 x2x 2. Solution The volume of a pyramid is height. The area of the base is V 1 pyramid is 25 1 3 75 x3 2x2 0 x3 2x2 75 x2x 2 Substitute 25 for V. Multiply each side by 3. Write in general form. Use The possible rational solutions are synthetic division to test some of the possible solutions. Note that in this case, it makes sense to test only positive -values. Using synthetic division, you can determine that is a solution. x ±15, ±25, ±75. ±5, ±3, x ±1, x 5 2 5 3 0 15 15 75 75 0 5 1 1 are imaginary and The other two solutions, which satisfy can be discarded. You can conclude that the base of the candle mold should be 5 inches by 5 inches and the height of the mold should be 5 2 3 inches. x2 3x 15 0, Now try Exercise 107. 333202_0205.qxd 12/7/05 9:36 AM Page 179 Section 2.5 Zeros of Polynomial Functions 179 2.5 Exercises VOCABULARY CHECK: Fill in the blanks. 1. The ________ ________ of ________ states that if at least one zero in the complex number system. f x is a polynomial of degree n n > 0, f then has 2. The ________ ________ ________ states that if n linear factors f x an x c1 x c2 f x... x cn is a polynomial of degree where c1, c2,..., cn n n > 0, then has precisely f are complex numbers. 3. The test that gives a list of the possible rational zeros of a polynomial function is called the ________ ________ Test. 4. If a bi is a complex zero of a polynomial with real coefficients, then so is its ________, a bi. 5. A quadratic factor that cannot be factored further as a product of linear factors containing real numbers is said to be ________ over the ________. 6. The theorem that can be
used to determine the possible numbers of positive real zeros and negative real zeros of a function is called ________ ________ of ________. 7. A real number b bound if no real zeros are greater than is a(n) ________ bound for the real zeros of b. f if no real zeros are less than b, and is a(n) ________ PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–6, find all the zeros of the function. 9. f x 2x4 17x3 35x 2 9x 45 1. 2. 3. 4. 5. 6. f x xx 62 f x x2x 3x2 1 gx) x 2x 43 f x x 5x 82 f x x 6x ix i ht t 3t 2t 3i t 3i In Exercises 7–10, use the Rational Zero Test to list f all possible rational zeros of Verify that the zeros of shown on the graph are contained in the list. f. 7. f x x3 3x 2 x 3 y 4 2 −4 −2 2 x −4 8. f x x3 4x 2 4x 16 y 18 −6 6 12 x y −8 −4 x 8 −20 −30 −40 10. f x 4x5 8x4 5x3 10x 2 x 2 y 2 −4 −2 x 4 In Exercises 11–20, find all the rational zeros of the function. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. f x x3 6x 2 11x 6 f x x3 7x 6 gx x3 4x 2 x 4 hx x3 9x 2 20x 12 ht t 3 12t 2 21t 10 px x3 9x 2 27x 27 Cx 2x3 3x 2 1 f x 3x3 19x 2 33x 9 f x 9x4 9x3 58x 2 4x 24 f x 2x4 15x3 23x 2 15x 25 333202_0205.qxd 12/7/05 9:36 AM Page 180 180 Chapter 2 Polynomial and Rational Functions In Exercises 21–24, find all real solutions of the polynomial equation. 21. 22
. 23. 24. z4 z3 2z 4 0 x 4 13x 2 12x 0 2y4 7y 3 26y 2 23y 6 0 x5 x4 3x3 5x 2 2x 0 f, In Exercises 25–28, (a) list the possible rational zeros of (b) sketch the graph of so that some of the possible zeros in part (a) can be disregarded, and then (c) determine all real zeros of f. f 25. 26. 27. 28. f x x3 x 2 4x 4 f x 3x3 20x 2 36x 16 f x 4x3 15x 2 8x 3 f x 4x3 12x 2 x 15 f, In Exercises 29–32, (a) list the possible rational zeros of so that some of the (b) use a graphing utility to graph possible zeros in part (a) can be disregarded, and then f. (c) determine all real zeros of f 29. 30. 31. 32. f x 2x4 13x 3 21x 2 2x 8 f x 4x4 17x 2 4 f x 32x3 52x 2 17x 3 f x 4x3 7x 2 11x 18 Graphical Analysis In Exercises 33–36, (a) use the zero or root feature of a graphing utility to approximate the zeros of the function accurate to three decimal places, (b) determine one of the exact zeros (use synthetic division to verify your result), and (c) factor the polynomial completely. 33. 35. 36. f x x 4 3x 2 2 hx x5 7x4 10x3 14x 2 24x gx 6x4 11x3 51x 2 99x 27 34. Pt t 4 7t 2 12 In Exercises 37– 42, find a polynomial function with real coefficients that has the given zeros. (There are many correct answers.) 37. 39. 41. 1, 5i, 5i 6, 5 2i, 5 2i 2 3, 1, 3 2 i 38. 40. 42. 4, 3i, 3i 2, 4 i, 4 i 5, 5, 1 3 i In Exercises 43– 46, write the polynomial (a) as the product of factors that are irreducible over the rationals,
(b) as the product of linear and quadratic factors that are irreducible over the reals, and (c) in completely factored form. 43. 44. f x x 4 6x 2 27 f x x 4 2x 3 3x 2 12x 18 (Hint: One factor is x 2 6. ) 45. 46. f x x 4 4x 3 5x 2 2x 6 x 2 2x 2. (Hint: One factor is ) f x x 4 3x 3 x 2 12x 20 (Hint: One factor is x 2 4. ) In Exercises 47–54, use the given zero to find all the zeros of the function. Function 47. 48. 49. 50. 51. 52. 53. 54. f x 2x 3 3x 2 50x 75 f x x3 x 2 9x 9 f x 2x 4 x 3 7x 2 4x 4 g x x 3 7x 2 x 87 g x 4x 3 23x 2 34x 10 h x 3x 3 4x 2 8x 8 f x x 4 3x 3 5x 2 21x 22 f x x 3 4x 2 14x 20 Zero 5i 3i 2i 5 2i 3 i 1 3 i 3 2 i 1 3i In Exercises 55–72, find all the zeros of the function and write the polynomial as a product of linear factors. f x x 2 x 56 gx x 2 10x 23 55. 57. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 58. 56. f x x 2 25 hx x 2 4x 1 f x x 4 81 f y y4 625 f z z 2 2z 2 h(x) x 3 3x 2 4x 2 gx x3 6x2 13x 10 f x x 3 2x 2 11x 52 h x x3 x 6 h x x3 9x2 27x 35 f x 5x 3 9x 2 28x 6 gx 3x3 4x2 8x 8 gx x 4 4x3 8x2 16x 16 h x x 4 6x3 10x2 6x 9 f x x 4 10x 2 9 72. f x x 4 29x 2 100 In Exercises 73–78, find all the zeros of the function. When there
is an extended list of possible rational zeros, use a graphing utility to graph the function in order to discard any rational zeros that are obviously not zeros of the function. 73. 74. 75. 76. f x x3 24x 2 214x 740 f s 2s3 5s2 12s 5 f x 16x 3 20x 2 4x 15 f x 9x 3 15x 2 11x 5 f x 2x4 5x 3 4x 2 5x 2 77. 78. gx x 5 8x 4 28x 3 56x 2 64x 32 333202_0205.qxd 12/7/05 9:36 AM Page 181 In Exercises 79– 86, use Descartes’s Rule of Signs to determine the possible numbers of positive and negative zeros of the function. 80. 82. hx 4x2 8x 3 hx 2x4 3x 2 79. 81. 83. 84. 85. 86. gx 5x5 10x hx 3x4 2x 2 1 gx 2x3 3x 2 3 f x 4x3 3x 2 2x 1 f x 5x3 x 2 x 5 f x 3x3 2x 2 x 3 In Exercises 87– 90, use synthetic division to verify the upper and lower bounds of the real zeros of f. (b) Lower: (b) Lower: 87. 88. 89. 90. f x x4 4x3 15 x 4 (a) Upper: f x 2x3 3x 2 12x 8 x 4 (a) Upper: f x x4 4x3 16x 16 x 5 (a) Upper: f x 2x4 8x 3 (a) Upper: x 3 (b) Lower: (b) Lower: x 1 x 3 x 3 x 4 In Exercises 91–94, find all the real zeros of the function. 91. 92. 93. 94. f x 4x3 3x 1 f z 12z3 4z 2 27z 9 f y 4y3 3y 2 8y 6 gx 3x3 2x 2 15x 10 In Exercises 95–98, find all the rational zeros of the polynomial function. 95. 96. 97. 98. 4 Px x 4 25 f x x3 3 f x x3 1 f z z3 11 x2 23 4 x
2 x 1 6 z2 1 2z 1 4x4 25x 2 36 2x33x 2 23x 12 4x3 x 2 4x 1 6z311z2 3z 2 3 4 In Exercises 99–102, match the cubic function with the numbers of rational and irrational zeros. (a) Rational zeros: 0; irrational zeros: 1 (b) Rational zeros: 3; irrational zeros: 0 (c) Rational zeros: 1; irrational zeros: 2 (d) Rational zeros: 1; irrational zeros: 0 99. 101. f x x3 1 f x x3 x 100. 102. f x x3 2 f x x3 2x 103. Geometry An open box is to be made from a rectangular piece of material, 15 centimeters by 9 centimeters, by cutting equal squares from the corners and turning up the sides. Section 2.5 Zeros of Polynomial Functions 181 (a) Let x represent the length of the sides of the squares removed. Draw a diagram showing the squares removed from the original piece of material and the resulting dimensions of the open box. (b) Use the diagram to write the volume of the box as a function of Determine the domain of the function. x. V (c) Sketch the graph of the function and approximate the dimensions of the box that will yield a maximum volume. (d) Find values of such that Which of these values is a physical impossibility in the construction of the box? Explain. x V 56. 104. Geometry A rectangular package to be sent by a delivery service (see figure) can have a maximum combined length and girth (perimeter of a cross section) of 120 inches. x x y (a) Show that the volume of the package is Vx 4x 230 x. (b) Use a graphing utility to graph the function and approximate the dimensions of the package that will yield a maximum volume. (c) Find values of Which of these values is a physical impossibility in the construction of the package? Explain. such that V 13,500. x 105. Advertising Cost A company that produces MP3 (in dollars) for selling a P players estimates that the profit particular model is given by P 76x 3 4830x 2 320,000, 0 ≤ x ≤ 60 x is the advertising expense (in tens of thousands where of dollars). Using this model, find the smaller of two advertising amounts that will yield a profit of $
2,500,000. 106. Advertising Cost A company that manufactures bicy(in dollars) for selling a P cles estimates that the profit particular model is given by P 45x 3 2500x 2 275,000, 0 ≤ x ≤ 50 x is the advertising expense (in tens of thousands where of dollars). Using this model, find the smaller of two advertising amounts that will yield a profit of $800,000. 333202_0205.qxd 12/7/05 9:36 AM Page 182 182 Chapter 2 Polynomial and Rational Functions 107. Geometry A bulk food storage bin with dimensions 2 feet by 3 feet by 4 feet needs to be increased in size to hold five times as much food as the current bin. (Assume each dimension is increased by the same amount.) (a) Write a function that represents the volume V of the new bin. (b) Find the dimensions of the new bin. 108. Geometry A rancher wants to enlarge an existing rectangular corral such that the total area of the new corral is 1.5 times that of the original corral. The current corral’s dimensions are 250 feet by 160 feet. The rancher wants to increase each dimension by the same amount. (a) Write a function that represents the area of the new A corral. (b) Find the dimensions of the new corral. (c) A rancher wants to add a length to the sides of the corral that are 160 feet, and twice the length to the sides that are 250 feet, such that the total area of the new corral is 1.5 times that of the original corral. Repeat parts (a) and (b). Explain your results. 109. Cost The ordering and transportation cost (in thousands of dollars) for the components used in manufacturing a product is given by C 100200 x 2 x x 30 x ≥ 1, C x is the order size (in hundreds). In calculus, it can where be shown that the cost is a minimum when 3x3 40x 2 2400x 36,000 0. Use a calculator to approximate the optimal order size to the nearest hundred units. 110. Height of a Baseball A baseball is thrown upward from a height of 6 feet with an initial velocity of 48 feet per second, and its height (in feet) is ht 16t 2 48t 6, 0 ≤ t ≤ 3 h t is the time (in seconds). You are told the ball where reaches a height
