Polynomial and Rational Functions

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1 Polnomial and Rational Functions. Quadratic Functions and Models. Polnomial Functions of Higher Degree. Polnomial and Snthetic Division. Comple Numbers.5 Zeros of Polnomial Functions.6 Rational Functions.7 Nonlinear Inequalities Quadratic functions are often used to model real-life phenomena, such as the path of a diver. Martin Rose/Bongarts/Gett Images SELECTED APPLICATIONS Polnomial and rational functions have man real-life applications. The applications listed below represent a small sample of the applications in this chapter. Path of a Diver, Eercise 77, page 6 Data Analsis: Home Prices, Eercises 9 96, page 5 Data Analsis: Cable Television, Eercise 7, page 6 Advertising Cost, Eercise 05, page 8 Athletics, Eercise 09, page 8 Reccling, Eercise, page 95 Average Speed, Eercise 79, page 96 Height of a Projectile, Eercise 67, page 05 7

2 8 Chapter Polnomial and Rational Functions. Quadratic Functions and Models What ou should learn Analze graphs of quadratic functions. Write quadratic functions in standard form and use the results to sketch graphs of functions. Use quadratic functions to model and solve real-life problems. Wh ou should learn it Quadratic functions can be used to model data to analze consumer behavior. For instance, in Eercise 8 on page 7, ou will use a quadratic function to model the revenue earned from manufacturing handheld video games. The Graph of a Quadratic Function In this and the net section, ou will stud the graphs of polnomial functions. In Section.6, ou were introduced to the following basic functions. f a b f c Linear function Constant function f Squaring function These functions are eamples of polnomial functions. Definition of Polnomial Function Let n be a nonnegative integer and let a n, a n,..., a, a, a 0 be real numbers with a n 0. The function given b f a n n a n n... a a a 0 is called a polnomial function of with degree n. Polnomial functions are classified b degree. For instance, a constant function has degree 0 and a linear function has degree. In this section, ou will stud second-degree polnomial functions, which are called quadratic functions. For instance, each of the following functions is a quadratic function. f 6 g h 9 k m Note that the squaring function is a simple quadratic function that has degree. John Henle/Corbis Definition of Quadratic Function Let a, b, and c be real numbers with a 0. The function given b f a b c Quadratic function is called a quadratic function. The graph of a quadratic function is a special tpe of U -shaped curve called a parabola. Parabolas occur in man real-life applications especiall those involving reflective properties of satellite dishes and flashlight reflectors. You will stud these properties in Section 0..

3 All parabolas are smmetric with respect to a line called the ais of smmetr, or simpl the ais of the parabola. The point where the ais intersects the parabola is the verte of the parabola, as shown in Figure.. If the leading coefficient is positive, the graph of f a b c is a parabola that opens upward. If the leading coefficient is negative, the graph of f a b c is a parabola that opens downward. Section. Quadratic Functions and Models 9 Opens upward Ais f ( ) = a + b + c, a < 0 Verte is highest point Verte is lowest point f ( ) = a + b + c, a > 0 Ais Opens downward Leading coefficient is positive. Leading coefficient is negative. FIGURE. The simplest tpe of quadratic function is f a. Its graph is a parabola whose verte is (0, 0). If a > 0, the verte is the point with the minimum -value on the graph, and if a < 0, the verte is the point with the maimum -value on the graph, as shown in Figure.. Eploration Graph a for a,, 0.5, 0.5,, and. How does changing the value of a affect the graph? Graph h for h,,, and. How does changing the value of h affect the graph? Graph k for k,,, and. How does changing the value of k affect the graph? f () = a, a> 0 Leading coefficient is positive. FIGURE. Minimum: (0, 0) Maimum: (0, 0) Leading coefficient is negative. When sketching the graph of f a, it is helpful to use the graph of as a reference, as discussed in Section.7. f () = a, a< 0

4 0 Chapter Polnomial and Rational Functions Eample Sketching Graphs of Quadratic Functions a. Compare the graphs of and f. b. Compare the graphs of and g. Solution a. Compared with, each output of f shrinks b a factor of creating the broader parabola shown in Figure.. b. Compared with, each output of g stretches b a factor of, creating the narrower parabola shown in Figure.., = g () = f () = = FIGURE. FIGURE. Now tr Eercise 9. In Eample, note that the coefficient a determines how widel the parabola given b f a opens. If is small, the parabola opens more widel than if a a is large. Recall from Section.7 that the graphs of f ± c, f ± c, f, and f are rigid transformations of the graph of f. For instance, in Figure.5, notice how the graph of can be transformed to produce the graphs of f and g. (0, ) = f() = + g() = ( + ) = (, ) Reflection in -ais followed b an upward shift of one unit FIGURE.5 Left shift of two units followed b a downward shift of three units

5 Section. Quadratic Functions and Models The standard form of a quadratic function identifies four basic transformations of the graph of. a. The factor a produces a vertical stretch or shrink. b. If a < 0, the graph is reflected in the -ais. c. The factor h represents a horizontal shift of h units. d. The term k represents a vertical shift of k units. The Standard Form of a Quadratic Function The standard form of a quadratic function is f a h k. This form is especiall convenient for sketching a parabola because it identifies the verte of the parabola as h, k. Standard Form of a Quadratic Function The quadratic function given b f a h k, a 0 is in standard form. The graph of f is a parabola whose ais is the vertical line h and whose verte is the point h, k. If a > 0, the parabola opens upward, and if a < 0, the parabola opens downward. To graph a parabola, it is helpful to begin b writing the quadratic function in standard form using the process of completing the square, as illustrated in Eample. In this eample, notice that when completing the square, ou add and subtract the square of half the coefficient of within the parentheses instead of adding the value to each side of the equation as is done in Appendi A.5. Eample Graphing a Parabola in Standard Form Sketch the graph of f 8 7 and identif the verte and the ais of the parabola. Solution Begin b writing the quadratic function in standard form. Notice that the first step in completing the square is to factor out an coefficient of that is not. f 8 7 Write original function. 7 Factor out of -terms. 7 Add and subtract within parentheses. f () = ( + ) (, ) = FIGURE.6 = After adding and subtracting within the parentheses, ou must now regroup the terms to form a perfect square trinomial. The can be removed from inside the parentheses; however, because of the outside of the parentheses, ou must multipl b, as shown below. f 7 Regroup terms. 8 7 Simplif. Write in standard form. From this form, ou can see that the graph of f is a parabola that opens upward and has its verte at,. This corresponds to a left shift of two units and a downward shift of one unit relative to the graph of, as shown in Figure.6. In the figure, ou can see that the ais of the parabola is the vertical line through the verte,. Now tr Eercise.

6 Chapter Polnomial and Rational Functions To find the -intercepts of the graph of f a b c, ou must solve the equation a b c 0. If a b c does not factor, ou can use the Quadratic Formula to find the -intercepts. Remember, however, that a parabola ma not have -intercepts. Eample Finding the Verte and -Intercepts of a Parabola f() = ( ) + (, ) (, 0) (, 0) 5 = FIGURE.7 Sketch the graph of f 6 8 and identif the verte and -intercepts. Solution f Write original function. Factor out of -terms Regroup terms. Write in standard form. From this form, ou can see that f is a parabola that opens downward with verte,. The -intercepts of the graph are determined as follows Add and subtract 9 within parentheses. Factor out. 0 0 Factor. Set st factor equal to 0. Set nd factor equal to 0. So, the -intercepts are, 0 and, 0, as shown in Figure.7. Now tr Eercise 9. Eample Writing the Equation of a Parabola (, ) = f() Write the standard form of the equation of the parabola whose verte is, and that passes through the point 0, 0, as shown in Figure.8. Solution Because the verte of the parabola is at h, k,, the equation has the form f a. Substitute for h and k in standard form. (0, 0) Because the parabola passes through the point 0, 0, it follows that f 0 0. So, 0 a 0 a Substitute 0 for ; solve for a. which implies that the equation in standard form is f. FIGURE.8 Now tr Eercise.

7 Applications Section. Quadratic Functions and Models Man applications involve finding the maimum or minimum value of a quadratic function. You can find the maimum or minimum value of a quadratic function b locating the verte of the graph of the function. Verte of a Parabola The verte of the graph of f is b a, f b a b c. If a > 0, has a minimum at b a.. If a < 0, has a maimum at b a. a. Eample 5 The Maimum Height of a Baseball Height (in feet) FIGURE.9 Baseball f() = (56.5, 8.5) Distance (in feet) A baseball is hit at a point feet above the ground at a velocit of 00 feet per second and at an angle of 5 with respect to the ground. The path of the baseball is given b the function f 0.00, where f is the height of the baseball (in feet) and is the horizontal distance from home plate (in feet). What is the maimum height reached b the baseball? Solution From the given function, ou can see that a 0.00 and b. Because the function has a maimum when b a, ou can conclude that the baseball reaches its maimum height when it is feet from home plate, where is b a b a feet. At this distance, the maimum height is f feet. The path of the baseball is shown in Figure.9. Now tr Eercise 77. Eample 6 Minimizing Cost A small local soft-drink manufacturer has dail production costs of C 70, , where C is the total cost (in dollars) and is the number of units produced. How man units should be produced each da to ield a minimum cost? Solution Use the fact that the function has a minimum when b a. From the given function ou can see that a and b 0. So, producing b a (0.075 each da will ield a minimum cost. units Now tr Eercise 8.

8 Chapter Polnomial and Rational Functions. Eercises VOCABULARY CHECK: Fill in the blanks.. A polnomial function of degree and leading coefficient is a function of the form f a n n a n n... n a n a a 0 a n 0 where n is a and a are numbers.. A function is a second-degree polnomial function, and its graph is called a.. The graph of a quadratic function is smmetric about its.. If the graph of a quadratic function opens upward, then its leading coefficient is and the verte of the graph is a. 5. If the graph of a quadratic function opens downward, then its leading coefficient is and the verte of the graph is a. In Eercises 8, match the quadratic function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), (f), (g), and (h).] (a) (c) (e) (g) (, ) 6 (, 0) (, ) (, 0) (b) (d) (f) (h) (0, ) 6 6 (, 0) 6 8 (, ) 6 (0, ). f. f. f. f 5. f ( ) 6. f 7. f 8. f k In Eercises 9, graph each function. Compare the graph of each function with the graph of. 9. (a) f (b) g 8 (c) h (d) k 0. (a) f (b) g (c) h (d). (a) f (b) g (c) h (d) k. (a) (b) (c) f g h (d) k In Eercises 8, sketch the graph of the quadratic function without using a graphing utilit. Identif the verte, ais of smmetr, and -intercept(s).. f 5. h 5 5. f 6. f 6 7. f f 6 9. h g. f 5. f. f 5. f 5. h f f 8. f 6

9 Section. Quadratic Functions and Models 5 In Eercises 9 6, use a graphing utilit to graph the quadratic function. Identif the verte, ais of smmetr, and -intercepts. Then check our results algebraicall b writing the quadratic function in standard form. 9. f 0. f 0. g 8. f 0. f 6. f 5. g 6. f In Eercises 7, find the standard form of the quadratic function (, ) (, 0).. (, ) (, 0) (0, ) (, 0) (, 0) 6 (, 0) (, 0) (, 0) (, ) 6 In Eercises 5, write the standard form of the equation of the parabola that has the indicated verte and whose graph passes through the given point.. Verte:, 5 ; point: 0, 9. Verte:, ; point:, 5. Verte:, ; point:, 6. Verte:, ; point: 0, 7. Verte: 5, ; point: 7, 5 8. Verte:, ; point:, 0 9. Verte:, ; point:, (0, ) (, 0) (0, ) 6 (, ) Verte: 5, ; point:, 5. Verte: 5 point: 7, 0 ;, 6 5. Verte: 6, 6 ; point: Graphical Reasoning In Eercises 5 56, determine the -intercept(s) of the graph visuall. Then find the -intercepts algebraicall to confirm our results In Eercises 57 6, use a graphing utilit to graph the quadratic function. Find the -intercepts of the graph and compare them with the solutions of the corresponding quadratic equation when f f 58. f f f f f 5 6. f f , 6 In Eercises 65 70, find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given -intercepts. (There are man correct answers.) 65., 0,, , 0, 5, , 0, 0, 0 68., 0, 8, 0 69., 0, 70. 5, 0, 0,,

10 6 Chapter Polnomial and Rational Functions In Eercises 7 7, find two positive real numbers whose product is a maimum. 7. The sum is The sum is S. 7. The sum of the first and twice the second is. 7. The sum of the first and three times the second is. 75. Numerical, Graphical, and Analtical Analsis A rancher has 00 feet of fencing to enclose two adjacent rectangular corrals (see figure). (c) Use the result of part (b) to write the area A of the rectangular region as a function of. What dimensions will produce a maimum area of the rectangle? 77. Path of a Diver The path of a diver is given b 9 9 where is the height (in feet) and is the horizontal distance from the end of the diving board (in feet). What is the maimum height of the diver? 78. Height of a Ball The height (in feet) of a punted football is given b where is the horizontal distance (in feet) from the point at which the ball is punted (see figure). (a) Write the area A of the corral as a function of. (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 maimum enclosed area. (c) Use a graphing utilit to graph the area function. Use the graph to approimate the dimensions that will produce the maimum enclosed area. (d) Write the area function in standard form to find analticall the dimensions that will produce the maimum area. (e) Compare our results from parts (b), (c), and (d). 76. Geometr An indoor phsical fitness room consists of a rectangular region with a semicircle on each end (see figure). The perimeter of the room is to be a 00-meter single-lane running track. (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. (b) Use the result of part (a) to write an equation, in terms of and, for the distance traveled in one lap around the track. Solve for. (a) How high is the ball when it is punted? (b) What is the maimum height of the punt? (c) How long is the punt? 79. Minimum Cost A manufacturer of lighting fitures has dail production costs of C where C is the total cost (in dollars) and is the number of units produced. How man fitures should be produced each da to ield a minimum cost? 80. Minimum Cost A tetile manufacturer has dail production costs of C 00, where C is the total cost (in dollars) and is the number of units produced. How man units should be produced each da to ield a minimum cost? 8. Maimum Profit The profit P (in dollars) for a compan that produces antivirus and sstem utilities software is P ,000 Not drawn to scale where is the number of units sold. What sales level will ield a maimum profit?