of 64 feet. Is this possible? p 111. Profit The demand equation for a certain product is where is the unit price (in dollars) is the number of units produced and is is the total cost (in dollars) is the number of units produced. The total profit p 140 0.0001x, x of the product and sold. The cost equation C 80x 150,000, C and obtained by producing and selling units is the product where for x x P R C xp C. You are working in the marketing department of the company that produces this product, and you are asked to determine a price that will yield a profit of 9 million dollars. Is this possible? Explain. p Model It 112. Athletics The attendance A (in millions) at NCAA women’s college basketball games for the years 1997 through 2003 is shown in the table, where represents the year, with to 1997. (Source: National Collegiate Athletic Association) t corresponding t 7 Year, t Attendance, A 7 8 9 10 11 12 13 6.7 7.4 8.0 8.7 8.8 9.5 10.2 (a) Use the regression feature of a graphing utility to find a cubic model for the data. (b) Use the graphing utility to create a scatter plot of the data. Then graph the model and the scatter plot in the same viewing window. How do they compare? (c) According to the model found in part (a), in what year did attendance reach 8.5 million? (d) According to the model found in part (a), in what year did attendance reach 9 million? (e) According to the right-hand behavior of the model, will the attendance continue to increase? Explain. Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 113 and 114, decide whether 113. It is possible for a third-degree polynomial function with integer coefficients to have no real zeros. 114. If x i is a zero of the function given by f x x 3 ix2 ix 1 then x i must also be a zero of f. Think About It ble) the zeros of the function x r1, x r2, x r3. and In Exercises 115–120, determine (if possiif the function has zeros at f g 115. gx f x 116. gx 3f x 333202_0205.qxd
12/7/05 9:36 AM Page 183 117. 119. gx f x 5 gx 3 f x 118. 120. gx f 2x gx f x 121. Exploration Use a graphing utility to graph the funcf x x 4 4x 2 k for different values of tion given by k. satisfy the specified characteristics. (Some parts do not have unique answers.) such that the zeros of Find values of k f (a) Four real zeros (b) Two real zeros, each of multiplicity 2 (c) Two real zeros and two complex zeros (d) Four complex zeros 122. Think About It Will the answers to Exercise 121 g? change for the function gx f x 2 (a) (b) gx f 2x 123. Think About It A third-degree polynomial function f and 3, and its leading coefficient is has real zeros f. negative. Write an equation for Sketch the graph of How many different polynomial functions are possible for f? 2,2, f. 1 124. Think About It Sketch the graph of a fifth-degree polynomial function whose leading coefficient is positive x 3 and that has one zero at 125. Writing Compile a list of all the various techniques for factoring a polynomial that have been covered so far in the text. Give an example illustrating each technique, and write a paragraph discussing when the use of each technique is appropriate. of multiplicity 2. 126. Use the information in the table to answer each question. Interval, 2 2, 1 1, 4 4, Value of f x Positive Negative Negative Positive (a) What are the three real zeros of the polynomial func- tion f? (b) What can be said about the behavior of the graph of f at x 1? (c) What is the least possible degree of the degree of ever be odd? Explain. f f? Explain. Can (d) Is the leading coefficient of f positive or negative? Explain. Section 2.5 Zeros of Polynomial Functions 183 (e) Write an equation for f. (There are many correct answers.) (f) Sketch a graph of the equation you wrote in part (e). 127. (a) Find a quadratic function f as zeros. Assume that (with integer coefficients) b is a positive ± b i that has integer. (b) Find a quadratic function a ± bi f as
zeros. Assume that (with integer coefficients) b is a positive that has integer. 128. Graphical Reasoning The graph of one of the following functions is shown below. Identify the function shown in the graph. Explain why each of the others is not the correct function. Use a graphing utility to verify your result. (a) (b) (c) (d) f x x 2x 2)x 3.5 g x x 2)x 3.5 h x x 2)x 3.5x 2 1 k x x 1)x 2x 3.5 x 2 4 y 10 –20 –30 –40 Skills Review In Exercises 129–132, perform the operation and simplify. 129. 130. 131. 132. 3 6i 8 3i 12 5i 16i 6 2i1 7i 9 5i9 5i In Exercises 133–138, use the graph of f to sketch the graph of g. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. 135. 134. 133. gx f x 2 gx f x 2 gx 2 f x gx f x gx f 2x 137. 2x 138. gx f 1 136. y 5 4 (4, 4) (0, 2) f (2, 2) (−2, 0) 21 43 x 333202_0206.qxd 12/7/05 9:56 AM Page 184 184 Chapter 2 Polynomial and Rational Functions 2.6 Rational Functions What you should learn • Find the domains of rational functions. • Find the horizontal and vertical asymptotes of graphs of rational functions. • Analyze and sketch graphs of rational functions. • Sketch graphs of rational functions that have slant asymptotes. • Use rational functions to model and solve real-life problems. Why you should learn it Rational functions can be used to model and solve real-life problems relating to business. For instance, in Exercise 79 on page 196, a rational function is used to model average speed over a distance. Introduction A rational function can be written in the form f x N(x) D(x) where Nx and Dx are polynomials and Dx is not the zero polynomial. In general, the domain of a rational function of includes all real numbers x -values that make the denominator zero. Much of the discussion of except rational functions will focus
on their graphical behavior near the -values excluded from the domain. x x Example 1 Finding the Domain of a Rational Function Find the domain of x -values. f x 1 x and discuss the behavior of f near any excluded Solution Because the denominator is zero when except f x to the left and right of x 0, x 0. is all real numbers To determine the behavior of near this excluded value, evaluate the domain of x 0, f f as indicated in the following tables. x f x 1 1 0.5 2 0.1 10 0.01 100 0.001 1000 0 x f x 0 0.001 0.01 0.1 0.5 1000 100 10 2 1 1 Note that as x contrast, as graph of f is shown in Figure 2.36. x approaches 0 from the left, approaches 0 from the right, f x f x decreases without bound. In increases without bound. The Mike Powell/Getty Images Note that the rational function f x 1x given by is also referred to as the reciprocal function discussed in Section 1.6. f x1 −1 FIGURE 2.36 Now try Exercise 1. 333202_0206.qxd 12/7/05 9:56 AM Page 185 Vertical asymptote: x = 0 2 1 −2 −1 −1 Section 2.6 Rational Functions 185 y f(x) = 1 x Horizontal and Vertical Asymptotes In Example 1, the behavior of near f f x as x 0 x 0 f x is denoted as follows. x as 0 1 2 Horizontal asymptote: y = 0 x f x decreases without bound x as approaches 0 from the left. f x x increases without bound as approaches 0 from the right. x 0 is a vertical asymptote of the graph of The line as shown in Figure 2.37. From this figure, you can see that the graph of also has a horizontal asymptote— x the line increases or decreases without bound. This means that the values of approach zero as f x 1x y 0. f, f FIGURE 2.37 f x 0 as x f x 0 as x x approaches 0 as f x decreases without bound. approaches 0 as f x x increases without bound. Definitions of Vertical and Horizontal Asymptotes x a 1. The line f x is a vertical asymptote of the graph of f x or f if either from the right or from the left. is a horizontal as
ymptote of the graph of f if a, y b as x 2. The line f x as x b x or. Eventually (as ), the distance between the horizontal asymptote and the points on the graph must approach zero. Figure 2.38 shows the horizontal and vertical asymptotes of the graphs of three rational functions. x x or f(x) = 2x + 1 x + 1 Vertical asymptote: x = −1 y 4 3 2 1 Horizontal asymptote: y = 2 f(x) = 4 x + 12 y 3 2 1 Horizontal asymptotex) = 2 (x −1)2 Vertical asymptote: x = 1 Horizontal asymptote: y = 0 −3 −2 −1 x 1 −2 −1 1 2 x (a) FIGURE 2.38 (b) 1 2 3 x −1 (c) The graphs of f x 1x in Figure 2.37 and f x 2x 1x 1 in Figure 2.38(a) are hyperbolas. You will study hyperbolas in Section 10.4. 333202_0206.qxd 12/7/05 9:56 AM Page 186 186 Chapter 2 Polynomial and Rational Functions Asymptotes of a Rational Function Let be the rational function given by f f x Nx Dx where Nx and Dx anx n an1x n1... a1x a 0 bmx m bm1x m1... b1x b0 have no common factors. 1. The graph of has vertical asymptotes at the zeros of f Dx. 2. The graph of has one or no horizontal asymptote determined by f comparing the degrees of Nx and Dx. a. If n < m, asymptote. n m, b. If the graph of the graph of f f has the line y 0 (the -axis) as a horizontal x has the line y an bm (ratio of the leading coefficients) as a horizontal asymptote. c. If n > m, the graph of f has no horizontal asymptote. Example 2 Finding Horizontal and Vertical Asymptotes Find all horizontal and vertical asymptotes of the graph of each rational function. a. f x 2x2 x2 1 b. f x x2 x 2 x2 x 6
Solution a. For this rational function, the degree of the numerator is equal to the degree of the denominator. The leading coefficient of the numerator is 2 and the leadas a ing coefficient of the denominator is 1, so the graph has the line horizontal asymptote. To find any vertical asymptotes, set the denominator equal to zero and solve the resulting equation for y 2 x. x2 1 0 x 1x 1 0 x 1 0 x 1 0 Set denominator equal to zero. Factor. Set 1st factor equal to 0. Set 2nd factor equal to 0. x 1 x 1 x 1 This equation has two real solutions x 1 lines shown in Figure 2.39. x 1 and so the graph has the as vertical asymptotes. The graph of the function is and x 1, b. For this rational function, the degree of the numerator is equal to the degree of the denominator. The leading coefficient of both the numerator and denomy 1 inator is 1, so the graph has the line as a horizontal asymptote. To find any vertical asymptotes, first factor the numerator and denominator as follows. f x x2 x 2 x2 x 6 x 1x 2 x 2x By setting the denominator you can determine that the graph has the line (of the simplified function) equal to zero, as a vertical asymptote. x 3 Now try Exercise 9. y 4 3 2 1 f(x) = 2x 2 x 2 − 1 Horizontal asymptote: y = 2 −4 −3 −2 −1 1 2 3 4 x Vertical asymptote: x = −1 FIGURE 2.39 Vertical asymptote: x = 1 333202_0206.qxd 12/7/05 9:56 AM Page 187 Section 2.6 Rational Functions 187 Analyzing Graphs of Rational Functions To sketch the graph of a rational function, use the following guidelines. You may also want to test for symmetry when graphing rational functions, especially for simple rational functions. Recall from Section 1.6 that the graph of f x 1 x is symmetric with respect to the origin. Guidelines for Analyzing Graphs of Rational Functions and Let f x NxDx, are polynomials. where Dx Nx 1. Simplify f, if possible. 2. Find and plot the -intercept (if any) by evaluating y
f 0. 3. Find the zeros of the numerator (if any) by solving the equation Then plot the corresponding -intercepts. Nx 0. x 4. Find the zeros of the denominator (if any) by solving the equation Dx 0. Then sketch the corresponding vertical asymptotes. 5. Find and sketch the horizontal asymptote (if any) by using the rule for finding the horizontal asymptote of a rational function. 6. Plot at least one point between and one point beyond each -intercept and x vertical asymptote. 7. Use smooth curves to complete the graph between and beyond the vertical asymptotes. Te c h n o l o g y Some graphing utilities have difficulty graphing rational functions that have vertical asymptotes. Often, the utility will connect parts of the graph that are not supposed to be connected. For instance, the top screen on the right shows the graph of f x 1 x 2. x 2 and the other to the right of Notice that the graph should consist of two unconnected portions—one to the left of problem, you can try changing the mode of the graphing utility to dot mode. The problem with this is that the graph is then represented as a collection of dots (as shown in the bottom screen on the right) rather than as a smooth curve. To eliminate this x 2. −5 −5 5 −5 5 −5 5 5 The concept of test intervals from Section 2.2 can be extended to graphing of rational functions. To do this, use the fact that a rational function can change signs only at its zeros and its undefined values (the -values for which its denominator is zero). Between two consecutive zeros of the numerator and the denominator, a rational function must be entirely positive or entirely negative. This means that when the zeros of the numerator and the denominator of a rational function are put in order, they divide the real number line into test intervals in which the function has no sign changes. A representative -value is chosen to determine if the value of the rational function is positive (the graph lies above the -axis) or negative (the graph lies below the -axis). x x x x 333202_0206.qxd 12/7/05 9:56 AM Page 188 188 Chapter 2 Polynomial and Rational Functions You can use transformations to help you sketch graphs of rational functions. For instance, the graph of
in Example 3 is a vertical stretch and a right shift of the graph of because f x 1x g gx 3 x 2 3 1 x 2 3f x 2. Horizontal asymptote: y = 0 y 4 2 −2 −4 g(x Vertical asymptote: x = 2 FIGURE 2.40 y 3 2 1 −1 Horizontal asymptote2 f x( ) = 2 x − x 1 −1 −4 −3 −2 Vertical asymptote: = 0 x Example 3 Sketching the Graph of a Rational Function Sketch the graph of gx 3 x 2 and state its domain. Solution y-intercept: x-intercept: Vertical asymptote: Horizontal asymptote: Additional points: g0 3 2 because 2, 0, 3 None, because x 2, y 0, 3 0 zero of denominator because degree of Nx < degree of Dx Test interval, 2 2, Representative x-value Value of g Sign 4 3 g4 0.5 g3 3 Negative Positive Point on graph 4, 0.5 3, 3 By plotting the intercepts, asymptotes, and a few additional points, you can obtain x 2. g the graph shown in Figure 2.40. The domain of is all real numbers except x Now try Exercise 27. Example 4 Sketching the Graph of a Rational Function Sketch the graph of f x 2x 1 x and state its domain. Solution y-intercept: x-intercept: Vertical asymptote: Horizontal asymptote: Additional points: is not in the domain because x 0 2x 1 0 zero of denominator None, because 2, 0, 1 x 0, y 2, because degree of Nx degree of Dx Test interval, 0 0, 1 2, 1 2 Representative Value of f Sign x-value.75 4 Positive Negative Positive Point on graph 1, 3 4, 2 1 4, 1.75 By plotting the intercepts, asymptotes, and a few additional points, you can obtain the graph shown in Figure 2.41. The domain of except x 0. is all real numbers x f FIGURE 2.41 Now try Exercise 31. 333202_0206.qxd 12/7/05 9:56 AM Page 189 Section 2.6 Rational Functions 189 Example 5 Sketching the Graph of a Rational Function Sketch the graph of f x xx2 x 2. Solution Fact
oring the denominator, you have f x x x 1x 2. because f 0 0 y-intercept: x-intercept: Vertical asymptotes: Horizontal asymptote: Additional points: 0, 0, 0, 0 x 1, y 0, x 2, zeros of denominator because degree of Nx < degree of Dx Test interval, 1 1, 0 0, 2 2, Representative x-value Value of f Sign 3 0.5 1 3 f 3 0.3 f 0.5 0.4 f 1 0.5 f 3 0.75 Negative Positive Negative Positive Point on graph 3, 0.3 0.5, 0.4 1, 0.5 3, 0.75 Vertical asymptote: x = −1 Vertical asymptote: x = 2 y Horizontal asymptote: y = 0 3 2 1 −1 −2 −3 − 1 x 2 3 f(x) = x x2 − x − 2 FIGURE 2.42 The graph is shown in Figure 2.42. Now try Exercise 35. If you are unsure of the shape of a portion of the graph of a rational function, plot some additional points. Also note that when the numerator and the denominator of a rational function have a common factor, the graph of the function has a hole at the zero of the common factor (see Example 6). f(x) = x2 − 9 x2 − 2x − 3 Horizontal asymptote: y = 1 y 3 2 1 −4 −3 −1 1 2 3 4 5 6 x Vertical asymptote: x = −1 −2 −3 −4 −5 Example 6 A Rational Function with Common Factors f x x2 9x2 2x 3. Sketch the graph of Solution By factoring the numerator and denominator, you have 2x 3 y-intercept: because, x 3. x 3x 3 x 3x 1 0, 3, 3, 0, x 1, y 1, because f 3 0 zero of (simplified) denominator because degree of Nx degree of Dx x-intercept: Vertical asymptote: Horizontal asymptote: Additional points: Test interval, 3 3, 1 1, Representative x-value Value of f Sign 4 2 2 f 4 0.33 f 2 1 f 2 1.67 Positive Negative Positive Point on graph 4, 0.33
2, 1 2, 1.67 The graph is shown in Figure 2.43. Notice that there is a hole in the graph at x 3 because the function is not defined when x 3. FIGURE 2.43 HOLE AT x 3 Now try Exercise 41. 333202_0206.qxd 12/7/05 9:56 AM Page 190 190 Chapter 2 Polynomial and Rational Functions f x Vertical asymptote: x − = 1 − 8 −6 −4 2 −2 −2 −4 FIGURE 2.44 8 4 6 Slant asymptote: y = x − 2 Slant Asymptotes Consider a rational function whose denominator is of degree 1 or greater. If the degree of the numerator is exactly one more than the degree of the denominator, the graph of the function has a slant (or oblique) asymptote. For example, the graph of x f x x 2 x x 1 has a slant asymptote, as shown in Figure 2.44. To find the equation of a slant asymptote, use long division. For instance, by dividing you obtain x2 x, x 1 into. Slant asymptote y x 2 increases or decreases without bound, the remainder term f approaches the line y x 2, x As approaches 0, so the graph of Figure 2.44. 2x 1 as shown in Example 7 A Rational Function with a Slant Asymptote Sketch the graph of f x x2 x 2x 1. Solution Factoring the numerator as x -intercepts. Using long division x 2x 1 allows you to recognize the allows you to recognize that the line y x y-intercept: x-intercepts: Vertical asymptote: Slant asymptote: Additional points: is a slant asymptote of the graph. f 0 2 because and 2, 0 0, 2, 1, 0 x 1, y x zero of denominator Test interval, 1 1, 1 1, 2 2, Representative x-value Value of f Sign 2 0.5 1.5 3 f 2 1.33 f 0.5 4.5 f 1.5 2.5 f 3 2 Negative Positive Negative Positive Point on graph 2, 1.33 0.5, 4.5 1.5, 2.5 3, 2 The graph is shown in Figure 2.45. Slant asym
ptote3 −2 11 3 4 5 x −2 −3 Vertical asymptote: x = 1 f(x) = x2 − x − 2 x − 1 FIGURE 2.45 Now try Exercise 61. 333202_0206.qxd 12/7/05 9:56 AM Page 191 Section 2.6 Rational Functions 191 Applications There are many examples of asymptotic behavior in real life. For instance, Example 8 shows how a vertical asymptote can be used to analyze the cost of removing pollutants from smokestack emissions. Example 8 Cost-Benefit Model A utility company burns coal to generate electricity. The cost removing % of the smokestack pollutants is given by p C 80,000p 100 p C (in dollars) of 0 ≤ p < 100. for Sketch the graph of this function. You are a member of a state legislature considering a law that would require utility companies to remove 90% of the pollutants from their smokestack emissions. The current law requires 85% removal. How much additional cost would the utility company incur as a result of the new law? Solution The graph of this function is shown in Figure 2.46. Note that the graph has a Because the current law requires 85% removal, vertical asymptote at the current cost to the utility company is p 100. C 80,000(85) 100 85 $453,333. Evaluate when C p 85. If the new law increases the percent removal to 90%, the cost will be C 80,000(90) 100 90 $720,000. Evaluate when C p 90. So, the new law would require the utility company to spend an additional 720,000 453,333 $266,667. Subtract 85% removal cost from 90% removal cost Smokestack Emissions 90% 85% C = 80,000 p 100 − p C 1000 800 600 400 200 20 40 60 80 100 Percent of pollutants removed p FIGURE 2.46 Now try Exercise 73. 333202_0206.qxd_pg 192 1/9/06 8:55 AM Page 192 192 Chapter 2 Polynomial and Rational Functions Example 9 Finding a Minimum Area A rectangular page is designed to contain 48 square inches of print. The margins at the top and bottom of the page are each 1 inch deep. The margins on each side inches wide. What should the dimensions of the page be so that the least are amount of paper is used
? 11 2 y 11 in. 2 1 in. x 1 in. 11 in. 2 FIGURE 2.47 Graphical Solution Let be the area to be minimized. From Figure 2.47, you can write A Numerical Solution A Let be the area to be minimized. From Figure 2.47, you can write A x 3y 2. A x 3 y 2. The printed area inside the margins is modeled by 48 xy y 48x. To find the minimum area, or A rewrite the equation for in terms of just one variable y. for by substituting 2 48x A x 348 x x 348 2x x, x > 0 x The graph of this rational function is shown in Figure 2.48. Because represents the width of the printed area, you need consider only the portion of the graph is positive. Using a graphing utility, you for which to occur can approximate the minimum value of x 8.5 when is 488.5 5.6 y inches. The corresponding value of inches. So, the dimensions should be A x x 3 11.5 inches by y 2 7.6 inches. 200 A = (x + 3)(48 + 2x) x, x > 0 The printed area inside the margins is modeled by y 48x. A or To find the minimum area, rewrite the equation for 48x for y. 48 xy in terms of just one variable by substituting 2 A x 348 x x 348 2x x, x > 0 Use the table feature of a graphing utility to create a table of values for the function y1 x 348 2x x x 1. y1 occurs when From the table, you can see that the minibeginning at x mum value of is somewhere between 8 and 9, as shown in Figure 2.49. To approximate the minimum value y1 to one decimal place, change the table so that it starts at of x 8 and increases by 0.1. The minimum value of occurs x 8.5, when as shown in Figure 2.50. The corresponding y inches. So, the dimensions should value of is x 3 11.5 y 2 7.6 inches. be 488.5 5.6 inches by y1 0 0 FIGURE 2.48 24 Now try Exercise 77. FIGURE 2.49 FIGURE 2.50 If you go on to take a course in calculus, you will learn an analytic technique that produces a minimum area. In this case, that x for finding
the exact value of value is x 62 8.485. 333202_0206.qxd 12/7/05 9:56 AM Page 193 2.6 Exercises Section 2.6 Rational Functions 193 VOCABULARY CHECK: Fill in the blanks. 1. Functions of the form f x NxDx, where Nx and Dx are polynomials and Dx is not the zero polynomial, are called ________ ________. 2. If 3. If x → a as x → ±, f x → ± f x → b as from the left or the right, then x a y b is a ________ ________ of the graph of f x NxDx, 4. For the rational function given by f Nx then the graph of has a ________ (or oblique) ________. if the degree of degree of Dx, then f. is a ________ ________ of the graph of f. is exactly one more than the PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1– 4, (a) complete each table for the function, (b) determine the vertical and horizontal asymptotes of the graph of the function, and (c) find the domain of the function. f x x 0.5 0.9 0.99 0.999 12 −4 −4 −2 3. f x 3x.5 1.1 1.01 1.001 f x x 5 10 100 1000 2. f x 5x x 1 y 12 8 −8 −4 −4 x 4 8 4. f x 4x x2 1 y 8 4 x 4 8 −8 x 4 8 −8 −4 −4 −8 In Exercises 5 –12, find the domain of the function and identify any horizontal and vertical asymptotes. 5. f x 1 x 2 6. f x 4 x 23 7. 9. 11 3x 2 1 x 2 x 9 8. f x 1 5x 1 2x 10. 12. f x 2x 2 x 1 f x 3x 2 x 5 x 2 1 In Exercises 13 –16, match the rational function with its graph. [The graphs are labeled (a), (b), (c), and (d).] (a) y (b) 4 2 −2 −4