11 Section. Quadratic Functions and Models 7 8. Maimum Profit The profit P (in hundreds of dollars) that a compan makes depends on the amount (in hundreds of dollars) the compan spends on advertising according to the model P What ependiture for advertising will ield a maimum profit? 8. Maimum Revenue The total revenue R earned (in thousands of dollars) from manufacturing handheld video games is given b R p 5p 00p where p is the price per unit (in dollars). (a) Find the revenue earned for each price per unit given below. $0 $5 $0 (b) Find the unit price that will ield a maimum revenue. What is the maimum revenue? Eplain our results. 8. Maimum Revenue The total revenue R earned per da (in dollars) from a pet-sitting service is given b R p p 50p where p is the price charged per pet (in dollars). (a) Find the revenue earned for each price per pet given below. $ $6 $8 (b) Find the price that will ield a maimum revenue. What is the maimum revenue? Eplain our results. 85. Graphical Analsis From 960 to 00, the per capita consumption C of cigarettes b Americans (age 8 and older) can be modeled b C 99.8t.6t, 0 t where t is the ear, with t 0 corresponding to 960. (Source: Tobacco Outlook Report) (a) Use a graphing utilit to graph the model. (b) Use the graph of the model to approimate the maimum average annual consumption. Beginning in 966, all cigarette packages were required b law to carr a health warning. Do ou think the warning had an effect? Eplain. (c) In 000, the U.S. population (age 8 and over) was 09,8,09. Of those, about 8,08,590 were smokers. What was the average annual cigarette consumption per smoker in 000? What was the average dail cigarette consumption per smoker? 86. Data Analsis The numbers (in thousands) of hairdressers and cosmetologists in the United States for the ears 99 through 00 are shown in the table. (Source: U.S. Bureau of Labor Statistics) Year Model It (a) Use a graphing utilit to create a scatter plot of the data. Let represent the ear, with corresponding to 99. (b) Use the regression feature of a graphing utilit to find a quadratic model for the data. (c) Use a graphing utilit 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 utilit to approimate the ear in which the number of hairdressers and cosmetologists was the least. (e) Verif our answer to part (d) algebraicall. (f) Use the model to predict the number of hairdressers and cosmetologists in Wind Drag The number of horsepower required to overcome wind drag on an automobile is approimated b 0.00s 0.005s 0.09, Number of hairdressers and cosmetologists, s 00 where s is the speed of the car (in miles per hour). (a) Use a graphing utilit to graph the function. (b) Graphicall estimate the maimum speed of the car if the power required to overcome wind drag is not to eceed 0 horsepower. Verif our estimate algebraicall.

12 8 Chapter Polnomial and Rational Functions 88. Maimum Fuel Econom A stud was done to compare the speed (in miles per hour) with the mileage (in miles per gallon) of an automobile. The results are shown in the table. (Source: Federal Highwa Administration) (a) Use a graphing utilit to create a scatter plot of the data. (b) Use the regression feature of a graphing utilit to find a quadratic model for the data. (c) Use a graphing utilit 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. Snthesis True or False? In Eercises 89 and 90, determine whether the statement is true or false. Justif our answer. 89. The function given b f has no -intercepts. 90. The graphs of and have the same ais of smmetr. 9. Write the quadratic function in standard form to verif that the verte occurs at b a, f b a. Speed, f 0 7 g 0 f a b c Mileage, Profit The profit P (in millions of dollars) for a recreational vehicle retailer is modeled b a quadratic function of the form where t represents the ear. If ou were president of the compan, which of the models below would ou prefer? Eplain our reasoning. (a) a is positive and b a t. (b) a is positive and t b a. (c) a is negative and b a t. (d) a is negative and t b a. 9. Is it possible for a quadratic equation to have onl one -intercept? Eplain. 9. Assume that the function given b has two real zeros. Show that the -coordinate of the verte of the graph is the average of the zeros of f. (Hint: Use the Quadratic Formula.) Skills Review In Eercises 95 98, find the equation of the line in slope-intercept form that has the given characteristics. 95. Passes through the points, and, 96. Passes through the point 7, and has a slope of 97. Passes through the point 0, and is perpendicular to the line Passes through the point 8, and is parallel to the line In Eercises 99 0, let f and let g 8. Find the indicated value. 99. f g P at bt c f a b c, g f fg 7 g f.5 0. f g 0. g f 0 a Make a Decision To work an etended application analzing the height of a basketball after it has been dropped, visit this tet s website at college.hmco.com.

13 Section. Polnomial Functions of Higher Degree 9. Polnomial Functions of Higher Degree What ou should learn Use transformations to sketch graphs of polnomial functions. Use the Leading Coefficient Test to determine the end behavior of graphs of polnomial functions. Find and use zeros of polnomial functions as sketching aids. Use the Intermediate Value Theorem to help locate zeros of polnomial functions. Wh ou should learn it You can use polnomial functions to analze business situations such as how revenue is related to advertising epenses, as discussed in Eercise 98 on page 5. Graphs of Polnomial Functions In this section, ou will stud basic features of the graphs of polnomial functions. The first feature is that the graph of a polnomial function is continuous. Essentiall, this means that the graph of a polnomial function has no breaks, holes, or gaps, as shown in Figure.0(a). The graph shown in Figure.0(b) is an eample of a piecewise-defined function that is not continuous. (a) Polnomial functions have continuous graphs. FIGURE.0 (b) Functions with graphs that are not continuous are not polnomial functions. The second feature is that the graph of a polnomial function has onl smooth, rounded turns, as shown in Figure.. A polnomial function cannot have a sharp turn. For instance, the function given b f, which has a sharp turn at the point 0, 0, as shown in Figure., is not a polnomial function. Bill Aron /PhotoEdit, Inc. 6 5 f() = (0, 0) Polnomial functions have graphs Graphs of polnomial functions with smooth rounded turns. cannot have sharp turns. FIGURE. FIGURE. The graphs of polnomial functions of degree greater than are more difficult to analze than the graphs of polnomials of degree 0,, or. However, using the features presented in this section, coupled with our knowledge of point plotting, intercepts, and smmetr, ou should be able to make reasonabl accurate sketches b hand.

14 0 Chapter Polnomial and Rational Functions For power functions given b f n, if n is even, then the graph of the function is smmetric with respect to the -ais, and if n is odd, then the graph of the function is smmetric with respect to the origin. The polnomial functions that have the simplest graphs are monomials of the form f n, where n is an integer greater than zero. From Figure., ou can see that when n is even, the graph is similar to the graph of f, and when n is odd, the graph is similar to the graph of f. Moreover, the greater the value of n, the flatter the graph near the origin. Polnomial functions of the form f n are often referred to as power functions. = (, ) = = 5 (, ) = (, ) (, ) (a) If n is even, the graph of n touches the ais at the -intercept. FIGURE. (b) If n is odd, the graph of n crosses the ais at the -intercept. Eample Sketching Transformations of Monomial Functions Sketch the graph of each function. a. f 5 b. h Solution a. Because the degree of f 5 is odd, its graph is similar to the graph of. In Figure., note that the negative coefficient has the effect of reflecting the graph in the -ais. b. The graph of h, as shown in Figure.5, is a left shift b one unit of the graph of. (, ) h() = ( + ) f() = 5 (, ) (0, ) (, ) (, 0) FIGURE. FIGURE.5 Now tr Eercise 9.

15 Section. Polnomial Functions of Higher Degree Eploration For each function, identif the degree of the function and whether the degree of the function is even or odd. Identif the leading coefficient and whether the leading coefficient is positive or negative. Use a graphing utilit 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. f b. f 5 5 c. f 5 5 d. f 5 e. f f. f g. f The Leading Coefficient Test In Eample, note that both graphs eventuall rise or fall without bound as moves to the right. Whether the graph of a polnomial function eventuall rises or falls can be determined b the function s degree (even or odd) and b its leading coefficient, as indicated in the Leading Coefficient Test. Leading Coefficient Test As moves without bound to the left or to the right, the graph of the polnomial function f a n n... a a 0 eventuall rises or falls in the following manner.. When n is odd: f() as f() as f() as f() as If the leading coefficient is positive a n > 0, the graph falls to the left and rises to the right.. When n is even: If the leading coefficient is negative a n < 0, the graph rises to the left and falls to the right. The notation f as indicates that the graph falls to the left. The notation f as indicates that the graph rises to the right. f() as f() as f() as f() as If the leading coefficient is positive a n > 0, the graph rises to the left and right. If the leading coefficient is negative a n < 0, the graph falls to the left and right. The dashed portions of the graphs indicate that the test determines onl the right-hand and left-hand behavior of the graph.

16 Chapter Polnomial and Rational Functions Eample Appling the Leading Coefficient Test A polnomial function is written in standard form if its terms are written in descending order of eponents from left to right. Before appling the Leading Coefficient Test to a polnomial function, it is a good idea to check that the polnomial function is written in standard form. Describe the right-hand and left-hand behavior of the graph of each function. a. f b. f 5 c. f 5 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.6. b. Because the degree is even and the leading coefficient is positive, the graph rises to the left and right, as shown in Figure.7. 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.8. Eploration For each of the graphs in Eample, count the number of zeros of the polnomial function and the number of relative minima and relative maima. Compare these numbers with the degree of the polnomial. What do ou observe? f() = + FIGURE.6 Now tr Eercise 5. f() = FIGURE.7 FIGURE.8 f() = 5 In Eample, note that the Leading Coefficient Test tells ou onl whether the graph eventuall rises or falls to the right or left. Other characteristics of the graph, such as intercepts and minimum and maimum points, must be determined b other tests. Remember that the zeros of a function of are the -values for which the function is zero. Zeros of Polnomial Functions It can be shown that for a polnomial function f of degree n, the following statements are true.. The function f has, at most, n real zeros. (You will stud this result in detail in the discussion of the Fundamental Theorem of Algebra in Section.5.). The graph of f has, at most, n turning points. (Turning points, also called relative minima or relative maima, are points at which the graph changes from increasing to decreasing or vice versa.) Finding the zeros of polnomial functions is one of the most important problems in algebra. There is a strong interpla between graphical and algebraic approaches to this problem. Sometimes ou can use information about the graph of a function to help find its zeros, and in other cases ou can use information about the zeros of a function to help sketch its graph. Finding zeros of polnomial functions is closel related to factoring and finding -intercepts.

17 Section. Polnomial Functions of Higher Degree Real Zeros of Polnomial Functions If f is a polnomial function and a is a real number, the following statements are equivalent.. a is a zero of the function f.. a is a solution of the polnomial equation f 0.. a is a factor of the polnomial f.. a, 0 is an -intercept of the graph of f. Eample Finding the Zeros of a Polnomial Function Find all real zeros of f (). Then determine the number of turning points of the graph of the function. Algebraic Solution To find the real zeros of the function, set f equal to zero and solve for Set f equal to 0. Remove common monomial factor. Factor completel. So, the real zeros are 0,, and. Because the function is a fourth-degree polnomial, the graph of f can have at most turning points. Graphical Solution Use a graphing utilit to graph. In Figure.9, the graph appears to have zeros at 0, 0,, 0, and, 0. Use the zero or root feature, or the zoom and trace features, of the graphing utilit to verif these zeros. So, the real zeros are 0,, and. From the figure, ou can see that the graph has three turning points. This is consistent with the fact that a fourth-degree polnomial can have at most three turning points. = + Now tr Eercise 7. FIGURE.9 In Eample, note that because k is even, the factor ields the repeated zero 0. The graph touches the -ais at 0, as shown in Figure.9. Repeated Zeros A factor a k, k >, ields a repeated zero a of multiplicit k.. If k is odd, the graph crosses the -ais at a.. If k is even, the graph touches the -ais (but does not cross the -ais) at a.

18 Chapter Polnomial and Rational Functions Eample uses an algebraic approach to describe the graph of the function. A graphing utilit is a complement to this approach. Remember that an important aspect of using a graphing utilit 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 Eample. a. b. Technolog To graph polnomial functions, ou can use the fact that a polnomial function can change signs onl at its zeros. Between two consecutive zeros, a polnomial must be entirel positive or entirel negative. This means that when the real zeros of a polnomial function are put in order, the 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 polnomial function is positive (the graph lies above the -ais) or negative (the graph lies below the -ais). Eample Sketching the Graph of a Polnomial Function Sketch the graph of f. Solution. Appl the Leading Coefficient Test. Because the leading coefficient is positive and the degree is even, ou know that the graph eventuall rises to the left and to the right (see Figure.0).. Find the Zeros of the Polnomial. B factoring f as f, ou can see that the zeros of f are 0 and (both of odd multiplicit). So, the -intercepts occur at 0, 0 and, 0. Add these points to our graph, as shown in Figure.0.. Plot a Few Additional Points. Use the zeros of the polnomial to find the test intervals. In each test interval, choose a representative -value and evaluate the polnomial function, as shown in the table. Test interval Representative Value of f Sign Point on -value graph, 0 f 7 Positive, , f Negative,,.5 f Positive.5, Draw the Graph. Draw a continuous curve through the points, as shown in Figure.. Because both zeros are of odd multiplicit, ou know that the graph should cross the -ais at 0 and. f() = If ou are unsure of the shape of a portion of the graph of a polnomial function, plot some additional points, such as the point 0.5, 0.5 as shown in Figure.. Up to left (0, 0) ) Up to right, 0) FIGURE.0 FIGURE. Now tr Eercise 67.