x 6 −8 −6 −4 −2 (c) y (d) y 4 2 −2 −4 y 4 2 −2 x 4 6 −4 −2 4 2 −2 −4 x x 13. 15 14. 16 In Exercises 17–20, find the zeros (if any) of the rational function. 17. 19. gx 18. hx 2 5 x 2 2 20. gx x3 8 x 2 1 333202_0206.qxd 12/7/05 9:56 AM Page 194 194 194 Chapter 2 Chapter 2 Polynomial and Rational Functions Polynomial and Rational Functions In Exercises 21– 26, find the domain of the function and identify any horizontal and vertical asymptotes. f x x 3 x2 9 f x x2 4 f x x 4 x2 16 f x x2 1 21. 23. 24. 22. x2 3x 2 x2 2x 3 f x x2 3x 4 2x2 x 1 25. 26. f x 6x2 11x 3 6x2 7x 3 In Exercises 27–46, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical and horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. 27. 29. 31. 33. 35. 37. 39. 40. 41. 43. 45. 28. 30. 32. 34. 36. 38. f x 1 x 3 gx 1 3 x Px 1 3x 1 x f t 1 2t t f x 1 x 22 gx x2 2x 8 x2 9 f x 1 hx x 2 1 x 2 Cx 5 2x 1 x f x x 2 x 2 9 gs s s 2 1 hx x2 5x 4 x2 4 f x 2x2 5x 3 x 3 2x2 x 2 x2 x 2 x 3 2x2 5x 6 f x f x x2 3x x2 x 6 f x 2x2 5x 2 2x2 x 6 f t t2 1 t 1 42. 44. 46. f x 5x 4 x2 x 12 f x 3x2 8x 4 2x2 3x 2 f x x2 16 x 4 Analytical, Numerical, and Graphical Analysis 47– 50, do
the following. In Exercises (a) Determine the domains of and f g. (b) Simplify graph of f f. and find any vertical asymptotes of the (c) Compare the functions by completing the table. (d) Use a graphing utility to graph and f g in the same viewing window. (e) Explain why the graphing utility may not show the difference in the domains of and f g. 47. f x x 2 1 x 1, gx x 1 3 2 1.5 1 0.5 0 1 x f x gx 48. f x x 2x 2 x 2 2x, gx x 1 0 1 1.5 2 2.5 3 x f x gx 49. f x x 2 x 2 2x, gx 1 x 0.5 0 0.5 1 1.5 2 3 x f x gx 50. f x 2x 6 x 2 7x 12, gx gx In Exercises 51–64, (a) state the domain of the function, (b) identify all intercepts, (c) identify any vertical and slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. gx x2 5 hx x2 4 51. 52. x x 53. 55. 57. 59. 61. f x 2x2 1 x gx x2 54. 56. 58. 60. 62. f x 1 x2 x hx x2 x 1 f x x2 3x 1 gx x 3 2x 2 8 f x 2x 2 5x 5 x 2 333202_0206.qxd 12/7/05 9:56 AM Page 195 63. 64. f x 2x3 x2 2x 1 x2 3x 2 f x 2x3 x2 8x 4 x2 3x 2 In Exercises 65– 68, use a graphing utility to graph the rational function. Give the domain of the function and identify any asymptotes. Then zoom out sufficiently far so that the graph appears as a line. Identify the line. 65. 66. 67. 68. f x x 2 5x 8 x 3 f x 2x 2 x x 1 gx 1 3x 2 x 3 x 2 hx 12 2x x 2 24 x Graphical Reasoning In Exercises 69–72, (a) use
the graph x to determine any -intercepts of the graph of the rational and solve the resulting equation function and (b) set to confirm your result in part (a). y 0 69. y x 1 x 3 y 70. y 2x x 3 y 6 4 2 −2 −4 x 4 6 8 6 4 2 −2 −4 x 2 4 6 8 71. y 1 x x 724 −2 −4 y 8 4 x 4 −8 −4 −4 x 4 8 73. Pollution The cost (in millions of dollars) of removing of the industrial and municipal pollutants discharged C p% into a river is given by C 255p 100 p, 0 ≤ p < 100. (a) Use a graphing utility to graph the cost function. Section 2.6 Rational Functions 195 (b) Find the costs of removing 10%, 40%, and 75% of the pollutants. (c) According to this model, would it be possible to remove 100% of the pollutants? Explain. 74. Recycling In a pilot project, a rural township is given recycling bins for separating and storing recyclable p% products. The cost of the population is given by (in dollars) for supplying bins to C C 25,000p 100 p, 0 ≤ p < 100. (a) Use a graphing utility to graph the cost function. (b) Find the costs of supplying bins to 15%, 50%, and 90% of the population. (c) According to this model, would it be possible to supply bins to 100% of the residents? Explain. 75. Population Growth The game commission introduces 100 deer into newly acquired state game lands. The popuN lation N 205 3t 1 0.04t of the herd is modeled by t ≥ 0, where t is the time in years (see figure). N 1400 1200 1000 800 600 400 200 100 150 200 50 Time (in years) (a) Find the populations when t 25. (b) What is the limiting size of the herd as time increases? t 10, t 5, and 76. Concentration of a Mixture A 1000-liter tank contains liters of a 75% 50 liters of a 25% brine solution. You add brine solution to the tank. x (a) Show that the concentration C, the proportion of brine to total solution, in the final mixture is C 3x 50 4x 50. (b) Determine the domain of the function based on
the physical constraints of the problem. (c) Sketch a graph of the concentration function. (d) As the tank is filled, what happens to the rate at which the concentration of brine is increasing? What percent does the concentration of brine appear to approach? 333202_0206.qxd 12/7/05 9:56 AM Page 196 196 196 Chapter 2 Chapter 2 Polynomial and Rational Functions Polynomial and Rational Functions 77. Page Design A page that is inches high contains 30 square inches of print. The top and bottom margins are 1 inch deep and the margins on each side are 2 inches wide (see figure). inches wide and y x 2 in. 1 in. 1 in. x 2 in. y (a) Show that the total area on the page is A A 2xx 11 x 4. (b) Determine the domain of the function based on the physical constraints of the problem. (c) Use a graphing utility to graph the area function and approximate the page size for which the least amount of paper will be used. Verify your answer numerically using the table feature of the graphing utility. 78. Page Design A rectangular page is designed to contain 64 square inches of print. The margins at the top and bottom of the page are each 1 inch deep. The margins on inches wide. What should the dimensions each side are of the page be so that the least amount of paper is used? 11 2 Model It 80. Sales The sales (in millions of dollars) for the Yankee Candle Company in the years 1998 through 2003 are shown in the table. (Source: The Yankee Candle Company) S 1998 184.5 2001 379.8 1999 256.6 2002 444.8 2000 338.8 2003 508.6 A model for these data is given by S 5.816t2 130.68 0.004t2 1.00 t represents the year, with, 8 ≤ t ≤ 13 where 1998. t 8 corresponding to (a) Use a graphing utility to plot the data and graph the model in the same viewing window. How well does the model fit the data? (b) Use the model to estimate the sales for the Yankee Candle Company in 2008. (c) Would this model be useful for estimating sales after 2008? Explain. Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 81 and 82, determine whether 81. A polynomial can have infinitely many vertical asym
ptotes. 82. The graph of a rational function can never cross one of its asymptotes. Think About It In Exercises 83 and 84, write a rational function that has the specified characteristics. (There are many correct answers.) f 83. Vertical asymptote: None 84. Vertical asymptote: Horizontal asymptote: y 2 x 2, Horizontal asymptote: None x 1 79. Average Speed A driver averaged 50 miles per hour on the round trip between Akron, Ohio, and Columbus, Ohio, 100 miles away. The average speeds for going and returning were and miles per hour, respectively. (a) Show that y x y 25x x 25. Skills Review (b) Determine the vertical and horizontal asymptotes of the graph of the function. (c) Use a graphing utility to graph the function. (d) Complete the table. 30 35 40 45 50 55 60 x y (e) Are the results in the table what you expected? Explain. (f) Is it possible to average 20 miles per hour in one direction and still average 50 miles per hour on the round trip? Explain. In Exercises 85– 88, completely factor the expression. 85. 87. x2 15x 56 x 3 5x2 4x 20 86. 88. 3x2 23x 36 x 3 6x2 2x 12 In Exercises 93–96, solve the inequality and graph the solution on the real number line. 89. 91. 10 3x ≤ 0 4x 2 < 20 90. 92. 5 2x > 5x 1 22x 3 ≥ 5 1 93. Make a Decision To work an extended application analyzing the total manpower of the Department of Defense, visit this text’s website at college.hmco.com. (Data Source: U.S. Census Bureau) 333202_0207.qxd 12/7/05 9:40 AM Page 197 2.7 Nonlinear Inequalities Section 2.7 Nonlinear Inequalities 197 What you should learn • Solve polynomial inequalities. • Solve rational inequalities. • Use inequalities to model and solve real-life problems. Why you should learn it Inequalities can be used to model and solve real-life problems. For instance, in Exercise 73 on page 205, a polynomial inequality is used to model the percent of households that own a television and have cable in the United States. Poly
nomial Inequalities x2 2x 3 < 0, x To solve a polynomial inequality such as you can use the fact that a polynomial can change signs only at its zeros (the -values that make the polynomial equal to zero). Between two consecutive zeros, a polynomial must be entirely positive or entirely negative. This means that when the real zeros of a polynomial are put in order, they divide the real number line into intervals in which the polynomial has no sign changes. These zeros are the critical numbers of the inequality, and the resulting intervals are the test intervals for the inequality. For instance, the polynomial above factors as x2 2x 3 x 1x 3 x 1 and has two zeros, into three test intervals:, 1, 1, 3, and x 3. These zeros divide the real number line and 3,. (See Figure 2.51.) you need only test one value from So, to solve the inequality each of these test intervals to determine whether the value satisfies the original inequality. If so, you can conclude that the interval is a solution of the inequality. x2 2x 3 < 0, Zero x − = 1 Zero x = 3 © Jose Luis Pelaez, Inc./Corbis Test Interval − ( −, 1) Test Interval −1, 3) ( Test Interval (3, ) x −4 −3 −2 −1 0 1 2 3 4 5 FIGURE 2.51 Three test intervals for x2 2x 3 You can use the same basic approach to determine the test intervals for any polynomial. Finding Test Intervals for a Polynomial To determine the intervals on which the values of a polynomial are entirely negative or entirely positive, use the following steps. 1. Find all real zeros of the polynomial, and arrange the zeros in increasing order (from smallest to largest). These zeros are the critical numbers of the polynomial. 2. Use the critical numbers of the polynomial to determine its test intervals. x 3. Choose one representative -value in each test interval and evaluate the polynomial at that value. If the value of the polynomial is negative, the polynomial will have negative values for every -value in the interval. If the value of the polynomial is positive, the polynomial will have positive values for every -value in the interval. x x 333202_0