19 Section. Polnomial Functions of Higher Degree 5 Eample 5 Sketching the Graph of a Polnomial Function Sketch the graph of Solution f Appl the Leading Coefficient Test. Because the leading coefficient is negative and the degree is odd, ou know that the graph eventuall rises to the left and falls to the right (see Figure.).. Find the Zeros of the Polnomial. B factoring f ou can see that the zeros of f are 0 (odd multiplicit) and (even multiplicit). So, the -intercepts occur at and 0, 0, 0. Add these points to our graph, as shown in Figure... Plot a Few Additional Points. Use the zeros of the polnomial to find the test intervals. In each test interval, choose a representative -value and evaluate the polnomial function, as shown in the table. Observe in Eample 5 that the sign of f is positive to the left of and negative to the right of the zero 0. Similarl, the sign of f is negative to the left and to the right of the zero. This suggests that if the zero of a polnomial function is of odd multiplicit, then the sign of f changes from one side of the zero to the other side. If the zero is of even multiplicit, then the sign of f does not change from one side of the zero to the other side.. Draw the Graph. Draw a continuous curve through the points, as shown in Figure.. As indicated b the multiplicities of the zeros, the graph crosses the -ais at 0, 0 but does not cross the -ais at, 0. Test interval Representative Value of f Sign Point on -value graph, 0 0,, Up to left 6 5 (0, 0) Down to right (, 0) 0.5 f 0.5 f () = + 6 Positive 0.5 f 0.5 Negative f Negative, 0.5, 0.5, 9 FIGURE. FIGURE. Now tr Eercise 69.

20 6 Chapter Polnomial and Rational Functions The Intermediate Value Theorem The net theorem, called the Intermediate Value Theorem, illustrates the eistence of real zeros of polnomial functions. This theorem implies that if a, f a and b, f b are two points on the graph of a polnomial function such that f a f b, then for an number d between f a and f b there must be a number c between a and b such that f c d. (See Figure..) fb ( ) fc () = d fa ( ) a c b FIGURE. Intermediate Value Theorem Let a and b be real numbers such that a < b. If f is a polnomial function such that f a f b, then, in the interval a, b, f takes on ever value between f a and f b. The Intermediate Value Theorem helps ou locate the real zeros of a polnomial function in the following wa. If ou can find a value a at which a polnomial function is positive, and another value b at which it is negative, ou can conclude that the function has at least one real zero between these two values. For eample, the function given b f is negative when and positive when. Therefore, it follows from the Intermediate Value Theorem that f must have a real zero somewhere between and, as shown in Figure.5. f () = + + (, ) f( ) = f has a zero between and. (, ) f( ) = FIGURE.5 B continuing this line of reasoning, ou can approimate an real zeros of a polnomial function to an desired accurac. This concept is further demonstrated in Eample 6.

21 Section. Polnomial Functions of Higher Degree 7 Eample 6 Approimating a Zero of a Polnomial Function Use the Intermediate Value Theorem to approimate the real zero of f. Solution Begin b computing a few function values, as follows. f () = + (0, ) (, ) f has a zero (, ) between 0.8 and 0.7. FIGURE.6 f 0 Because f is negative and f 0 is positive, ou can appl the Intermediate Value Theorem to conclude that the function has a zero between and 0. To pinpoint this zero more closel, divide the interval, 0 into tenths and evaluate the function at each point. When ou do this, ou will find that f and f So, f must have a zero between 0.8 and 0.7, as shown in Figure.6. For a more accurate approimation, compute function values between f 0.8 and f 0.7 and appl the Intermediate Value Theorem again. B continuing this process, ou can approimate this zero to an desired accurac. Now tr Eercise 85. You can use the table feature of a graphing utilit to approimate the zeros of a polnomial function. For instance, for the function given b f Technolog create a table that shows the function values for 0 0, as shown in the first table at the right. Scroll through the table looking for consecutive function values that differ in sign. From the table, ou can see that f 0 and f differ in sign. So, ou can conclude from the Intermediate Value Theorem that the function has a zero between 0 and. You can adjust our table to show function values for 0 using increments of 0., as shown in the second table at the right. B scrolling through the table ou can see that f 0.8 and f 0.9 differ in sign. So, the function has a zero between 0.8 and 0.9. If ou repeat this process several times, ou should obtain as the zero of the function. Use the zero or root feature of a graphing utilit to confirm this result.

22 8 Chapter Polnomial and Rational Functions. Eercises VOCABULARY CHECK: Fill in the blanks.. The graphs of all polnomial functions are, which means that the graphs have no breaks, holes, or gaps.. The is used to determine the left-hand and right-hand behavior of the graph of a polnomial function.. A polnomial function of degree n has at most real zeros and at most turning points.. If a is a zero of a polnomial function f, then the following three statements are true. (a) a is a of the polnomial equation f 0. (b) is a factor of the polnomial f. (c) a, 0 is an of the graph f. 5. If a real zero of a polnomial function is of even multiplicit, then the graph of f the -ais at a, and if it is of odd multiplicit then the graph of f the -ais at a. 6. A polnomial function is written in form if its terms are written in descending order of eponents from left to right. 7. The Theorem states that if f is a polnomial function such that f a f b, then in the interval a, b, f takes on ever value between f a and f b. In Eercises 8, match the polnomial function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), (f), (g), and (h).] (g) (h) (a) (b) 8 6 (c) (e) (d) (f) f. f. f 5. f 5. f 6. f 7. f 8. f In Eercises 9, sketch the graph of n and each transformation. f 5 9. (a) f (b) f (c) f (d) f 0. 5 (a) f 5 (b) (c) f 5 (d) f 5. (a) f (b) f (c) f (d) f (e) f (f) f

23 Section. Polnomial Functions of Higher Degree 9. 6 (a) f 8 6 (c) f 6 (e) f 6 (b) (d) f 6 (f) f 6 In Eercises, describe the right-hand and left-hand behavior of the graph of the polnomial function. f. f g h f. 5 f f 6 5 f 5. h t t 5t. f s 7 8 s 5s 7s Graphical Analsis In Eercises 6, use a graphing utilit to graph the functions f and g in the same viewing window. Zoom out sufficientl far to show that the right-hand and left-hand behaviors of f and g appear identical f 6 f 9, g f, g f 6, g f 6, g In Eercises 7, (a) find all the real zeros of the polnomial function, (b) determine the multiplicit of each zero and the number of turning points of the graph of the function, and (c) use a graphing utilit to graph the function and verif our answers. 7. f 5 8. f 9 9. h t t 6t 9 0. f f f 5 f. g f t t t t f 0 g t t 5 6t 9t f 5 6 f f 0. g. f 5 00 Graphical Analsis In Eercises 6, (a) use a graphing utilit to graph the function, (b) use the graph to approimate an -intercepts of the graph, (c) set 0 and solve the resulting equation, and (d) compare the results of part (c) with an -intercepts of the graph In Eercises 7 56, find a polnomial function that has the given zeros. (There are man correct answers.) 7. 0, , 9., 6 50., ,, 5. 0,, 5 5.,,, 0 5.,, 0,, 55., 56., 5, 5 In Eercises 57 66, find a polnomial of degree n that has the given zero(s). (There are man correct answers.) Zero(s) , 59., 0, 60.,, ,, ,, 6.,,, , 66.,, 5, 6 Degree n n n n n n n n n 5 n 5 In Eercises 67 80, sketch the graph of the function b (a) appling the Leading Coefficient Test, (b) finding the zeros of the polnomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points. 67. f g 69. f t t t g f 7. f f 5 8 f 5 f f f 78. h 79. g t t t 80. g 0

24 50 Chapter Polnomial and Rational Functions In Eercises 8 8, use a graphing utilit to graph the function. Use the zero or root feature to approimate the real zeros of the function. Then determine the multiplicit of each zero. 8. f 8. f 8. g h 5 5 In Eercises 85 88, use the Intermediate Value Theorem and the table feature of a graphing utilit to find intervals one unit in length in which the polnomial function is guaranteed to have a zero. Adjust the table to approimate the zeros of the function. Use the zero or root feature of a graphing utilit to verif our results. 85. f 86. f g 88. h Numerical and Graphical Analsis An open bo is to be made from a square piece of material, 6 inches on a side, b cutting equal squares with sides of length from the corners and turning up the sides (see figure). 90. Maimum Volume An open bo with locking tabs is to be made from a square piece of material inches on a side. This is to be done b cutting equal squares from the corners and folding along the dashed lines shown in the figure. in. in. (a) Verif that the volume of the bo is given b the function V 8 6. (b) Determine the domain of the function V. (c) Sketch a graph of the function and estimate the value of for which V is maimum. 9. Construction A roofing contractor is fabricating gutters from -inch aluminum sheeting. The contractor plans to use an aluminum siding folding press to create the gutter b creasing equal lengths for the sidewalls (see figure). 6 (a) Verif that the volume of the bo is given b the function V 6. (b) Determine the domain of the function. (c) Use a graphing utilit to create a table that shows the bo height and the corresponding volumes V. Use the table to estimate the dimensions that will produce a maimum volume. (d) Use a graphing utilit to graph V and use the graph to estimate the value of for which V is maimum. Compare our result with that of part (c). (a) Let represent the height of the sidewall of the gutter. Write a function A that represents the cross-sectional area of the gutter. (b) The length of the aluminum sheeting is 6 feet. Write a function V that represents the volume of one run of gutter in terms of. (c) Determine the domain of the function in part (b). (d) Use a graphing utilit to create a table that shows the sidewall height and the corresponding volumes V. Use the table to estimate the dimensions that will produce a maimum volume. (e) Use a graphing utilit to graph V. Use the graph to estimate the value of for which V is a maimum. Compare our result with that of part (d). (f) Would the value of change if the aluminum sheeting were of different lengths? Eplain.

25 Section. Polnomial Functions of Higher Degree 5 9. Construction An industrial propane tank is formed b adjoining two hemispheres to the ends of a right circular clinder. The length of the clindrical portion of the tank is four times the radius of the hemispherical components (see figure). r r (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 utilit to graph the function. (d) The total volume of the tank is to be 0 cubic feet. Use the graph from part (c) to estimate the radius and length of the clindrical portion of the tank. Data Analsis: Home Prices In Eercise 9 96, use the table, which shows the median prices (in thousands of dollars) of new privatel owned U.S. homes in the Midwest and in the South for the ears 997 through 00.The data can be approimated b the following models. 0.9t.t 5.t t.7t.8t 9 In the models, t represents the ear, with t 7 corresponding to 997. (Source: U.S. Census Bureau; U.S. Department of Housing and Urban Development) Year, t Use a graphing utilit to plot the data and graph the model for in the same viewing window. How closel does the model represent the data? 9. Use a graphing utilit to plot the data and graph the model for in the same viewing window. How closel does the model represent the data? 95. Use the models to predict the median prices of a new privatel owned home in both regions in 008. Do our answers seem reasonable? Eplain. 96. Use the graphs of the models in Eercises 9 and 9 to write a short paragraph about the relationship between the median prices of homes in the two regions. 97. Tree Growth The growth of a red oak tree is approimated b the function 98. Revenue The total revenue R (in millions of dollars) for a compan is related to its advertising epense b the function R 00, , where is the amount spent on advertising (in tens of thousands 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 rapidl. This point is called the point of diminishing returns because an epense above this amount will ield less return per dollar invested in advertising. Revenue (in millions of dollars) (a) Use a graphing utilit to graph the function. (Hint: Use a viewing window in which 0 5 and 5 60.) (b) Estimate the age of the tree when it is growing most rapidl. This point is called the point of diminishing returns because the increase in size will be less with each additional ear. (c) Using calculus, the point of diminishing returns can also be found b finding the verte of the parabola given b 0.009t 0.7t Find the verte of this parabola. (d) Compare our results from parts (b) and (c) R Model It G 0.00t 0.7t 0.58t 0.89 where G is the height of the tree (in feet) and t t is its age (in ears) Advertising epense (in tens of thousands of dollars)

26 5 Chapter Polnomial and Rational Functions Snthesis True or False? In Eercises 99 0, determine whether the statement is true or false. Justif our answer. 99. A fifth-degree polnomial can have five turning points in its graph. 00. It is possible for a sith-degree polnomial to have onl one solution. 0. The graph of the function given b f rises to the left and falls to the right. 0. Graphical Analsis For each graph, describe a polnomial function that could represent the graph. (Indicate the degree of the function and the sign of its leading coefficient.) (a) (b) 0. Eploration Eplore the transformations of the form g a h 5 k. (a) Use a graphing utilit to graph the functions given b 5 and 5 5. Determine whether the graphs are increasing or decreasing. Eplain. (b) Will the graph of g alwas be increasing or decreasing? If so, is this behavior determined b a, h, or k? Eplain. (c) Use a graphing utilit to graph the function given b H 5. Use the graph and the result of part (b) to determine whether H can be written in the form H a h 5 k. Eplain. Skills Review In Eercises 05 08, factor the epression completel. (c) (d) In Eercises 09, solve the equation b factoring Graphical Reasoning Sketch a graph of the function given b f. Eplain how the graph of each function g differs (if it does) from the graph of each function f. Determine whether g is odd, even, or neither. (a) g f (b) g f (c) g f (d) (e) (f ) (g) (h) g f g f g f g f g f f In Eercises 6, solve the equation b completing the square In Eercises 7, describe the transformation from a common function that occurs in f. Then sketch its graph. 7. f 8. f 9. f 5 0. f 7 6. f 9. f 0

27 Section. Polnomial and Snthetic Division 5. Polnomial and Snthetic Division What ou should learn Use long division to divide polnomials b other polnomials. Use snthetic division to divide polnomials b binomials of the form k. Use the Remainder Theorem and the Factor Theorem. Wh ou should learn it Snthetic division can help ou evaluate polnomial functions. For instance, in Eercise 7 on page 60, ou will use snthetic division to determine the number of U.S. militar personnel in 008. (, 0 ) (, 0) FIGURE.7 Kevin Fleming/Corbis f() = Long Division of Polnomials In this section, ou will stud two procedures for dividing polnomials. These procedures are especiall valuable in factoring and finding the zeros of polnomial functions. To begin, suppose ou are given the graph of f Notice that a zero of f occurs at, as shown in Figure.7. Because is a zero of f, ou know that is a factor of f. This means that there eists a second-degree polnomial q such that f q. To find q, ou can use long division, as illustrated in Eample. Eample Long Division of Polnomials Divide b, and use the result to factor the polnomial completel. Solution 6 7 ) From this division, ou can conclude that and b factoring the quadratic 6 7, ou have Note that this factorization agrees with the graph shown in Figure.7 in that the three -intercepts occur at, and,. Now tr Eercise 5. 6 Think 6. 7 Think 7. Think. Multipl: 6. Subtract. Multipl: 7. Subtract. Multipl:. Subtract.

28 5 Chapter Polnomial and Rational Functions In Eample, is a factor of the polnomial 6 9 6, and the long division process produces a remainder of zero. Often, long division will produce a nonzero remainder. For instance, if ou divide 5 b, ou obtain the following. Divisor Quotient Dividend Remainder In fractional form, ou can write this result as follows. Dividend 5 Divisor This implies that ) 5 5 Remainder Quotient Divisor 5 ( Multipl each side b. which illustrates the following theorem, called the Division Algorithm. The Division Algorithm If f and d are polnomials such that d 0, and the degree of d is less than or equal to the degree of f, there eist unique polnomials q and r such that f d q r Dividend Quotient Divisor Remainder where r 0 or the degree of r is less than the degree of d. If the remainder r is zero, d divides evenl into f. The Division Algorithm can also be written as f r q d d. In the Division Algorithm, the rational epression f d is improper because the degree of f is greater than or equal to the degree of d. On the other hand, the rational epression r d is proper because the degree of r is less than the degree of d.

29 Section. Polnomial and Snthetic Division 55 Before ou appl the Division Algorithm, follow these steps.. Write the dividend and divisor in descending powers of the variable.. Insert placeholders with zero coefficients for missing powers of the variable. Eample Long Division of Polnomials Divide b. Solution Because there is no -term or -term in the dividend, ou need to line up the subtraction b using zero coefficients (or leaving spaces) for the missing terms. ) So, divides evenl into, and ou can write, 0. Now tr Eercise. You can check the result of Eample b multipling. Eample Long Division of Polnomials Divide 5 b. Solution ) 5 6 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 5 Now tr Eercise 5..

30 56 Chapter Polnomial and Rational Functions Snthetic Division There is a nice shortcut for long division of polnomials when dividing b divisors of the form k. This shortcut is called snthetic division. The pattern for snthetic division of a cubic polnomial is summarized as follows. (The pattern for higher-degree polnomials is similar.) Snthetic Division (for a Cubic Polnomial) To divide a b c d b k, use the following pattern. k a b c d ka a r Remainder Coefficients of quotient Coefficients of dividend Vertical pattern: Add terms. Diagonal pattern: Multipl b k. Snthetic division works onl for divisors of the form k. [Remember that k k. ]You cannot use snthetic division to divide a polnomial b a quadratic such as. Eample Using Snthetic Division Use snthetic division to divide 0 b. Solution You should set up the arra as follows. Note that a zero is included for the missing -term in the dividend. 0 0 Then, use the snthetic division pattern b adding terms in columns and multipling the results b. Divisor: Dividend: Remainder: So, ou have 0 Quotient:. Now tr Eercise 9.

31 Section. Polnomial and Snthetic Division 57 The Remainder and Factor Theorems The remainder obtained in the snthetic division process has an important interpretation, as described in the Remainder Theorem. The Remainder Theorem If a polnomial f is divided b k, the remainder is r f k. For a proof of the Remainder Theorem, see Proofs in Mathematics on page. The Remainder Theorem tells ou that snthetic division can be used to evaluate a polnomial function. That is, to evaluate a polnomial function f when k, divide f b k. The remainder will be f k, as illustrated in Eample 5. Eample 5 Using the Remainder Theorem Use the Remainder Theorem to evaluate the following function at. f Solution Using snthetic division, ou obtain the following. 8 6 Because the remainder is r 9, ou can conclude that f r f k This means that, 9 is a point on the graph of f. You can check this b substituting in the original function. Check f Now tr Eercise 5. Another important theorem is the Factor Theorem, stated below. This theorem states that ou can test to see whether a polnomial has k as a factor b evaluating the polnomial at k. If the result is 0, k is a factor. The Factor Theorem A polnomial f has a factor k if and onl if f k 0. For a proof of the Factor Theorem, see Proofs in Mathematics on page.

32 58 Chapter Polnomial and Rational Functions Eample 6 Factoring a Polnomial: Repeated Division f() = (, 0 ( (, 0) (, 0) 0 FIGURE (, 0) Show that and are factors of f Then find the remaining factors of f. Solution Using snthetic division with the factor, ou obtain the following remainder, so f 0 and is a factor. Take the result of this division and perform snthetic division again using the factor. 6 Because the resulting quadratic epression factors as 5 the complete factorization of f is f. Note that this factorization implies that f,,, and. has four real zeros: This is confirmed b the graph of f, which is shown in Figure.8. Now tr Eercise remainder, so f 0 and is a factor. Uses of the Remainder in Snthetic Division The remainder r, obtained in the snthetic division of f b k, provides the following information.. The remainder r gives the value of f at k. That is, r f k.. If r 0, k is a factor of f.. If r 0, k, 0 is an -intercept of the graph of f. Throughout this tet, the importance of developing several problem-solving strategies is emphasized. In the eercises for this section, tr using more than one strateg to solve several of the eercises. For instance, if ou find that k divides evenl into f (with no remainder), tr sketching the graph of f. You should find that k, 0 is an -intercept of the graph.

33 Section. Polnomial and Snthetic Division 59. Eercises VOCABULARY CHECK:. Two forms of the Division Algorithm are shown below. Identif and label each term or function. f d q r f r q d d In Eercises 5, fill in the blanks.. The rational epression p q 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.. An alternative method to long division of polnomials is called, in which the divisor must be of the form k.. The Theorem states that a polnomial f has a factor k if and onl if f k The Theorem states that if a polnomial f is divided b k, the remainder is r f k. Analtical Analsis In Eercises and, use long division to verif that... Graphical Analsis In Eercises and, (a) use a graphing utilit to graph the two equations in the same viewing window, (b) use the graphs to verif that the epressions are equivalent, and (c) use long division to verif the results algebraicall...,, 5 5, 5, In Eercises 5 8, use long division to divide In Eercises 9 6, use snthetic division to divide In Eercises 7, write the function in the form f k q r for the given value of k, and demonstrate that f k r. Function Value of k 7. f k 8. f 5 8 k

34 60 Chapter Polnomial and Rational Functions Function 9. f f 0. f. f 5. f 6. f Value of k k k 5 k k 5 k k In Eercises 5 8, use snthetic division to find each function value. Verif our answers using another method. 5. f 0 (a) f (b) f (c) f (d) f 8 6. g 6 (a) g (b) g (c) g (d) g 7. h 5 0 (a) h (b) h (c) h (d) h 5 8. f (a) f (b) f (c) f 5 (d) f 0 In Eercises 9 56, use snthetic division to show that is a solution of the third-degree polnomial equation, and use the result to factor the polnomial completel. List all real solutions of the equation. Polnomial Equation Value of In Eercises 57 6, (a) verif the given factors of the function f, (b) find the remaining factors of f, (c) use our results to write the complete factorization of f, (d) list all real zeros of f, and (e) confirm our results b using a graphing utilit to graph the function. Function 57. f f f f Factors,, 5,, Function 6. f f f f 8 Factors, 5, 5, 5, Graphical Analsis In Eercises 65 68, (a) use the zero or root feature of a graphing utilit to approimate the zeros of the function accurate to three decimal places, (b) determine one of the eact zeros, and (c) use snthetic division to verif our result from part (b), and then factor the polnomial completel. 65. f g h t t t 7t 68. f s s s 0s In Eercises 69 7, simplif the rational epression b using long division or snthetic division Model It 7. Data Analsis: Militar Personnel The numbers M (in thousands) of United States militar personnel on active dut for the ears 99 through 00 are shown in the table, where t represents the ear, with t corresponding to 99. (Source: U.S. Department of Defense) Year,t Militar personnel, M

35 Section. Polnomial and Snthetic Division 6 7. Data Analsis: Cable Television The average monthl basic rates R (in dollars) for cable television in the United States for the ears 99 through 00 are shown in the table, where t represents the ear, with t corresponding to 99. (Source: Kagan Research LLC) (a) Use a graphing utilit to create a scatter plot of the data. (b) Use the regression feature of the graphing utilit 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 snthetic division to evaluate the model for the ear 008. Snthesis True or False? In Eercises 75 77, determine whether the statement is true or false. Justif our answer. 75. If 7 is a factor of some polnomial function f, then is a zero of f. 7 Model It (continued) (a) Use a graphing utilit to create a scatter plot of the data. (b) Use the regression feature of the graphing utilit 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 snthetic division to evaluate the model for the ear 008. Even though the model is relativel accurate for estimating the given data, would ou use this model to predict the number of militar personnel in the future? Eplain. Year, t Basic rate, R is a factor of the polnomial The rational epression 0 is improper. 78. Eploration Use the form f k q r to create a cubic function that (a) passes through the point, 5 and rises to the right, and (b) passes through the point, and falls to the right. (There are man correct answers.) Think About It In Eercises 79 and 80, perform the division b assuming that n is a positive integer. n 9 n 7 n n 8. Writing Briefl eplain what it means for a divisor to divide evenl into a dividend. 8. Writing Briefl eplain how to check polnomial division, and justif our reasoning. Give an eample. Eploration In Eercises 8 and 8, find the constant c such that the denominator will divide evenl into the numerator. c 5 c Think About It In Eercises 85 and 86, answer the questions about the division f k, where f. 85. What is the remainder when k? Eplain. 86. If it is necessar to find f, is it easier to evaluate the function directl or to use snthetic division? Eplain. Skills Review n n 5 n 6 n In Eercises 87 9, use an method to solve the quadratic equation In Eercises 9 96, find a polnomial function that has the given zeros. (There are man correct answers.) 9. 0,, 9. 6, 95.,, 96.,,,

36 6 Chapter Polnomial and Rational Functions. Comple Numbers What ou should learn Use the imaginar unit i to write comple numbers. Add,subtract, and multipl comple numbers. Use comple conjugates to write the quotient of two comple numbers in standard form. Find comple solutions of quadratic equations. Wh ou should learn it You can use comple numbers to model and solve real-life problems in electronics. For instance, in Eercise 8 on page 68, ou will learn how to use comple numbers to find the impedance of an electrical circuit. The Imaginar Unit i You have learned that some quadratic equations have no real solutions. For instance, the quadratic equation 0 has no real solution because there is no real number that can be squared to produce. To overcome this deficienc, mathematicians created an epanded sstem of numbers using the imaginar unit i, defined as i Imaginar unit where i. B adding real numbers to real multiples of this imaginar unit, the set of comple numbers is obtained. Each comple number can be written in the standard form a bi. For instance, the standard form of the comple number 5 9 is 5 i because i. In the standard form a bi, the real number a is called the real part of the comple number a bi, and the number bi (where b is a real number) is called the imaginar part of the comple number. Definition of a Comple Number If a and b are real numbers, the number a bi is a comple number, and it is said to be written in standard form. If b 0, the number a bi a is a real number. If b 0, the number a bi is called an imaginar number. A number of the form bi, where b 0, is called a pure imaginar number. The set of real numbers is a subset of the set of comple numbers, as shown in Figure.9. This is true because ever real number a can be written as a comple number using b 0. That is, for ever real number a, ou can write a a 0i. Real numbers Imaginar numbers Comple numbers Richard Megna/Fundamental Photographs FIGURE.9 Equalit of Comple Numbers Two comple numbers a bi and c di, written in standard form, are equal to each other a bi c di if and onl if a c and b d. Equalit of two comple numbers

37 Operations with Comple Numbers Section. Comple Numbers 6 To add (or subtract) two comple numbers, ou add (or subtract) the real and imaginar parts of the numbers separatel. Addition and Subtraction of Comple Numbers If a bi and c di are two comple numbers written in standard form, their sum and difference are defined as follows. Sum: a bi c di a c b d i Difference: a bi c di a c b d i The additive identit in the comple number sstem is zero (the same as in the real number sstem). Furthermore, the additive inverse of the comple number a bi is (a bi) a bi. Additive inverse So, ou have a bi a bi 0 0i 0. Eample Adding and Subtracting Comple Numbers a. 7i 6i 7i 6i ( ) (7i 6i) 5 i Remove parentheses. Group like terms. Write in standard form. b. ( i) i i i Remove parentheses. c. d. i i 0 i i 5i i i 5i 0 5i 5i 0 0i 0 Now tr Eercise 7. Group like terms. Simplif. i i 5i i i 7 i i i 7 i Write in standard form. 7 i i i Note in Eamples (b) and (d) that the sum of two comple numbers can be a real number.