207.qxd 12/7/05 9:40 AM Page 198 198 Chapter 2 Polynomial and Rational Functions Example 1 Solving a Polynomial Inequality Solve x2 x 6 < 0. Solution By factoring the polynomial as x2 x 6 x 2x 3 you can see that the critical numbers are polynomial’s test intervals are x 2 and x 3. So, the, 2, 2, 3, and 3,. Test intervals In each test interval, choose a representative -value and evaluate the polynomial. x Test Interval, 2 2, 3 3, x-Value x 3 x 0 x 4 Polynomial Value 32 3 6 6 02 0 6 6 42 4 6 6 Conclusion Positive Negative Positive This implies that the solution of the inequality 2, 3, From this you can conclude that the inequality is satisfied for all -values in 2, 3. is the interval as shown in Figure 2.52. Note that the original inequality contains a less than symbol. This means that the solution set does not contain the endpoints of the test interval x2 x 6 < 0 2, 3. x − x Choose = 3. − x x 3) > 0 ( + 2)( Choose = x x ( + 2)( x − 4. 3) > 0 −6 −5 −4 −3 −2 −4 −3 −1 1 2 4 5 x −2 −3 −6 −7 FIGURE 2.53 y = 2 − − x x 6 Choose = x x ( + 2)( x − 0. 3) < 0 FIGURE 2.52 Now try Exercise 13. As with linear inequalities, you can check the reasonableness of a solution by substituting -values into the original inequality. For instance, to check the solution found in Example 1, try substituting several -values from the interval 2, 3 into the inequality x x x2 x 6 < 0. Regardless of which -values you choose, the inequality should be satisfied. x You can also use a graph to check the result of Example 1. Sketch the graph as shown in Figure 2.53. Notice that the graph is below the y x2 x 6, of -axis on the interval 2, 3. x 333202_0207.qxd 12/7/05 9:40 AM Page 199 Section 2.7 Nonlinear Inequalities 199 In Example 1, the polynomial inequality was given in general form (with the polynomial
on one side and zero on the other). Whenever this is not the case, you should begin the solution process by writing the inequality in general form. Example 2 Solving a Polynomial Inequality Solve 2x3 3x2 32x > 48. Solution Begin by writing the inequality in general form. 2x 3 3x2 32x > 48 Write original inequality. 2x 3 3x2 32x 48 > 0 x 4x 42x 3 > 0 Write in general form. Factor. You may find it easier to determine the sign of a polynomial from its factored form. For instance, in Example 2, if the test value is substituted into the factored form x 2 x 4x 42x 3 you can see that the sign pattern of the factors is which yields a negative result. Try using the factored forms of the polynomials to determine the signs of the polynomials in the test intervals of the other examples in this section. 2, 3 The critical numbers are 2, 4,, 4, 4, 3 x-Value x 5 x 0 x 2 x 5 Test Interval, 4 4, 3 3 2, 4 4, 2 x 4, x 3 2, 4,. and and x 4, and the test intervals are Polynomial Value 253 352 325 48 203 302 320 48 223 322 322 48 253 352 325 48 Conclusion Negative Positive Negative Positive From this you can conclude that the inequality is satisfied on the open intervals 4, 3 Therefore, the solution set consists of all real numbers in the 4,, intervals as shown in Figure 2.54. and 4, 3 4,. and 2 2 x Choose = x 4)( + 4)(2 x − ( x 0. − 3) > 0 − ( x x Choose = 5. − 4)( + 4)(2 x x 3) > 0 −7 −6 −5 −4 −3 −2 − Choose = x 4)( + 4)(2 x − ( x −5. − 3) < 0 − ( x FIGURE 2.54 Now try Exercise 21. x Choose = 2 − 4)( + 4)(2 x x. 3) < 0 When solving a polynomial inequality, be sure you have accounted for the particular type of inequality symbol given in the inequality. For instance, in Example 2, note that the original inequality contained a “greater than” symbol and the solution consisted of two open intervals. If the original inequality had been 2x3
3x2 32x ≥ 48 the solution would have consisted of the closed interval 4,. 4, 3 2 and the interval 333202_0207.qxd 12/7/05 9:40 AM Page 200 200 Chapter 2 Polynomial and Rational Functions Each of the polynomial inequalities in Examples 1 and 2 has a solution set that consists of a single interval or the union of two intervals. When solving the exercises for this section, watch for unusual solution sets, as illustrated in Example 3. Example 3 Unusual Solution Sets a. The solution set of the following inequality consists of the entire set of real x2 2x 4 In other words, the value of the quadratic,. numbers, is positive for every real value of x. x2 2x 4 > 0 b. The solution set of the following inequality consists of the single real number x2 2x 1 has only one critical number, because the quadratic and it is the only value that satisfies the inequality. 1, x 1, x2 2x 1 ≤ 0 c. The solution set of the following inequality is empty. In other words, the quadis not less than zero for any value of x2 3x 5 ratic x. x2 3x 5 < 0 d. The solution set of the following inequality consists of all real numbers this solution set can be written as In interval notation, x 2. except, 2 2,. x2 4x 4 > 0 Now try Exercise 25. Exploration You can use a graphing utility to verify the results in Example 3. For instance, the graph of y -values are greater than 0 for all values of Use the graphing utility to graph the following: y x 2 3x 5 is shown below. Notice that the as stated in Example 3(a). y x 2 4x 4 y x 2 2x 1 y x 2 2x 4 x, Explain how you can use the graphs to verify the results of parts (b), (c), and (d) of Example 3. 10 −2 9 −9 333202_0207.qxd 12/7/05 9:40 AM Page 201 Section 2.7 Nonlinear Inequalities 201 Rational Inequalities The concepts of critical numbers and test intervals can be extended to rational inequalities. To do this, use the fact that the value of a rational expression can change sign only at its zeros (the -values for which its numerator is zero) and its undefined values (the -values
for which its denominator is zero). These two types of numbers make up the critical numbers of a rational inequality. When solving a rational inequality, begin by writing the inequality in general form with the rational expression on the left and zero on the right. x x Example 4 Solving a Rational Inequality Solve 2x 7 x 5 ≤ 3. Solution 2x 7 x 5 ≤ 3 3 ≤ 0 2x 7 x 5 2x 7 3x 15 x 5 Write original inequality. Write in general form. ≤ 0 ≤ 0 Find the LCD and add fractions. Simplify. x 8 x 5 Critical numbers: Test intervals: Test: x 5, x 8, 5, 5, 8, 8, Zeros and undefined values of rational expression Is x 8 x 5 ≤ 0? After testing these intervals, as shown in Figure 2.55, you can see that the (, 5) inequality is satisfied on the open intervals Moreover, x 8, you can conclude that the solution because, 5 8,. set consists of all real numbers in the intervals (Be sure to use a closed interval to indicate that can equal 8.) x 8x 5 0 8,. when and x x Choose = 6. − Choose = 4. x −x + 8 x − 5 < 0 FIGURE 2.55 Now try Exercise 39. x Choose = 9. −x + 8 x − 5 < 0 333202_0207.qxd 12/7/05 9:40 AM Page 202 202 Chapter 2 Polynomial and Rational Functions Applications One common application of inequalities comes from business and involves profit, revenue, and cost. The formula that relates these three quantities is Profit Revenue Cost P R C. Example 5 Increasing the Profit for a Product Calculators R The marketing department of a calculator manufacturer has determined that the demand for a new model of calculator is p 100 0.00001x, 0 ≤ x ≤ 10,000,000 Demand equation 250 200 150 100 50 ) FIGURE 2.56 x 0 2 4 6 8 10 Number of units sold (in millions) p is the price per calculator (in dollars) and where represents the number of calculators sold. (If this model is accurate, no one would be willing to pay $100 for the calculator. At the other extreme, the company couldn’t sell more than 10 million calculators.) The revenue for selling calculators is x x R xp x100 0.00001x Revenue equation as shown in Figure 2.56. The total cost
of producing calculator plus a development cost of $2,500,000. So, the total cost is x calculators is $10 per C 10x 2,500,000. Cost equation What price should the company charge per calculator to obtain a profit of at least $190,000,000? Solution Verbal Model: Equation: Profit Revenue Cost P R C P 100x 0.00001x2 10x 2,500,000 P 0.00001x 2 90x 2,500,000 Calculators P To answer the question, solve the inequality P ≥ 190,000,000 0.00001x2 90x 2,500,000 ≥ 190,000,000. When you write the inequality in general form, find the critical numbers and the test intervals, and then test a value in each test interval, you can find the solution to be x 3,500,000 ≤ x ≤ 5,500,000 as shown in Figure 2.57. Substituting the -values in the original price equation shows that prices of 200 150 100 50 0 −50 −100 0 2 4 6 8 10 Number of units sold (in millions) $45.00 ≤ p ≤ $65.00 will yield a profit of at least $190,000,000. FIGURE 2.57 Now try Exercise 71. 333202_0207.qxd 12/7/05 9:40 AM Page 203 Section 2.7 Nonlinear Inequalities 203 Another common application of inequalities is finding the domain of an expression that involves a square root, as shown in Example 6. Example 6 Finding the Domain of an Expression Find the domain of 64 4x2. Algebraic Solution Remember that the domain of an expression is the set of all -values 64 4x2 for which the expression is defined. Because is defined is nonnegative, the domain is (has real values) only if given by 64 4x2 x 64 4x2 ≥ 0. 64 4x2 ≥ 0 16 x2 ≥ 0 4 x4 x ≥ 0 Write in general form. Divide each side by 4. Write in factored form. So, the inequality has two critical numbers: can use these two numbers to test the inequality as follows. x 4, x 4, 4, 4, 4, 4, Critical numbers: Test intervals: and x 4 x 4. You Test: For what values of x is 64 4x2 ≥ 0? A test shows that the inequality is satisfied in the closed
interval 4, 4. is the interval 4, 4. So, the domain of the expression 64 4x2 Graphical Solution Begin by sketching the graph of the equation y 64 4x2, as shown in Figure 2.58. From the graph, you can determine that the -values 4 and 4). So, extend from to 4 (including the domain of the expression is the interval 4 64 4x2 4, 4. x y = 64 − 4x 2 y 10 6 4 2 −6 −4 −2 2 4 6 −2 x FIGURE 2.58 Now try Exercise 55. Complex Number Nonnegative Radicand Complex Number −4 FIGURE 2.59 4 x To analyze a test interval, choose a representative -value in the interval and evaluate the expression at that value. For instance, in Example 6, if you substitute any number from the interval you will obtain a nonnegative number under the radical symbol that simplifies to a real number. If you substitute any number from the intervals you will obtain a complex number. It might be helpful to draw a visual representation of the intervals as shown in Figure 2.59. into the expression 64 4x2, 4 4, 4 4, and W RITING ABOUT MATHEMATICS Profit Analysis Consider the relationship P R C described on page 202. Write a paragraph discussing why it might be beneficial to solve illustrate your reasoning. if you owned a business. Use the situation described in Example 5 to P < 0 333202_0207.qxd 12/7/05 9:40 AM Page 204 204 Chapter 2 Polynomial and Rational Functions 2.7 Exercises VOCABULARY CHECK: Fill in the blanks. 1. To solve a polynomial inequality, find the ________ numbers of the polynomial, and use these numbers to create ________ ________ for the inequality. 2. The critical numbers of a rational expression are its ________ and its ________ ________. 3. The formula that relates cost, revenue, and profit is ________. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1– 4, determine whether each value of solution of the inequality. x is a In Exercises 27–32, solve the inequality and write the solution set in interval notation. Inequality 1. x2 3 < 0 2. x2 x 12 ≥ 0 3.