38 6 Chapter Polnomial and Rational Functions Eploration Complete the following. i i i i i i i 5 i 6 i 7 i 8 i 9 i 0 i i What pattern do ou see? Write a brief description of how ou would find i raised to an positive integer power. Man of the properties of real numbers are valid for comple numbers as well. Here are some eamples. Associative Properties of Addition and Multiplication Commutative Properties of Addition and Multiplication Distributive Propert of Multiplication Over Addition Notice below how these properties are used when two comple numbers are multiplied. a bi c di a c di bi c di Distributive Propert ac ad i bc i bd i Distributive Propert ac ad i bc i bd i ac bd ad i bc i Commutative Propert ac bd ad bc i Associative Propert Rather than tring to memorize this multiplication rule, ou should simpl remember how the Distributive Propert is used to multipl two comple numbers. Eample Multipling Comple Numbers The procedure described above is similar to multipling two polnomials and combining like terms, as in the FOIL Method shown in Appendi A.. For instance, ou can use the FOIL Method to multipl the two comple numbers from Eample (b). F O I L i i 8 6i i i a. i i Distributive Propert 8 i Simplif. b. Distributive Propert i i i i i 8 6i i i 8 6i i Distributive Propert i 8 6i i i Group like terms. Write in standard form. c. ( i)( i) i i i 9 6i 6i i 9 6i 6i Distributive Propert Distributive Propert i 9 Simplif. Write in standard form. d. i i i Square of a binomial i i i 9 6i 6i i 9 6i 6i Distributive Propert Distributive Propert i 9 i 5 i Simplif. Write in standard form. Now tr Eercise 7.

39 Comple Conjugates Section. Comple Numbers 65 Notice in Eample (c) that the product of two comple numbers can be a real number. This occurs with pairs of comple numbers of the form a bi and a bi, called comple conjugates. a bi a bi a abi abi b i a b a b Eample Multipling Conjugates Multipl each comple number b its comple conjugate. a. i b. i Solution a. The comple conjugate of i is i. i i i b. The comple conjugate of i is i. i i i 6 9i Now tr Eercise 7. Note that when ou multipl the numerator and denominator of a quotient of comple numbers b c di c di ou are actuall multipling the quotient b a form of. You are not changing the original epression, ou are onl creating an epression that is equivalent to the original epression. To write the quotient of a bi and c di in standard form, where c and d are not both zero, multipl the numerator and denominator b the comple conjugate of the denominator to obtain a bi a bi c di c di c di c di Eample ac bd bc ad i c d. Standard form Writing a Quotient of Comple Numbers in Standard Form i i i i i i 8 i i 6i 6 i 8 6 6i 6 6i i Now tr Eercise 9. Multipl numerator and denominator b comple conjugate of denominator. Epand. i Simplif. Write in standard form.

40 66 Chapter Polnomial and Rational Functions Comple Solutions of Quadratic Equations When using the Quadratic Formula to solve a quadratic equation, ou often obtain a result such as, which ou know is not a real number. B factoring out i, ou can write this number in standard form. i The number i is called the principal square root of. The definition of principal square root uses the rule ab a b for a > 0 and b < 0. This rule is not valid if both a and b are negative. For eample, whereas 5i 5i 5i 5i To avoid problems with square roots of negative numbers, be sure to convert comple numbers to standard form before multipling. a. b. c. Principal Square Root of a Negative Number If a is a positive number, the principal square root of the negative number a is defined as a ai. Eample 5 Writing Comple Numbers in Standard Form i i 6 i i 7 i i i i i Eample 6 i Now tr Eercise 59. Comple Solutions of a Quadratic Equation Solve (a) 0 and (b) 5 0. Solution a. 0 ±i Write original equation. Subtract from each side. Etract square roots. b. 5 0 Write original equation. ± 5 ± 56 6 ± i 6 ± i i i i Now tr Eercise 65. Quadratic Formula Simplif. Write 56 in standard form. Write in standard form.

41 Section. Comple Numbers 67. Eercises VOCABULARY CHECK:. Match the tpe of comple number with its definition. (a) Real Number (i) a bi, a 0, (b) Imaginar number (ii) a bi, a 0, (c) Pure imaginar number (iii) a bi, b 0 b 0 b 0 In Eercises, fill in the blanks.. The imaginar unit i is defined as i, where i.. If a is a positive number, the root of the negative number a is defined as a a i.. The numbers a bi and a bi are called, and their product is a real number a b. In Eercises, find real numbers a and b such that the equation is true.. a bi 0 6i. a bi i. a b i 5 8i. a 6 bi 6 5i In Eercises 5 6, write the comple number in standard form i i. i i In Eercises 7 6, perform the addition or subtraction and write the result in standard form i 6 i 8. i 5 6i 9. 8 i i 0. i 6 i i. i 7i. 5 8i 0i 5. 5 i 5 i 6..6.i 5.8.i In Eercises 7 6, perform the operation and write the result in standard form. 7. i i 8. 6 i i 9. 6i 5 i 0. 8i 9 i. 0i 0i. 5i 5i. 5i. i 5. i i 6. i i In Eercises 7, write the comple conjugate of the comple number.then multipl the number b its comple conjugate i 8. 7 i 9. 5i 0. i In Eercises 5 5, write the quotient in standard form i i i i 9. i 6 7i 50. i i i 8 6i 5. i i 5. i 5i 5. 5i i In Eercises 55 58, perform the operation and write the result in standard form. 55. i i i i i 8i 58. i i 5 i i i i

42 68 Chapter Polnomial and Rational Functions In Eercises 59 6, write the comple number in standard form In Eercises 65 7, use the Quadratic Formula to solve the quadratic equation t t In Eercises 75 8, simplif the comple number and write it in standard form i i 76. i i 77. 5i i i i Resistor Inductor Capacitor Smbol aω bω cω Impedance Model It 8. Impedance The opposition to current in an electrical circuit is called its impedance. The impedance z in a parallel circuit with two pathwas satisfies the equation z z z where z is the impedance (in ohms) of pathwa and z is the impedance of pathwa. (a) The impedance of each pathwa in a parallel circuit is found b adding the impedances of all components in the pathwa. Use the table to find z and z. (b) Find the impedance z. a bi ci 8. Cube each comple number. (a) (b) i (c) i 85. Raise each comple number to the fourth power. (a) (b) (c) i (d) i 86. Write each of the powers of i as i, i,, or. (a) i 0 (b) i 5 (c) i 50 (d) i 67 Snthesis True or False? In Eercises 87 89, determine whether the statement is true or false. Justif our answer. 87. There is no comple number that is equal to its comple conjugate. 88. i 6 is a solution of i i 50 i 7 i 09 i Error Analsis Describe the error Proof Prove that the comple conjugate of the product of two comple numbers a b i and a b i is the product of their comple conjugates. 9. Proof Prove that the comple conjugate of the sum of two comple numbers a b i and a b i is the sum of their comple conjugates. Skills Review In Eercises 9 96, perform the operation and write the result in standard form In Eercises 97 00, solve the equation and check our solution Volume of an Oblate Spheroid Solve for a: V a b 0. Newton s Law of Universal Gravitation 6 Ω 9 Ω 0 Ω 0 Ω Solve for r: m m F r 0. Miture Problem A five-liter container contains a miture with a concentration of 50%. How much of this miture must be withdrawn and replaced b 00% concentrate to bring the miture up to 60% concentration?

43 Section.5 Zeros of Polnomial Functions 69.5 Zeros of Polnomial Functions What ou should learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polnomial functions. Find rational zeros of polnomial functions. Find conjugate pairs of comple zeros. Find zeros of polnomials b factoring. Use Descartes s Rule of Signs and the Upper and Lower Bound Rules to find zeros of polnomials. Wh ou should learn it Finding zeros of polnomial functions is an important part of solving real-life problems. For instance, in Eercise on page 8, the zeros of a polnomial function can help ou analze the attendance at women s college basketball games. The Fundamental Theorem of Algebra You know that an nth-degree polnomial can have at most n real zeros. In the comple number sstem, this statement can be improved. That is, in the comple number sstem, ever nth-degree polnomial function has precisel n zeros. This important result is derived from the Fundamental Theorem of Algebra, first proved b the German mathematician Carl Friedrich Gauss ( ). The Fundamental Theorem of Algebra If f is a polnomial of degree n, where n > 0, then f has at least one zero in the comple number sstem. Using the Fundamental Theorem of Algebra and the equivalence of zeros and factors, ou obtain the Linear Factorization Theorem. Linear Factorization Theorem If f is a polnomial of degree n, where n > 0, then f has precisel n linear factors f a n c c... c n where c, c,..., c n are comple numbers. For a proof of the Linear Factorization Theorem, see Proofs in Mathematics on page. Note that the Fundamental Theorem of Algebra and the Linear Factorization Theorem tell ou onl that the zeros or factors of a polnomial eist, not how to find them. Such theorems are called eistence theorems. Eample Zeros of Polnomial Functions Recall that in order to find the zeros of a function f, set f equal to 0 and solve the resulting equation for. For instance, the function in Eample (a) has a zero at because 0. a. The first-degree polnomial f has eactl one zero:. b. Counting multiplicit, the second-degree polnomial function f 6 9 has eactl two zeros: and. (This is called a repeated zero.) c. The third-degree polnomial function f i i has eactl three zeros: 0, i, and i. d. The fourth-degree polnomial function f i i has eactl four zeros:,, i, and i. Now tr Eercise.

44 70 Chapter Polnomial and Rational Functions The Rational Zero Test The Rational Zero Test relates the possible rational zeros of a polnomial (having integer coefficients) to the leading coefficient and to the constant term of the polnomial. The Rational Zero Test If the polnomial f a n n a n n... a a a 0 has integer coefficients, ever rational zero of f has the form Fogg Art Museum Historical Note Although the were not contemporaries,jean Le Rond d Alembert (77 78) worked independentl of Carl Gauss in tring to prove the Fundamental Theorem of Algebra. His efforts were such that, in France, the Fundamental Theorem of Algebra is frequentl known as the Theorem of d Alembert. Rational zero p q where p and q have no common factors other than, and p a factor of the constant term a 0 q a factor of the leading coefficient a n. To use the Rational Zero Test, ou 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 an, are actual zeros of the polnomial. Note that when the leading coefficient is, the possible rational zeros are simpl the factors of the constant term. Eample Rational Zero Test with Leading Coefficient of f() = + + FIGURE.0 Find the rational zeros of f. Solution Because the leading coefficient is, the possible rational zeros are ±, the factors of the constant term. B testing these possible zeros, ou can see that neither works. f f So, ou can conclude that the given polnomial has no rational zeros. Note from the graph of f in Figure.0 that f does have one real zero between and 0. However, b the Rational Zero Test, ou know that this real zero is not a rational number. Now tr Eercise 7.

45 Section.5 Zeros of Polnomial Functions 7 Eample Rational Zero Test with Leading Coefficient of When the list of possible rational zeros is small, as in Eample, it ma be quicker to test the zeros b evaluating the function. When the list of possible rational zeros is large, as in Eample, it ma be quicker to use a different approach to test the zeros, such as using snthetic division or sketching a graph. Find the rational zeros of f 6. Solution Because the leading coefficient is, the possible rational zeros are the factors of the constant term. Possible rational zeros: ±, ±, ±, ±6 B appling snthetic division successivel, ou can determine that and are the onl two rational zeros. 0 So, f factors as 0 f remainder, so is a zero. 0 remainder, so is a zero. Because the factor produces no real zeros, ou can conclude that and are the onl real zeros of f, which is verified in Figure f () = (, 0) (, 0) FIGURE. Now tr Eercise. If the leading coefficient of a polnomial is not, the list of possible rational zeros can increase dramaticall. In such cases, the search can be shortened in several was: () a programmable calculator can be used to speed up the calculations; () a graph, drawn either b hand or with a graphing utilit, can give a good estimate of the locations of the zeros; () the Intermediate Value Theorem along with a table generated b a graphing utilit can give approimations of zeros; and () snthetic 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 b working with the lower-degree polnomial obtained in snthetic division, as shown in Eample.

46 7 Chapter Polnomial and Rational Functions Eample Using the Rational Zero Test Remember that when ou tr to find the rational zeros of a polnomial function with man possible rational zeros, as in Eample, ou 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 ma be helpful. Find the rational zeros of f 8. Solution The leading coefficient is and the constant term is. Factors of ±, ± Possible rational zeros: Factors of ±, ± ±, ±, ±, ± B snthetic division, ou can determine that is a rational zero. 5 So, f factors as 8 5 f 5 0 and ou can conclude that the rational zeros of f are,, and. Now tr Eercise 7. 5 Recall from Section. that if a is a zero of the polnomial function then a is a solution of the polnomial equation f 0. f, f () = FIGURE Eample 5 Solving a Polnomial Equation Find all the real solutions of Solution The leading coefficient is 0 and the constant term is. Possible rational solutions: With so man possibilities (, in fact), it is worth our time to stop and sketch a graph. From Figure., it looks like three reasonable solutions would be 6 5,, and. Testing these b snthetic division shows that is the onl rational solution. So, ou have Using the Quadratic Formula for the second factor, ou find that the two additional solutions are irrational numbers. and Now tr Eercise. Factors of ±, ±, ±, ±, ±6, ± Factors of 0 ±, ±, ±5, ±0

47 Conjugate Pairs Section.5 Zeros of Polnomial Functions 7 In Eample (c) and (d), note that the pairs of comple zeros are conjugates. That is, the are of the form a bi and a bi. Comple Zeros Occur in Conjugate Pairs Let f be a polnomial function that has real coefficients. If a bi, where b 0, is a zero of the function, the conjugate a bi is also a zero of the function. Be sure ou see that this result is true onl if the polnomial function has real coefficients. For instance, the result applies to the function given b f but not to the function given b g i. Eample 6 Finding a Polnomial with Given Zeros Find a fourth-degree polnomial function with real coefficients that has,, and i as zeros. Solution Because i is a zero and the polnomial is stated to have real coefficients, ou know that the conjugate i must also be a zero. So, from the Linear Factorization Theorem, f can be written as f a i i. For simplicit, let a to obtain f Now tr Eercise 7. Factoring a Polnomial The Linear Factorization Theorem shows that ou can write an nth-degree polnomial as the product of n linear factors. f a n c c c... c n However, this result includes the possibilit that some of the values of are comple. The following theorem sas that even if ou do not want to get involved with comple factors, ou can still write f as the product of linear and/or quadratic factors. For a proof of this theorem, see Proofs in Mathematics on page. Factors of a Polnomial Ever polnomial of degree n > 0 with real coefficients can be written as the product of linear and quadratic factors with real coefficients, where the quadratic factors have no real zeros. c i

48 7 Chapter Polnomial and Rational Functions A quadratic factor with no real zeros is said to be prime or irreducible over the reals. Be sure ou see that this is not the same as being irreducible over the rationals. For eample, the quadratic i i is irreducible over the reals (and therefore over the rationals). On the other hand, the quadratic is irreducible over the rationals but reducible over the reals. Eample 7 Finding the Zeros of a Polnomial Function Find all the zeros of f 6 60 given that i is a zero of f. Algebraic Solution Graphical Solution Because comple zeros occur in conjugate pairs, ou know that Because comple zeros alwas occur in conjugate i is also a zero of f. This means that both pairs, ou know that i is also a zero of i and i f. Because the polnomial is a fourth-degree polnomial, ou know that there are at most two are factors of f. Multipling these two factors produces other zeros of the function. Use a graphing utilit i i i i to graph 9i as shown in Figure.. Using long division, ou can divide 0 into f to obtain the following. 0 ) 6 60 So, ou have 0 0 f and ou can conclude that the zeros of i,, and. Now tr Eercise f 0 are i, = FIGURE. You can see that and appear to be zeros of the graph of the function. Use the zero or root feature or the zoom and trace features of the graphing utilit to confirm that and are zeros of the graph. So, ou can conclude that the zeros of f are i, i,, and. In Eample 7, if ou were not told that i is a zero of f, ou could still find all zeros of the function b using snthetic division to find the real zeros and. Then ou could factor the polnomial as 0. Finall, b using the Quadratic Formula, ou could determine that the zeros are,, i, and i

49 Section.5 Zeros of Polnomial Functions 75 Eample 8 shows how to find all the zeros of a polnomial function, including comple zeros. In Eample 8, the fifth-degree polnomial function has three real zeros. In such cases, ou can use the zoom and trace features or the zero or root feature of a graphing utilit to approimate the real zeros. You can then use these real zeros to determine the comple zeros algebraicall. f() = (, 0) (, 0) FIGURE. Eample 8 Finding the Zeros of a Polnomial Function Write f 5 8 as the product of linear factors, and list all of its zeros. Solution The possible rational zeros are ±, ±, ±, and ±8. Snthetic division produces the following. So, ou have f 5 8 You can factor as, and b factoring as ou obtain i i f i i which gives the following five zeros of f.,,, i, 8 and From the graph of f shown in Figure., ou can see that the real zeros are the onl ones that appear as -intercepts. Note that is a repeated zero. Now tr Eercise is a zero. is a zero. i You can use the table feature of a graphing utilit to help ou determine which of the possible rational zeros are zeros of the polnomial in Eample 8. The table should be set to ask mode. Then enter each of the possible rational zeros in the table. When ou do this, ou will see that there are two rational zeros, and, as shown at the right. Technolog

50 76 Chapter Polnomial and Rational Functions Other Tests for Zeros of Polnomials You know that an nth-degree polnomial function can have at most n real zeros. Of course, man nth-degree polnomials do not have that man real zeros. For instance, f has no real zeros, and f has onl one real zero. The following theorem, called Descartes s Rule of Signs, sheds more light on the number of real zeros of a polnomial. Descartes s Rule of Signs Let f () a n n a n n... a a a 0 be a polnomial with real coefficients and a The number of positive real zeros of f is either equal to the number of variations in sign of f or less than that number b an even integer.. The number of negative real zeros of f is either equal to the number of variations in sign of f or less than that number b an even integer. A variation in sign means that two consecutive coefficients have opposite signs. When using Descartes s Rule of Signs, a zero of multiplicit k should be counted as k zeros. For instance, the polnomial has two variations in sign, and so has either two positive or no positive real zeros. Because ou can see that the two positive real zeros are of multiplicit. Eample 9 Using Descartes s Rule of Signs Describe the possible real zeros of f 5 6. Solution The original polnomial has three variations in sign. to to f() = f 5 6 FIGURE.5 to The polnomial f has no variations in sign. So, from Descartes s Rule of Signs, the polnomial f 5 6 has either three positive real zeros or one positive real zero, and has no negative real zeros. From the graph in Figure.5, ou can see that the function has onl one real zero (it is a positive number, near ). Now tr Eercise 79.

51 Section.5 Zeros of Polnomial Functions 77 Another test for zeros of a polnomial function is related to the sign pattern in the last row of the snthetic division arra. This test can give ou an upper or lower bound of the real zeros of f. A real number b is an upper bound for the real zeros of f if no zeros are greater than b. Similarl, b is a lower bound if no real zeros of f are less than b. Upper and Lower Bound Rules Let f be a polnomial with real coefficients and a positive leading coefficient. Suppose f is divided b c, using snthetic division.. 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.. If c < 0 and the numbers in the last row are alternatel positive and negative (zero entries count as positive or negative), c is a lower bound for the real zeros of f. Eample 0 Finding the Zeros of a Polnomial Function Find the real zeros of f 6. Solution The possible real zeros are as follows. Factors of ±, ± Factors of 6 ±, ± ±, ±, ±, ±6, ±, ± 6, ±, ± The original polnomial f has three variations in sign. The polnomial f 6 6 has no variations in sign. As a result of these two findings, ou can appl Descartes s Rule of Signs to conclude that there are three positive real zeros or one positive real zero, and no negative zeros. Tring produces the following So, is not a zero, but because the last row has all positive entries, ou know that is an upper bound for the real zeros. So, ou can restrict the search to zeros between 0 and. B trial and error, ou can determine that is a zero. So, f Because 6 has no real zeros, it follows that is the onl real zero. Now tr Eercise 87.

52 78 Chapter Polnomial and Rational Functions Before concluding this section, here are two additional hints that can help ou find the real zeros of a polnomial.. If the terms of f have a common monomial factor, it should be factored out before appling the tests in this section. For instance, b writing f 5 5 ou can see that 0 is a zero of f and that the remaining zeros can be obtained b analzing the cubic factor.. If ou are able to find all but two zeros of f, ou can alwas use the Quadratic Formula on the remaining quadratic factor. For instance, if ou succeeded in writing f 5 ou can appl the Quadratic Formula to to conclude that the two remaining zeros are 5 and 5. Eample Using a Polnomial Model You are designing candle-making kits. Each kit contains 5 cubic inches of candle wa and a mold for making a pramid-shaped candle. You want the height of the candle to be inches less than the length of each side of the candle s square base. What should the dimensions of our candle mold be? Solution The volume of a pramid is V Bh, where B is the area of the base and h is the height. The area of the base is and the height is. So, the volume of the pramid is V. Substituting 5 for the volume ields the following. 5 Substitute 5 for V Multipl each side b. Write in general form. The possible rational solutions are ±, ±, ±5, ±5, ±5, ±75. Use snthetic division to test some of the possible solutions. Note that in this case, it makes sense to test onl positive -values. Using snthetic division, ou can determine that 5 is a solution The other two solutions, which satisf 5 0, are imaginar and can be discarded. You can conclude that the base of the candle mold should be 5 inches b 5 inches and the height of the mold should be 5 inches. Now tr Eercise 07.

53 Section.5 Zeros of Polnomial Functions 79.5 Eercises VOCABULARY CHECK: Fill in the blanks.. The of states that if f is a polnomial of degree n n > 0, then f has at least one zero in the comple number sstem.. The states that if is a polnomial of degree then has precisel linear factors f a n c c... f n n > 0, f n c n where c, c,..., c n are comple numbers.. The test that gives a list of the possible rational zeros of a polnomial function is called the Test.. If a bi is a comple zero of a polnomial 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 is a(n) bound for the real zeros of f if no real zeros are less than b, and is a(n) bound if no real zeros are greater than b. In Eercises 6, find all the zeros of the function.. f 6. f. g ). f f 6 i i 6. h t t t t i t i In Eercises 7 0, use the Rational Zero Test to list all possible rational zeros of f. Verif that the zeros of f shown on the graph are contained in the list. 7. f f f f In Eercises 0, find all the rational zeros of the function.. f 6 6. f 7 6. g. h h t t t t 0 6. p C 8. f f f 5 5 5

54 80 Chapter Polnomial and Rational Functions In Eercises, find all real solutions of the polnomial equation.. z z z In Eercises 5 8, (a) list the possible rational zeros of f, (b) sketch the graph of f so that some of the possible zeros in part (a) can be disregarded, and then (c) determine all real zeros of f. 5. f 6. f f f 5 In Eercises 9, (a) list the possible rational zeros of f, (b) use a graphing utilit to graph f so that some of the possible zeros in part (a) can be disregarded, and then (c) determine all real zeros of f. 9. f 8 0. f 7. f 5 7. f 7 8 Graphical Analsis In Eercises 6, (a) use the zero or root feature of a graphing utilit to approimate the zeros of the function accurate to three decimal places, (b) determine one of the eact zeros (use snthetic division to verif our result), and (c) factor the polnomial completel.. f. P t t 7t h g In Eercises 7, find a polnomial function with real coefficients that has the given zeros. (There are man correct answers.) 7., 5i, 5i 8., i, i 9. 6, 5 i, 5 i 0., i, i.,, i. 5, 5, i In Eercises 6, write the polnomial (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 completel factored form.. f 6 7. f 8 (Hint: One factor is 6. ) 5. f 5 6 (Hint: One factor is. ) 6. f 0 (Hint: One factor is. ) In Eercises 7 5, use the given zero to find all the zeros of the function. Function 7. f f f g g 0 5. h f 5 5. f 0 Zero 5i i i 5 i i i i i In Eercises 55 7, find all the zeros of the function and write the polnomial as a product of linear factors. 55. f f h 58. g f 8 f 65 f z z z h() g 6 0 f 5 h 6 h f g 8 8 g h f f 9 00 In Eercises 7 78, find all the zeros of the function. When there is an etended list of possible rational zeros, use a graphing utilit to graph the function in order to discard an rational zeros that are obviousl not zeros of the function. 7. f f s s 5s s f f f g

55 Section.5 Zeros of Polnomial Functions 8 In Eercises 79 86, use Descartes s Rule of Signs to determine the possible numbers of positive and negative zeros of the function. 79. g h 8 8. h 8. h g f f 5 5 f In Eercises 87 90, use snthetic division to verif the upper and lower bounds of the real zeros of f. 87. f 5 (a) Upper: (b) Lower: 88. f 8 (a) Upper: (b) Lower: 89. f 6 6 (a) Upper: 5 (b) Lower: 90. f 8 (a) Upper: (b) Lower: In Eercises 9 9, find all the real zeros of the function. 9. f 9. f z z z 7z 9 9. f g 5 0 In Eercises 95 98, find all the rational zeros of the polnomial function P f 6 f f z z 6 z z 6 6z z z In Eercises 99 0, match the cubic function with the numbers of rational and irrational zeros. (a) Rational zeros: 0; irrational zeros: (b) Rational zeros: ; irrational zeros: 0 (c) Rational zeros: ; irrational zeros: (d) Rational zeros: ; irrational zeros: f 00. f 0. f 0. f 0. Geometr An open bo is to be made from a rectangular piece of material, 5 centimeters b 9 centimeters, b cutting equal squares from the corners and turning up the sides. (a) Let 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 bo. (b) Use the diagram to write the volume V of the bo as a function of. Determine the domain of the function. (c) Sketch the graph of the function and approimate the dimensions of the bo that will ield a maimum volume. (d) Find values of such that V 56. Which of these values is a phsical impossibilit in the construction of the bo? Eplain. 0. Geometr A rectangular package to be sent b a deliver service (see figure) can have a maimum combined length and girth (perimeter of a cross section) of 0 inches. (a) Show that the volume of the package is V 0. (b) Use a graphing utilit to graph the function and approimate the dimensions of the package that will ield a maimum volume. (c) Find values of such that V,500. Which of these values is a phsical impossibilit in the construction of the package? Eplain. 05. Advertising Cost A compan that produces MP plaers estimates that the profit P (in dollars) for selling a particular model is given b P ,000, where is the advertising epense (in tens of thousands of dollars). Using this model, find the smaller of two advertising amounts that will ield a profit of $,500, Advertising Cost A compan that manufactures biccles estimates that the profit P (in dollars) for selling a particular model is given b P ,000, where is the advertising epense (in tens of thousands of dollars). Using this model, find the smaller of two advertising amounts that will ield a profit of $800,000.