4. x 2 x 4 ≥ 3 3x2 x2 4 < 1 Values (a) (c) (a) (c) (a) (c) (a) (cb) (d) (b) (d) (b) (d) (b) (d In Exercises 5–8, find the critical numbers of the expression. 5. 7. 2x2 x 6 2 3 x 5 6. 8. 9x3 25x 2 2 x x 2 x 1 27. 29. 31. 4x3 6x2 < 0 x3 4x ≥ 0 x 12x 23 ≥ 0 28. 30. 32. 4x3 12x 2 > 0 2x3 x 4 ≤ 0 x4x 3 ≤ 0 Graphical Analysis In Exercises 33–36, use a graphing utility to graph the equation. Use the graph to approximate that satisfy each inequality. the values of x Equation Inequalities 33. 34. 35. 36. y x 2 2x 3 y 1 2x 2 2x 1 8x3 1 y 1 2x y x3 x 2 16x 16 (a) (a) (a) (ab) (b) (b) (b ≥ 36 In Exercises 37–50, solve the inequality and graph the solution on the real number line. In Exercises 9–26, solve the inequality and graph the solution on the real number line. 10. 12. 14. 16. x2 < 36 x 32 ≥ 1 x2 6x 9 < 16 x2 2x > 3 9. 11. 13. 15. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. x2 ≤ 9 x 22 < 25 x2 4x 4 ≥ 9 x2 x < 6 x2 2x 3 < 0 x2 4x 1 > 0 x2 8x 5 ≥ 0 2x2 6x 15 ≤ 0 x3 3x2 x 3 > 0 x3 2x2 4x 8 ≤ 0 x3 2x2 9x 2 ≥ 20 2x3 13x2 8x 46 ≥ 6 4x2 4x 1 ≤ 0 x2 3x 8 > 0 37. 39. 41. 43. 45. 47. 48. 49. 50. 4 < 0 38. 40. 42. 44. 46. 1 x x 12 x 2 5 7x 1 2x 3x 5 x 5 4
x 5 1 x 3 x2 2x x2 9 x2 x 6 x > ≤ ≤ 0 1 2x 3 9 4x 3 ≥ 0 5 x 1 3x x 1 2x x 1 x x 4 ≤ < 1 3 333202_0207.qxd 12/7/05 9:40 AM Page 205 Graphical Analysis In Exercises 51–54, use a graphing utility to graph the equation. Use the graph to approximate that satisfy each inequality. the values of x 71. Cost, Revenue, and Profit The revenue and cost equations for a product are R x75 0.0005x and C 30x 250,000 Section 2.7 Nonlinear Inequalities 205 Equation y 3x x 2 y 2x 2 x 1 y 2x2 x2 4 y 5x x2 4 51. 52. 53. 54. Inequalities (a) y ≤ 0 (b) y ≥ 6 (a) y ≤ 0 (b) y ≥ 8 (a) y ≥ 1 (b) y ≤ 2 (a) y ≥ 1 (b) y ≤ 0 In Exercises 55–60, find the domain of Use a graphing utility to verify your result. x in the expression. 55. 57. 59. 4 x2 x2 7x 12 x x2 2x 35 56. 58. 60. x2 4 144 9x2 x x2 9 In Exercises 61–66, solve the inequality. (Round your answers to two decimal places.) 61. 62. 63. 64. 65. 66. 0.4x2 5.26 < 10.2 1.3x2 3.78 > 2.12 0.5x2 12.5x 1.6 > 0 1.2x2 4.8x 3.1 < 5.3 1 2.3x 5.2 2 3.1x 3.7 > 3.4 > 5.8 67. Height of a Projectile A projectile is fired straight upward from ground level with an initial velocity of 160 feet per second. (a) At what instant will it be back at ground level? (b) When will the height exceed 384 feet? 68. Height of a Projectile A projectile is fired straight upward from ground level with an initial velocity of 128 feet per second. (a) At what instant will it be back at ground level? (b) When will the height be less than 128 feet? 69.
Geometry A rectangular playing field with a perimeter of 100 meters is to have an area of at least 500 square meters. Within what bounds must the length of the rectangle lie? 70. Geometry A rectangular parking lot with a perimeter of 440 feet is to have an area of at least 8000 square feet. Within what bounds must the length of the rectangle lie? R C and represents the are measured in dollars and where number of units sold. How many units must be sold to obtain a profit of at least $750,000? What is the price per unit? x 72. Cost, Revenue, and Profit The revenue and cost equations for a product are R x50 0.0002x and C 12x 150,000 R C and where represents the are measured in dollars and number of units sold. How many units must be sold to obtain a profit of at least $1,650,000? What is the price per unit? x Model It 73. Cable Television The percents of households in the United States that owned a television and had cable from 1980 to 2003 can be modeled by C C 0.0031t3 0.216t2 5.54t 19.1, 0 ≤ t ≤ 23 t where is the year, with (Source: Nielsen Media Research) t 0 corresponding to 1980. (a) Use a graphing utility to graph the equation. (b) Complete the table to determine the year in which the percent of households that own a television and have cable will exceed 75%. 24 26 28 30 32 34 t C (c) Use the trace feature of a graphing utility to verify your answer to part (b). (d) Complete the table to determine the years during which the percent of households that own a television and have cable will be between 85% and 100%. 36 37 38 39 40 41 42 43 t C (e) Use the trace feature of a graphing utility to verify your answer to part (d). (f) Explain why the model may give values greater than 100% even though such values are not reasonable. 333202_0207.qxd 12/7/05 9:40 AM Page 206 206 Chapter 2 Polynomial and Rational Functions 74. Safe Load The maximum safe load uniformly distributed over a one-foot section of a two-inch-wide wooden beam Load 168.5d 2 472.1, is approximated by the model where is the depth of the beam. d (a) Evaluate the model for d 4,
d 6, d 8, d 10, and d 12. Use the results to create a bar graph. (b) Determine the minimum depth of the beam that will safely support a load of 2000 pounds. 75. Resistors When two resistors of resistances R1 and R2 connected in parallel (see figure), the total resistance satisfies the equation are R Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 77 and 78, determine whether 77. The zeros of the polynomial x3 2x2 11x 12 ≥ 0 divide the real number line into four test intervals. 78. The solution set of the inequality 3 2x2 3x 6 ≥ 0 is the entire set of real numbers. Exploration In Exercises 79–82, find the interval for such that the equation has at least one real solution. b 1 R 1 R1 1 R2. for a parallel circuit in which R1 Find must be at least 1 ohm. R2 2 ohms and R 79. 80. 81. 82. x2 bx 4 0 x2 bx 4 0 3x2 bx 10 0 2x2 bx 5 0 + _ E R1 R2 76. Education The numbers N (in thousands) of master’s degrees earned by women in the United States from 1990 to 2002 are approximated by the model N 0.03t2 9.6t 172 represents the year, with corresponding to (Source: U.S. National Center for t 0 t where 1990 (see figure). Education Statistics) 83. (a) Write a conjecture about the intervals for Exercises 79–82. Explain your reasoning. b in (b) What is the center of each interval for b in Exercises 79–82? 84. Consider the polynomial line shown below. x ax b and the real number a b x (a) Identify the points on the line at which the polynomial is zero. (b) In each of the three subintervals of the line, write the sign of each factor and the sign of the product. (c) For what -values does the polynomial change signs? x Skills Review In Exercises 85–88, factor the expression completely ( 320 300 280 260 240 220 200 180 160 140 2 4 85. 86. 87. 88. 4x2 20x 25 x 32 16 x2x 3 4x 3 2x
4 54x t 14 16 18 In Exercises 89 and 90, write an expression for the area of the region. 8 10 6 Year (0 ↔ 1990) 12 (a) According to the model, during what year did the number of master’s degrees earned by women exceed 220,000? (b) Use the graph to verify the result of part (a). (c) According to the model, during what year will the number of master’s degrees earned by women exceed 320,000? (d) Use the graph to verify the result of part (c). 89. x x2 + 1 90. 3 + 2 b b 333202_020R.qxd 12/7/05 9:43 AM Page 207 2 Chapter Summary What did you learn? Section 2.1 Analyze graphs of quadratic functions (p. 128). Write quadratic functions in standard form and use the results to sketch graphs of functions (p. 131). Use quadratic functions to model and solve real-life problems (p. 133). Section 2.2 Use transformations to sketch graphs of polynomial functions (p. 139). Use the Leading Coefficient Test to determine the end behavior of graphs of polynomial functions (p. 141). Find and use zeros of polynomial functions as sketching aids (p. 142). Use the Intermediate Value Theorem to help locate zeros of polynomial functions (p. 146). Section 2.3 Use long division to divide polynomials by other polynomials (p. 153). Use synthetic division to divide polynomials by binomials of the form x k (p. 156). Use the Remainder Theorem and the Factor Theorem (p. 157). Section 2.4 Use the imaginary unit to write complex numbers (p. 162). Add, subtract, and multiply complex numbers (p. 163). Use complex conjugates to write the quotient of two complex numbers i in standard form (p. 165). Find complex solutions of quadratic equations (p. 166). Section 2.5 Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions (p. 169). Find rational zeros of polynomial functions (p. 170). Find conjugate pairs of complex zeros (p. 173). Use factoring (p. 173), Descartes’s Rule of Signs (p. 176), and
the Upper and Lower Bound Rules (p. 177), to find zeros of polynomials. Section 2.6 Find the domains of rational functions (p. 184). Find the horizontal and vertical asymptotes of graphs of rational functions (p. 185). Analyze and sketch graphs of rational functions (p. 187). Sketch graphs of rational functions that have slant asymptotes (p. 190). Use rational functions to model and solve real-life problems (p. 191). Section 2.7 Solve polynomial inequalities (p. 197), and rational inequalities (p. 201). Use inequalities to model and solve real-life problems (p. 202). Chapter Summary 207 Review Exercises 1, 2 3–18 19–22 23–28 29–32 33–42 43–46 47–52 53–60 61–64 65–68 69–74 75–78 79–82 83–88 89–96 97, 98 99 –110 111–114 115–118 119–130 131–134 135–138 139–146 147, 148 333202_020R.qxd 12/7/05 9:43 AM Page 208 208 Chapter 2 Polynomial and Rational Functions 2 Review Exercises In Exercises 1 and 2, graph each function. Compare 2.1 the graph of each function with the graph of y x2. 1. (a) (b) (c) (d) 2. (a) (b) (c) (d) f x 2x 2 gx 2x 2 hx x 2 2 kx x 22 f x x 2 4 gx 4 x 2 hx x 32 kx 1 2x 2 1 In Exercises 3–14, write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and -intercept(s). x 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. gx x2 2x f x 6x x2 f x x2 8x 10 hx 3 4x x2 f t 2t 2 4t 1 f x x2 8x 12 hx 4x2 4x 13 f x x2 6x 1 hx x2 5x 4 f x 4x 2 4x 5 f x 1 x2 5x 4 3 6x2 24x 22 f x 1 2 In Exercises 15–
18, write the standard form of the equation of the parabola that has the indicated vertex and whose graph passes through the given point. 15. y 16. y 2 −2 −4 −6 17. Vertex: 18. Vertex: (4, 1) 4 (2, −1) x 8 6 2 (0, 3) (2, 2) −2 2 4 6 x 1, 4; 2, 3; point: 2, 3 point: 1, 6 19. Geometry The perimeter of a rectangle is 200 meters. (a) Draw a diagram that gives a visual representation of y, the problem. Label the length and width as respectively. and x x. (b) Write as a function of Use the result to write the y area as a function of x. (c) Of all possible rectangles with perimeters of 200 meters, find the dimensions of the one with the maximum area. 20. Maximum Revenue The total revenue R earned (in dollars) from producing a gift box of candles is given by R p 10p2 800p where p is the price per unit (in dollars). (a) Find the revenues when the prices per box are $20, $25, and $30. (b) Find the unit price that will yield a maximum revenue. What is the maximum revenue? Explain your results. 21. Minimum Cost A soft-drink manufacturer has daily production costs of C 70,000 120x 0.055x 2 C is the total cost (in dollars) and where is the number of units produced. How many units should be produced each day to yield a minimum cost? x 22. Sociology The average age of the groom at a first marriage for a given age of the bride can be approximated by the model y 0.107x2 5.68x 48.5, 20 ≤ x ≤ 25 y is the age of the groom and is the age of the where bride. Sketch a graph of the model. For what age of the bride is the average age of the groom 26? (Source: U.S. Census Bureau) x In Exercises 23–28, sketch the graphs of 2.2 the transformation. y x n and 23. 24. 25. 26. 27. 28. y x3, y x3, y x4, y x 4, y x5, y x5, f x x 43 f x 4x3 f x 2 x 4 f x 2x 24 f
x x 35 f x 1 2x5 3 333202_020R.qxd 12/7/05 9:43 AM Page 209 In Exercises 29–32, describe the right-hand and left-hand behavior of the graph of the polynomial function. 29. 30. 31. 32. f x x 2 6x 9 f x 1 2 x3 2x x4 3x 2 2 gx 3 4 hx x5 7x 2 10x In Exercises 33–38, find all the real zeros of the polynomial function. Determine the multiplicity of each zero and the number of turning points of the graph of the function. Use a graphing utility to verify your answers. 33. 35. 37. f x 2x2 11x 21 f t t 3 3t f x 12x3 20x2 34. 36. 38. f x xx 32 f x x3 8x2 gx x4 x3 2x2 In Exercises 39– 42, sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points. 39. 40. 41. 42. f x x3 x2 2 gx 2x3 4x2 f x xx3 x2 5x 3 hx 3x2 x4 In Exercises 43– 46, (a) use the Intermediate Value Theorem and the table feature of a graphing utility to find intervals one unit in length in which the polynomial function is guaranteed to have a zero. (b) Adjust the table to approximate the zeros of the function. Use the zero or root feature of the graphing utility to verify your results. 43. 44. 45. 46. f x 3x 3 x2 3 f x 0.25x 3 3.65x 6.12 f x x 4 5x 1 f x 7x 4 3x 3 8x2 2 2.3 In Exercises 47–52, use long division to divide. 47. 48. 49. 50. 51. 52. 24x 2 x 8 3x 2 4x 7 3x 2 5x 3 13x 2 x 2 x2 3x 1 3x4 x 2 1 x4 3x3 4x 2 6x 3 x2 2 6x4 10x3
13x 2 5x 2 2x2 1 Review Exercises 209 In Exercises 53–56, use synthetic division to divide. 53. 54. 55. 56. 6x4 4x3 27x 2 18x x 2 0.1x3 0.3x 2 0.5 x 5 2x3 19x 2 38x 24 x 4 3x3 20x2 29x 12 x 3 In Exercises 57 and 58, use synthetic division to determine x whether the given values of are zeros of the function. 57. 58. x 1 f x 20x4 9x3 14x 2 3x (a) f x 3x3 8x 2 20x 16 (a) x 4 x 4 x 3 4 (b) (b) (c) (c) x 0 x 2 3 (d) x 1 (d) x 1 In Exercises 59 and 60, use synthetic division to find each function value. 59. 60. f 3 f x x4 10x3 24x 2 20x 44 (a) (b) gt 2t 5 5t 4 8t 20 (a) (b) g2 g4 f 1 f, In Exercises 61– 64, (a) verify the given factor(s) of the funcf, (b) find the remaining factors of (c) use your results tion f, to write the complete factorization of (d) list all real zeros of and (e) confirm your results by using a graphing utility to graph the function. f, Function 61. 62. 63. 64. f x x3 4x2 25x 28 f x 2x3 11x2 21x 90 f x x 4 4x 3 7x2 22x 24 f x x4 11x3 41x2 61x 30 Factor(s) x 4 x 6 x 2x 3 x 2x 5 In Exercises 65– 68, write the complex number in 2.4 standard form. 65. 67. 6 4 i2 3i 66. 68. 3 25 5i i2 In Exercises 69–74, perform the operation and write the result in standard form. 69. 70. 71. 73. 7 5i 4 2i i 2 2 2 2 2 2 5i13 8i 10 8i2 3i i 2 2 1 6i5 2i 72. 74. i6 i3 2i 333202_020
R.qxd 12/7/05 9:43 AM Page 210 210 Chapter 2 Polynomial and Rational Functions In Exercises 75 and 76, write the quotient in standard form. 75. 6 i 4 i 76. 3 2i 5 i In Exercises 77 and 78, perform the operation and write the result in standard form. 77. 4 2 3i 2 1 i 78. 1 2 i 5 1 4i In Exercises 79– 82, find all solutions of the equation. 79. 81. 3x2 1 0 x2 2x 10 0 80. 82. 2 8x2 0 6x2 3x 27 0 2.5 83. 84. 85. 86. 87. 88. In Exercises 83–88, find all the zeros of the function. f x 3xx 22 f x x 4x 92 f x x2 9x 8 f x x 3 6x f x x 4x 6x 2ix 2i f x x 8x 52x 3 ix 3 i In Exercises 89 and 90, use the Rational Zero Test to list all possible rational zeros of f. 89. 90. f x 4x3 8x 2 3x 15 f x 3x4 4x3 5x 2 8 In Exercises 91–96, find all the rational zeros of the function. 113. f x 91. 92. 93. 94. 95. 96. f x x3 2x2 21x 18 f x 3x3 20x2 7x 30 f x x3 10x2 17x 8 f x x3 9x2 24x 20 f x x 4 x3 11x2 x 12 f x 25x4 25x3 154x2 4x 24 In Exercises 97 and 98, find a polynomial function with real coefficients that has the given zeros. (There are many correct answers.) 97. 2 3, 4, 3 i 98. 2, 3, 1 2i In Exercises 99–102, use the given zero to find all the zeros of the function. Function 99. 100. 101. 102. f x x3 4x2 x 4 h x x3 2x2 16x 32 gx 2x 4 3x 3 13x2 37x 15 f x 4x4 11x3 14x2 6x Zero i 4i 2 i 1 i In Exercises 103–106, find all the z
eros of the function and write the polynomial as a product of linear factors. 103. 104. 105. 106. f x x3 4x2 5x gx x3 7x2 36 gx x4 4x3 3x2 40x 208 f x x4 8x3 8x2 72x 153 In Exercises 107 and 108, use Descartes’s Rule of Signs to determine the possible numbers of positive and negative zeros of the function. 107. 108. gx 5x3 3x 2 6x 9 hx 2x5 4x3 2x 2 5 In Exercises 109 and 110, use synthetic division to verify the upper and lower bounds of the real zeros of f. 109. 110. f x 4x3 3x2 4x 3 x 1 (a) Upper: x 1 (b) Lower: 4 f x 2x3 5x2 14x 8 x 8 (a) Upper: x 4 (b) Lower: In Exercises 111–114, find the domain of the rational 2.6 function. 111. f x 5x x 12 8 x2 10x 24 112. 114. f x 3x2 1 3x f x x2 x 2 x2 4 In Exercises 115–118, identify any horizontal or vertical asymptotes. 115. 117. f x 4 x 3 hx 2x 10 x2 2x 15 116. 118. f x 2x2 5x 3 x2 2 hx x3 4x2 x2 3x 2 In Exercises 119–130, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical and horizontal asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. 119. f x 5 x 2 121. 123. 125. gx 2 x 1 x px 120. 122. 124. f x 4 x hx x 3 x 2 f x 2x 126. hx x 2 4 4 x 12 333202_020R.qxd 12/7/05 9:43 AM Page 211 Review Exercises 211 127. 129. f x 6x2 x 2 1 f x 6x2 11x 3 3x2 x 128. 130. y 2x 2 x 2 4 f x 6x2 7x 2 4x2 1 In Exercises
131–134, (a) state the domain of the function, (b) identify all intercepts, (c) identify any vertical and slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. f x x2 1 x 1 f x 2x3 x2 1 131. 132. 133. 134. f x 3x3 2x2 3x 2 3x2 x 4 f x 3x3 4x2 12x 16 3x2 5x 2 135. Average Cost A business has a production cost of for producing units of a product. The x C 0.5x 500 average cost per unit, 0.5x 500 x C C x C, is given by, x > 0. x Determine the average cost per unit as bound. (Find the horizontal asymptote.) increases without 136. Seizure of Illegal Drugs The cost C dollars) for the federal government to seize illegal drug as it enters the country is given by C 528p 100 p 0 ≤ p < 100., (in millions of p% of an (a) Use a graphing utility to graph the cost function. (b) Find the costs of seizing 25%, 50%, and 75% of the drug. (c) According to this model, would it be possible to seize 100% of the drug? 137. Page Design A page that is inches high contains 30 square inches of print. The top and bottom margins are 2 inches deep and the margins on each side are 2 inches wide. inches wide and x y (a) Draw a diagram that gives a visual representation of the problem. (b) Show that the total area on the page is A A 2x2x 7 x 4. (c) Determine the domain of the function based on the physical constraints of the problem. (d) Use a graphing utility to graph the area function and approximate the page size for which the least amount of paper will be used. Verify your answer numerically using the table feature of the graphing utility. 138. Photosynthesis The amount y of CO2 uptake (in milligrams per square decimeter per hour) at optimal temperatures and with the natural supply of is approximated by the model y 18.47x 2.96 0.23x 1 x > 0 CO2, x where is the light intensity (in watts per square meter). Use a graphing utility to graph the function and determine the limiting