56 8 Chapter Polnomial and Rational Functions 07. Geometr A bulk food storage bin with dimensions feet b feet b feet needs to be increased in size to hold five times as much food as the current bin. (Assume each dimension is increased b the same amount.) (a) Write a function that represents the volume V of the new bin. (b) Find the dimensions of the new bin. 08. Geometr A rancher wants to enlarge an eisting rectangular corral such that the total area of the new corral is.5 times that of the original corral. The current corral s dimensions are 50 feet b 60 feet. The rancher wants to increase each dimension b the same amount. (a) Write a function that represents the area A of the new 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 60 feet, and twice the length to the sides that are 50 feet, such that the total area of the new corral is.5 times that of the original corral. Repeat parts (a) and (b). Eplain our results. 09. Cost The ordering and transportation cost C (in thousands of dollars) for the components used in manufacturing a product is given b C , where is the order size (in hundreds). In calculus, it can be shown that the cost is a minimum when , Use a calculator to approimate the optimal order size to the nearest hundred units. 0. Height of a Baseball A baseball is thrown upward from a height of 6 feet with an initial velocit of 8 feet per second, and its height h (in feet) is h t 6t 8t 6, 0 t where t is the time (in seconds). You are told the ball reaches a height of 6 feet. Is this possible?. Profit The demand equation for a certain product is p , where p is the unit price (in dollars) of the product and is the number of units produced and sold. The cost equation for the product is C 80 50,000, where C is the total cost (in dollars) and is the number of units produced. The total profit obtained b producing and selling units is P R C p C. You are working in the marketing department of the compan that produces this product, and ou are asked to determine a price p that will ield a profit of 9 million dollars. Is this possible? Eplain.. Athletics The attendance A (in millions) at NCAA women s college basketball games for the ears 997 through 00 is shown in the table, where t represents the ear, with t 7 corresponding to 997. (Source: National Collegiate Athletic Association) Snthesis Year, t (a) Use the regression feature of a graphing utilit to find a cubic model for the data. (b) Use the graphing utilit to create a scatter plot of the data. Then graph the model and the scatter plot in the same viewing window. How do the compare? (c) According to the model found in part (a), in what ear did attendance reach 8.5 million? (d) According to the model found in part (a), in what ear did attendance reach 9 million? (e) According to the right-hand behavior of the model, will the attendance continue to increase? Eplain. True or False? In Eercises and, decide whether the statement is true or false. Justif our answer.. It is possible for a third-degree polnomial function with integer coefficients to have no real zeros.. If iis a zero of the function given b f i i Model It Attendance, A then i must also be a zero of f. Think About It In Eercises 5 0, determine (if possible) the zeros of the function g if the function f has zeros at r, r, and r. 5. g f 6. g f

57 Section.5 Zeros of Polnomial Functions 8 7. g f 5 8. g f 9. g f 0. g f. Eploration Use a graphing utilit to graph the function given b f k for different values of k. Find values of k such that the zeros of f satisf the specified characteristics. (Some parts do not have unique answers.) (a) Four real zeros (b) Two real zeros, each of multiplicit (c) Two real zeros and two comple zeros (d) Four comple zeros. Think About It Will the answers to Eercise change for the function g? (a) g f (b) g f. Think About It A third-degree polnomial function f has real zeros,, and, and its leading coefficient is negative. Write an equation for f. Sketch the graph of f. How man different polnomial functions are possible for f?. Think About It Sketch the graph of a fifth-degree polnomial function whose leading coefficient is positive and that has one zero at of multiplicit. 5. Writing Compile a list of all the various techniques for factoring a polnomial that have been covered so far in the tet. Give an eample illustrating each technique, and write a paragraph discussing when the use of each technique is appropriate. 6. Use the information in the table to answer each question. Interval,,,, Value of f Positive Negative Negative Positive (a) What are the three real zeros of the polnomial function f? (b) What can be said about the behavior of the graph of f at? (c) What is the least possible degree of f? Eplain. Can the degree of f ever be odd? Eplain. (d) Is the leading coefficient of f positive or negative? Eplain. (e) Write an equation for f. (There are man correct answers.) (f) Sketch a graph of the equation ou wrote in part (e). 7. (a) Find a quadratic function f (with integer coefficients) that has ± bi as zeros. Assume that b is a positive integer. (b) Find a quadratic function f (with integer coefficients) that has a ± bi as zeros. Assume that b is a positive integer. 8. Graphical Reasoning The graph of one of the following functions is shown below. Identif the function shown in the graph. Eplain wh each of the others is not the correct function. Use a graphing utilit to verif our result. (a) f ).5 (b) g ).5 (c) h ).5 (d) k ).5 Skills Review In Eercises 9, perform the operation and simplif. 9. 6i 8 i 0. 5i 6i. 6 i 7i. 9 5i 9 5i In Eercises 8, use the graph of f to sketch the graph of g. To print an enlarged cop of the graph, go to the website g f. g f 5. g f 6. g f 5 (0, ) (, ) f 7. g f 8. g f (, 0) (, )

58 8 Chapter Polnomial and Rational Functions.6 Rational Functions What ou should learn Find the domains of rational functions. Find the horizontal and vertical asmptotes of graphs of rational functions. Analze and sketch graphs of rational functions. Sketch graphs of rational functions that have slant asmptotes. Use rational functions to model and solve real-life problems. Wh ou should learn it Rational functions can be used to model and solve real-life problems relating to business. For instance, in Eercise 79 on page 96, a rational function is used to model average speed over a distance. Introduction A rational function can be written in the form f N() D() where N and D are polnomials and D is not the zero polnomial. In general, the domain of a rational function of includes all real numbers ecept -values that make the denominator zero. Much of the discussion of rational functions will focus on their graphical behavior near the -values ecluded from the domain. Eample Finding the Domain of a Rational Function Find the domain of f and discuss the behavior of f near an ecluded -values. Solution Because the denominator is zero when 0, the domain of f is all real numbers ecept 0. To determine the behavior of f near this ecluded value, evaluate f to the left and right of 0, as indicated in the following tables f f Mike Powell/Gett Images From the table, note that as approaches 0 from the left, f decreases without bound. In contrast, as approaches 0 from the right, f increases without bound. Because f decreases without bound from the left and increases without bound from the right, ou can conclude that f is not continuous. The graph of f is shown in Figure.6. f () = Eploration Use the table and trace features of a graphing utilit to verif that the function f in Eample is not continuous. FIGURE.6 Now tr Eercise.

59 Vertical asmptote: = 0 f() = FIGURE.7 Horizontal asmptote: = 0 Horizontal and Vertical Asmptotes In Eample, the behavior of f near 0 is denoted as follows. f as 0 Section.6 Rational Functions 85 f as f decreases without bound f increases without bound as approaches 0 from the left. as approaches 0 from the right. The line 0 is a vertical asmptote of the graph of f, as shown in Figure.7. From this figure, ou can see that the graph of f also has a horizontal asmptote the line 0. This means that the values of f approach zero as increases or decreases without bound. f 0 as f 0 as 0 Eploration Use a table of values to determine whether the functions in Figure.8 are continuous. If the graph of a function has an asmptote, can ou conclude that the function is not continuous? Eplain. f approaches 0 as f approaches 0 as decreases without bound. increases without bound. Definitions of Vertical and Horizontal Asmptotes. The line a is a vertical asmptote of the graph of f if f or f as a, either from the right or from the left.. The line b is a horizontal asmptote of the graph of f if f b as or. Eventuall (as or ), the distance between the horizontal asmptote and the points on the graph must approach zero. Figure.8 shows the horizontal and vertical asmptotes of the graphs of three rational functions. f() = + + Vertical asmptote: = Horizontal asmptote: = f() = + Horizontal asmptote: = 0 f() = ( ) Vertical asmptote: = Horizontal asmptote: = 0 (a) FIGURE.8 (b) (c) The graphs of f in Figure.7 and f in Figure.8(a) are hperbolas. You will stud hperbolas in Section 0..

60 86 Chapter Polnomial and Rational Functions Asmptotes of a Rational Function Let f be the rational function given b f N D a n n a n n... a a 0 b m m b m m... b b 0 where N and D have no common factors.. The graph of f has vertical asmptotes at the zeros of D.. The graph of f has one or no horizontal asmptote determined b comparing the degrees of N and D. a. If n < m, the graph of f has the line 0 (the -ais) as a horizontal asmptote. b. If n m, the graph of f has the line a n b m (ratio of the leading coefficients) as a horizontal asmptote. c. If n > m, the graph of f has no horizontal asmptote. Eample Finding Horizontal and Vertical Asmptotes Horizontal asmptote: = Vertical asmptote: = FIGURE.9 f() = Vertical asmptote: = Find all horizontal and vertical asmptotes of the graph of each rational function. a. b. f f 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 and the leading coefficient of the denominator is, so the graph has the line as a horizontal asmptote. To find an vertical asmptotes, set the denominator equal to zero and solve the resulting equation for. 0 0 Set denominator equal to zero. Factor. 0 Set st factor equal to 0. 0 Set nd factor equal to 0. This equation has two real solutions and, so the graph has the lines and as vertical asmptotes. The graph of the function is shown in Figure.9. 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 denominator is, so the graph has the line as a horizontal asmptote. To find an vertical asmptotes, first factor the numerator and denominator as follows. f 6, B setting the denominator (of the simplified function) equal to zero, ou can determine that the graph has the line as a vertical asmptote. Now tr Eercise 9.

61 Section.6 Rational Functions 87 Analzing Graphs of Rational Functions To sketch the graph of a rational function, use the following guidelines. You ma also want to test for smmetr when graphing rational functions, especiall for simple rational functions. Recall from Section.6 that the graph of f is smmetric with respect to the origin. Guidelines for Analzing Graphs of Rational Functions Let f N D, where N and D are polnomials.. Simplif f, if possible.. Find and plot the -intercept (if an) b evaluating f 0.. Find the zeros of the numerator (if an) b solving the equation N 0. Then plot the corresponding -intercepts.. Find the zeros of the denominator (if an) b solving the equation D 0. Then sketch the corresponding vertical asmptotes. 5. Find and sketch the horizontal asmptote (if an) b using the rule for finding the horizontal asmptote of a rational function. 6. Plot at least one point between and one point beond each -intercept and vertical asmptote. 7. Use smooth curves to complete the graph between and beond the vertical asmptotes. Technolog Some graphing utilities have difficult graphing rational functions that have vertical asmptotes. Often, the utilit 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. Notice that the graph should consist of two unconnected portions one to the left of and the other to the right of. To eliminate this problem, ou can tr changing the mode of the graphing utilit 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 The concept of test intervals from Section. can be etended to graphing of rational functions. To do this, use the fact that a rational function can change signs onl 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 entirel positive or entirel negative. This means that when the zeros of the numerator and the denominator of a rational function are put in order, the 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 -ais) or negative (the graph lies below the -ais).