amount of uptake. CO2 2.7 In Exercises 139–146, solve the inequality. 139. 141. 143. 145. 6x2 5x < 4 x3 16x ≥ 0 3 2 x 1 x 1 x2 7x 12 x ≤ ≥ 0 140. 142. 144. 146. 2x2 x ≥ 15 12x3 20x2 < > 147. Investment P dollars invested at interest rate r compounded annually increases to an amount A P1 r2 in 2 years. An investment of $5000 is to increase to an amount greater than $5500 in 2 years. The interest rate must be greater than what percent? 148. Population of a Species A biologist introduces 200 of the P ladybugs into a crop field. The population ladybugs is approximated by the model P 10001 3t 5 t t is the time in days. Find the time required for the where population to increase to at least 2000 ladybugs. Synthesis True or False? In Exercises 149 and 150, determine whether the statement is true or false. Justify your answer. 149. A fourth-degree polynomial with real coefficients can have 5, 8i, 4i, and 5 as its zeros. 150. The domain of a rational function can never be the set of all real numbers. 151. Writing Explain how to determine the maximum or minimum value of a quadratic function. 152. Writing Explain the connections among factors of a polynomial, zeros of a polynomial function, and solutions of a polynomial equation. 153. Writing Describe what is meant by an asymptote of a graph. 333202_020R.qxd 12/7/05 9:43 AM Page 212 212 Chapter 2 Polynomial and Rational Functions 2 y 6 4 2 −4 −2 −4 −6 Chapter Test (0, 3) x 2 4 6 8 (3, −6) FIGURE FOR 2 Take this test as you would take a test in class. When you are finished, check your work against the answers given in the back of the book. 1. Describe how the graph of differs from the graph of g f x x 2. (a) gx 2 x 2 (b) gx x 3 2 2 2. Find an equation of the parabola shown in the figure at the left. 3. The path of a ball is given by x the ball and y 1 20 x 2 3x 5, where
y is the height (in feet) of is the horizontal distance (in feet) from where the ball was thrown. (a) Find the maximum height of the ball. (b) Which number determines the height at which the ball was thrown? Does changing this value change the coordinates of the maximum height of the ball? Explain. 4. Determine the right-hand and left-hand behavior of the graph of the function h t 3 4t 5 2t 2. Then sketch its graph. 5. Divide using long division. 6. Divide using synthetic division. 3x 3 4x 1 x 2 1 2x4 5x 2 3 x 2 7. Use synthetic division to show that f x 4x3 x 2 12x 3. x 3 is a zero of the function given by Use the result to factor the polynomial function completely and list all the real zeros of the function. 8. Perform each operation and write the result in standard form. 10i 3 25 (a) 2 3 i2 3 i (b) 9. Write the quotient in standard form: 5 2 i. In Exercises 10 and 11, find a polynomial function with real coefficients that has the given zeros. (There are many correct answers.) 10. 0, 3, 3 i, 3 i 11. 1 3 i, 1 3 i, 2, 2 In Exercises 12 and 13, find all the zeros of the function. 12. f x x3 2x2 5x 10 13. f x x4 9x2 22x 24 In Exercises 14–16, identify any intercepts and asymptotes of the graph the function. Then sketch a graph of the function. 14. hx 4 x 2 1 15. f x 2x2 5x 12 x2 16 16. gx x 2 2 x 1 In Exercises 17 and 18, solve the inequality. Sketch the solution set on the real number line. 17. 2x2 5x > 12 18. 2 x > 5 x 6 333202_020R.qxd 12/7/05 9:43 AM Page 213 Proofs in Mathematics These two pages contain proofs of four important theorems about polynomial functions. The first two theorems are from Section 2.3, and the second two theorems are from Section 2.5. The Remainder Theorem (p. 157) If a polynomial r f k. is divided
by x k, f x the remainder is Proof From the Division Algorithm, you have f x x kqx rx and because either rx you know that you have at x k, rx 0 must be a constant. That is, or the degree of rx is less than the degree of rx r. Now, by evaluating x k, f x f k k kqk r 0qk r r. To be successful in algebra, it is important that you understand the connection among factors of a polynomial, zeros of a polynomial function, and solutions or roots of a polynomial equation. The Factor Theorem is the basis for this connection. The Factor Theorem (p. 157) A polynomial has a factor x k f x if and only if f k 0. Proof Using the Division Algorithm with the factor x k, you have f x x kqx rx. By the Remainder Theorem, rx r f k, and you have f x x kqx fk where qx is a polynomial of lesser degree than f x. If f k 0, then f x x kqx x k f x by and you see that f x, division of Theorem, you have f k 0. is a factor of x k is a factor of yields a remainder of 0. So, by the Remainder Conversely, if x k f x. 213 333202_020R.qxd 12/7/05 9:43 AM Page 214 Proofs in Mathematics The Fundamental Theorem of Algebra The Linear Factorization Theorem is closely related to the Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra has a long and interesting history. In the early work with polynomial equations, The Fundamental Theorem of Algebra was thought to have been not true, because imaginary solutions were not considered. In fact, in the very early work by mathematicians such as Abu al-Khwarizmi (c. 800 A.D.), negative solutions were also not considered. Once imaginary numbers were accepted, several mathematicians attempted to give a general proof of the Fundamental Theorem of Algebra. These included Gottfried von Leibniz (1702), Jean d’Alembert (1746), Leonhard Euler (1749), JosephLouis Lagrange (1772), and Pierre Simon Laplace (1795). The mathematician usually credited with the first correct proof of the
Fundamental Theorem of Algebra is Carl Friedrich Gauss, who published the proof in his doctoral thesis in 1799. f x Linear Factorization Theorem (p. 169) n > 0, is a polynomial of degree where If linear factors f x an n, x c1 c1, c2,..., cn... x cn x c2 are complex numbers. where then has precisely f n Proof Using the Fundamental Theorem of Algebra, you know that must have at least one zero, and you have is a factor of f x, f x c1 c1. f x x c1 f1 Consequently, f1 x x. If the degree of Theorem to conclude that must have a zero which implies that is greater than zero, you again apply the Fundamental f x x c1 f1 x c2 f2 f1 It is clear that the degree of n that you can repeatedly apply the Fundamental Theorem x. x n 1, c2, is that the degree of f2 x and times until you obtain n 2, is x c1 x c2... x cn f x an an where is the leading coefficient of the polynomial f x. Factors of a Polynomial with real coefficients can be written as Every polynomial of degree the product of linear and quadratic factors with real coefficients, where the quadratic factors have no real zeros. (p. 173) n > 0 Proof To begin, you use the Linear Factorization Theorem to conclude that completely factored in the form f x can be f x dx c1 x c2 x c3... x cn. ci If each is real, there is nothing more to prove. If any b 0, then, because the coefficients of a bi cj gate obtain a bi, are real, you know that the conjuis also a zero. By multiplying the corresponding factors, you is complex f x ci ci x ci x cj x a bix a bi x2 2ax a2 b2 where each coefficient is real. 214 333202_020R.qxd 12/7/05 9:43 AM Page 215 P.S. Problem Solving This collection of thought-provoking and challenging exercises further explores and expands upon concepts learned in this chapter. 1. Show that if r ak3 bk2 ck d f x ax3 bx2 cx d f k r
, then using long division. In where other words, verify the Remainder Theorem for a thirddegree polynomial function. 2. In 2000 B.C., the Babylonians solved polynomial equations by referring to tables of values. One such table gave the values of To be able to use this table, the Babylonians sometimes had to manipulate the equation as shown below. y3 y2. ax3 bx2 c Original equation a3 x3 b3 3 a2 x2 b2 2 ax b ax b a2 c b3 a2 c b3 Multiply each side by a2 b3. Rewrite. a2cb3 Then they would find column of the table. Because they knew that the corresponding -value was equal to y x bya. 2, 3,..., 10. Record the they could conclude that y 1, axb, (a) Calculate in the for y3 y2 y3 y2 values in a table. Use the table from part (a) and the method above to solve each equation. (b) (c) (d) (e) (f) (g) x3 x2 252 x3 2x2 288 3x3 x2 90 2x3 5x2 2500 7x3 6x2 1728 10x3 3x2 297 Using the methods from this chapter, verify your solution to each equation. 3. At a glassware factory, molten cobalt glass is poured into molds to make paperweights. Each mold is a rectangular prism whose height is 3 inches greater than the length of each side of the square base. A machine pours 20 cubic inches of liquid glass into each mold. What are the dimensions of the mold? 4. Determine whether the statement is true or false. If false, provide one or more reasons why the statement is false and f x ax3 bx2 cx d, correct the statement. Let a 0, f x x 1 qx 2 f 2 1. and let x 1 Then where qx is a second-degree polynomial. y ax2 bx c. 5. The parabola shown in the figure has an equation of the form Find the equation of this parabola by the following methods. (a) Find the equation analytically. (b) Use the regression feature of a graphing utility to find the equation. x y 2 (2, 2) (4, 0) −4 −2

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