62 88 Chapter Polnomial and Rational Functions Eample Sketching the Graph of a Rational Function You can use transformations to help ou sketch graphs of rational functions. For instance, the graph of g in Eample is a vertical stretch and a right shift of the graph of f because g f. Sketch the graph of Solution -intercept: -intercept: and state its domain. 0,, because g 0 None, because 0 Vertical asmptote:, zero of denominator Horizontal asmptote: 0, because degree of N < degree of D Additional points: g Test Representative Value of g Sign Point on interval -value graph,, g 0.5 Negative g Positive,, 0.5 Horizontal asmptote: = 0 FIGURE.0 Vertical asmptote: = 0 FIGURE. g() = 6 Vertical asmptote: = Horizontal asmptote: = f () = B plotting the intercepts, asmptotes, and a few additional points, ou can obtain the graph shown in Figure.0. The domain of g is all real numbers ecept. Eample Sketch the graph of f and state its domain. Solution Now tr Eercise 7. Sketching the Graph of a Rational Function -intercept:, 0, because 0 -intercept: None, because 0 is not in the domain Vertical asmptote: 0, zero of denominator Horizontal asmptote:, because degree of N degree of D Additional points: Test Representative Value of f Sign Point on interval -value graph, 0 0,, B plotting the intercepts, asmptotes, and a few additional points, ou can obtain the graph shown in Figure.. The domain of f is all real numbers ecept 0. Now tr Eercise. f f Positive Negative,, f.75 Positive,.75

63 Section.6 Rational Functions 89 Eample 5 Sketching the Graph of a Rational Function Sketch the graph of f. Vertical asmptote: = Horizontal asmptote: = 0 Vertical asmptote: = Solution Factoring the denominator, ou have f. -intercept: 0, 0, because f 0 0 -intercept: 0, 0 Vertical asmptotes:,, zeros of denominator Horizontal asmptote: 0, because degree of N < degree of D Additional points: f() = Test Representative Value of f Sign Point on interval -value graph,, 0 0,, 0.5 f 0. f Negative Positive, , 0. f 0.5 Negative, 0.5 f 0.75 Positive, 0.75 FIGURE. The graph is shown in Figure.. Now tr Eercise 5. If ou 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 Eample 6). Horizontal asmptote: = f() = Vertical asmptote: = 5 FIGURE. HOLE AT Eample 6 Sketch the graph of Solution A Rational Function with Common Factors B factoring the numerator and denominator, ou have f 9 -intercept: -intercept: f 9. 0,, because f 0, 0, because f 0 Vertical asmptote:, zero of (simplified) denominator Horizontal asmptote:, because degree of N degree of D Additional points: Test Representative Value of f Sign Point on interval -value graph,,, The graph is shown in Figure.. Notice that there is a hole in the graph at because the function is not defined when. Now tr Eercise.,. f 0. Positive, 0. f Negative, f.67 Positive,.67

64 90 Chapter Polnomial and Rational Functions Vertical asmptote: = f () = Slant asmptote: = FIGURE. Slant Asmptotes Consider a rational function whose denominator is of degree or greater. If the degree of the numerator is eactl one more than the degree of the denominator, the graph of the function has a slant (or oblique) asmptote. For eample, the graph of f has a slant asmptote, as shown in Figure.. To find the equation of a slant asmptote, use long division. For instance, b dividing into, ou obtain f. Slant asmptote As increases or decreases without bound, the remainder term approaches 0, so the graph of f approaches the line, as shown in Figure.. Eample 7 A Rational Function with a Slant Asmptote FIGURE.5 5 Vertical asmptote: = Slant asmptote: = f() = 5 Sketch the graph of f. Solution Factoring the numerator as allows ou to recognize the -intercepts. Using long division f allows ou to recognize that the line is a slant asmptote of the graph. -intercept: 0,, because f 0 -intercepts:, 0 and, 0 Vertical asmptote:, zero of denominator Slant asmptote: Additional points: Test Representative Value of f Sign Point on interval -value graph,,,, The graph is shown in Figure.5. Now tr Eercise 6. f. Negative 0.5 f Positive.5 f.5.5 Negative 0.5,.5 f Positive,,..5,.5

65 Applications Section.6 Rational Functions 9 There are man eamples of asmptotic behavior in real life. For instance, Eample 8 shows how a vertical asmptote can be used to analze the cost of removing pollutants from smokestack emissions. Eample 8 Cost-Benefit Model A utilit compan burns coal to generate electricit. The cost C (in dollars) of removing p% of the smokestack pollutants is given b C 80,000p 00 p for 0 p < 00. Sketch the graph of this function. You are a member of a state legislature considering a law that would require utilit companies to remove 90% of the pollutants from their smokestack emissions. The current law requires 85% removal. How much additional cost would the utilit compan incur as a result of the new law? Solution The graph of this function is shown in Figure.6. Note that the graph has a vertical asmptote at p 00. Because the current law requires 85% removal, the current cost to the utilit compan is C 80,000(85) Evaluate C when p 85. If the new law increases the percent removal to 90%, the cost will be C 80,000(90) $5,. $70,000. Evaluate C when p 90. So, the new law would require the utilit compan to spend an additional 70,000 5, $66,667. Subtract 85% removal cost from 90% removal cost. C Smokestack Emissions Cost (in thousands of dollars) C = 80,000 p 00 p 90% 85% p Percent of pollutants removed FIGURE.6 Now tr Eercise 7.

66 9 Chapter Polnomial and Rational Functions Eample 9 Finding a Minimum Area A rectangular page is designed to contain 8 square inches of print. The margins at the top and bottom of the page are each inch deep. The margins on each side are inches wide. What should the dimensions of the page be so that the least amount of paper is used? Graphical Solution Let A be the area to be minimized. From Figure.7, ou can write A. The printed area inside the margins is modeled b 8 or 8. To find the minimum area, rewrite the equation for A in terms of just one variable b substituting 8 for. A 8 8 The graph of this rational function is shown in Figure.8. Because represents the width of the printed area, ou need consider onl the portion of the graph for which is positive. Using a graphing utilit, ou can approimate the minimum value of A to occur when 8.5 inches. The corresponding value of is inches. So, the dimensions should be.5 inches 00 A =, b > 0 ( + )(8 + ), > inches. Numerical Solution Let A be the area to be minimized. From Figure.7, ou can write A. The printed area inside the margins is modeled b 8 or 8. To find the minimum area, rewrite the equation for A in terms of just one variable b substituting 8 for. A 8 Use the table feature of a graphing utilit to create a table of values for the function 8, > 0 8 in. FIGURE.7 in. in. beginning at. From the table, ou can see that the minimum value of occurs when is somewhere between 8 and 9, as shown in Figure.9. To approimate the minimum value of to one decimal place, change the table so that it starts at 8 and increases b 0.. The minimum value of occurs when 8.5, as shown in Figure.50. The corresponding value of is inches. So, the dimensions should be.5 inches b 7.6 inches. in. 0 0 FIGURE.8 Now tr Eercise 77. FIGURE.9 FIGURE.50 If ou go on to take a course in calculus, ou will learn an analtic technique for finding the eact value of that produces a minimum area. In this case, that value is

67 Section.6 Rational Functions 9.6 Eercises VOCABULARY CHECK: Fill in the blanks.. Functions of the form f N D, where N and D are polnomials and D is not the zero polnomial, are called.. If f ± as a from the left or the right, then a is a of the graph of f.. If f b as ±, then b is a of the graph of f.. For the rational function given b f N D, if the degree of N is eactl one more than the degree of D, then the graph of f has a (or oblique). In Eercises, (a) complete each table for the function, (b) determine the vertical and horizontal asmptotes of the graph of the function, and (c) find the domain of the function f f. f 8 f. f. f In Eercises 5, (a) find the domain of the function, (b) decide if the function is continuous, and (c) identif an horizontal and vertical asmptotes. 5. f 6. f f 7. f 8. f 9. f 0.. f 9. In Eercises 6, match the rational function with its graph. [The graphs are labeled (a), (b), (c), and (d).] (a) (c) (b) (d). f. 5. f f f 5 f 5 f In Eercises 7 0, find the zeros (if an) of the rational function. 7. g 8. h g 8 f 8 6 5

68 9 Chapter Polnomial and Rational Functions In Eercises 6, find the domain of the function and identif an horizontal and vertical asmptotes.. f. f f f f 6 f 6 7 In Eercises 7 6, (a) find the domain of the function, (b) decide if the function is continuous, (c) identif all intercepts, (d) identif an horizontal and vertical asmptotes, and (e) plot additional solution points as needed to sketch the graph of the function. 7. f 8. f 9. h 0. g. 5 C. P. t f. f t 9 t 5. g s s 6. f s g 8 h f 5 f f. f 6.. f 8 f f 6 f t t t Analtical, Numerical, and Graphical Analsis In Eercises 7 50, do the following. (a) Determine the domains of f and g. (b) Simplif f and find an vertical asmptotes of the graph of f. (c) Compare the functions b completing the table. (d) Use a graphing utilit to graph f and g in the same viewing window. (e) Eplain wh the graphing utilit ma not show the difference in the domains of f and g f, f, f, f 6 7, g f g g f g g f g g f g In Eercises 5 6, (a) state the domain of the function, (b) identif all intercepts, (c) identif an vertical and slant asmptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function g 5 h 5. f 5. f 55. g 56. h 57. f t t 58. f t g f f 5 5 f

69 Section.6 Rational Functions In Eercises 65 68, use a graphing utilit to graph the rational function. Give the domain of the function and identif an asmptotes. Then zoom out sufficientl far so that the graph appears as a line. Identif the line f f 8 f 5 8 f g h Graphical Reasoning In Eercises 69 7, (a) use the graph to determine an -intercepts of the graph of the rational function and (b) set 0 and solve the resulting equation to confirm our result in part (a) Pollution The cost C (in millions of dollars) of removing p% of the industrial and municipal pollutants discharged into a river is given b C 55p 00 p, p < 00. (a) Use a graphing utilit to graph the cost function (b) Find the costs of removing 0%, 0%, and 75% of the pollutants. (c) According to this model, would it be possible to remove 00% of the pollutants? Eplain. 7. Reccling In a pilot project, a rural township is given reccling bins for separating and storing recclable products. The cost C (in dollars) for suppling bins to p% of the population is given b C 5,000p 00 p, (a) Use a graphing utilit to graph the cost function. (b) Find the costs of suppling bins to 5%, 50%, and 90% of the population. (c) According to this model, would it be possible to suppl bins to 00% of the residents? Eplain. 75. Population Growth The game commission introduces 00 deer into newl acquired state game lands. The population N of the herd is modeled b N 0 5 t 0.0t, where t is the time in ears (see figure). (a) Find the populations when t 5, t 0, and t 5. (b) What is the limiting size of the herd as time increases? 76. Concentration of a Miture A 000-liter tank contains 50 liters of a 5% brine solution. You add liters of a 75% brine solution to the tank. (a) Show that the concentration C, the proportion of brine to total solution, in the final miture is C Deer population p < 00. t 0 N Time (in ears) (b) Determine the domain of the function based on the phsical 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? t

70 96 Chapter Polnomial and Rational Functions 77. Page Design A page that is inches wide and inches high contains 0 square inches of print. The top and bottom margins are inch deep and the margins on each side are inches wide (see figure). (a) Show that the total area A on the page is A. in. in. in. in. (b) Determine the domain of the function based on the phsical constraints of the problem. (c) Use a graphing utilit to graph the area function and approimate the page size for which the least amount of paper will be used. Verif our answer numericall using the table feature of the graphing utilit. 78. Page Design A rectangular page is designed to contain 6 square inches of print. The margins at the top and bottom of the page are each inch deep. The margins on each side are inches wide. What should the dimensions of the page be so that the least amount of paper is used? Model It 79. Average Speed A driver averaged 50 miles per hour on the round trip between Akron, Ohio, and Columbus, Ohio, 00 miles awa. The average speeds for going and returning were and miles per hour, respectivel. (a) Show that 5 5. (b) Determine the vertical and horizontal asmptotes of the graph of the function. (c) Use a graphing utilit to graph the function. (d) Complete the table (e) Are the results in the table what ou epected? Eplain. (f) Is it possible to average 0 miles per hour in one direction and still average 50 miles per hour on the round trip? Eplain. 80. Sales The sales S (in millions of dollars) for the Yankee Candle Compan in the ears 998 through 00 are shown in the table. (Source: The Yankee Candle Compan) A model for these data is given b S 5.86t t.00, 8 t where t represents the ear, with t 8 corresponding to 998. (a) Use a graphing utilit 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 Compan in 008. (c) Would this model be useful for estimating sales after 008? Eplain. Snthesis True or False? In Eercises 8 and 8, determine whether the statement is true or false. Justif our answer. 8. A polnomial can have infinitel man vertical asmptotes. 8. The graph of a rational function can never cross one of its asmptotes. Think About It In Eercises 8 and 8, write a rational function f that has the specified characteristics. (There are man correct answers.) 8. Vertical asmptote: None Horizontal asmptote: 8. Vertical asmptote:, Horizontal asmptote: None Skills Review In Eercises 85 88, completel factor the epression In Eercises 89 9, solve the inequalit and graph the solution on the real number line > < Make a Decision To work an etended application analzing the total manpower of the Department of Defense, visit this tet s website at college.hmco.com. (Data Source: U.S. Department of Defense)

71 Section.7 Nonlinear Inequalities 97.7 Nonlinear Inequalities What ou should learn Solve polnomial inequalities. Solve rational inequalities. Use inequalities to model and solve real-life problems. Wh ou should learn it Inequalities can be used to model and solve real-life problems. For instance, in Eercise 7 on page 05, a polnomial inequalit is used to model the percent of households that own a television and have cable in the United States. Polnomial Inequalities To solve a polnomial inequalit such as < 0, ou can use the fact that a polnomial can change signs onl at its zeros (the -values that make the polnomial equal to zero). Between two consecutive zeros, a polnomial must be entirel positive or entirel negative. This means that when the real zeros of a polnomial are put in order, the divide the real number line into intervals in which the polnomial has no sign changes. These zeros are the critical numbers of the inequalit, and the resulting intervals are the test intervals for the inequalit. For instance, the polnomial above factors as and has two zeros, and. These zeros divide the real number line into three test intervals:,,,, and,. (See Figure.5.) So, to solve the inequalit < 0, ou need onl test one value from each of these test intervals to determine whether the value satisfies the original inequalit. If so, ou can conclude that the interval is a solution of the inequalit. Test Interval (, ) Zero = Test Interval (, ) Zero = Test Interval (, ) Jose Luis Pelaez, Inc./Corbis 0 5 FIGURE.5 Three test intervals for You can use the same basic approach to determine the test intervals for an polnomial. Finding Test Intervals for a Polnomial To determine the intervals on which the values of a polnomial are entirel negative or entirel positive, use the following steps.. Find all real zeros of the polnomial, and arrange the zeros in increasing order (from smallest to largest). These zeros are the critical numbers of the polnomial.. Use the critical numbers of the polnomial to determine its test intervals.. Choose one representative -value in each test interval and evaluate the polnomial at that value. If the value of the polnomial is negative, the polnomial will have negative values for ever -value in the interval. If the value of the polnomial is positive, the polnomial will have positive values for ever -value in the interval.

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