Chapter 3

Write a rational function \(f\) that has the specified characteristics. (There are many correct answers.) Vertical asymptotes: \(x=0, x=\frac{5}{2}\) Horizontal asymptote: \(y=-3\)

Use the given zero of \(f\) to find all the zeros of \(f\). $$f(x)=25 x^{3}-55 x^{2}-54 x-18, \frac{1}{5}(-2+\sqrt{2} i)$$

Population The numbers \(P\) (in millions) of people age 18 and over in the United States for the years 1996 to 2005 are shown in the table. (Source: U.S. Census Bureau) $$ \begin{array}{|c|c|} \hline \text { Year } & \text { Population, } P \\ \hline 1996 & 199.2 \\ \hline 1997 & 201.7 \\ \hline 1998 & 204.4 \\ \hline 1999 & 207.1 \\ \hline 2000 & 209.1 \\ \hline \end{array} $$ $$ \begin{array}{|c|c|} \hline \text { Year } & \text { Population, } P \\ \hline 2001 & 212.5 \\ \hline 2002 & 215.1 \\ \hline 2003 & 217.8 \\ \hline 2004 & 220.4 \\ \hline 2005 & 222.9 \\ \hline \end{array} $$ (a) Use a graphing utility to create a scatter plot of the data. Let \(t=6\) correspond to \(1996 .\) (b) Use the regression feature of a graphing utility to find a linear model, a quadratic model, and a cubic model for the data. (c) Use a graphing utility to graph each model separately with the data in the same viewing window. How well does each model fit the data? (d) Use each model to predict the year in which the population is about \(231,000,000 .\) Explain any differences in the predictions.

Suppose that the remainder obtained in a polynomial division by \(x-k\) is zero. How is the divisor related to the graph of the dividend?

Modeling Polynomials Sketch the graph of a polynomial function that is of fourth degree, has a zero of multiplicity 2, and has a negative leading coefficient. Sketch another graph under the same conditions but with a positive leading coefficient.

Tuition and Fees The table shows the average values of tuition and fees \(F\) (in dollars) for in-state students at public institutions of higher education in the years 1996 to \(2005 .\) (Source: U.S. National Center for Educational Statistics) $$ \begin{aligned} &\begin{array}{|l|l|l|l|l|l|} \hline \text { Year } & 1996 & 1997 & 1998 & 1999 & 2000 \\ \hline \begin{array}{l} \text { Tuition and } \\ \text { fees, } F \end{array} & 2179 & 2271 & 2360 & 2430 & 2506 \\ \hline \end{array}\\\ &\begin{array}{|l|c|c|c|c|c|} \hline \text { Year } & 2001 & 2002 & 2003 & 2004 & 2005 \\ \hline \begin{array}{l} \text { Tuition and } \\ \text { fees, } F \end{array} & 2562 & 2700 & 2903 & 3319 & 3638 \\ \hline \end{array} \end{aligned} $$ (a) Use a graphing utility to create a scatter plot of the data. Let \(t\) represent the year, with \(t=6\) corresponding to \(1996 .\) (b) Use the regression feature of a graphing utility to find a quadratic model for the data. (c) Use a graphing utility to graph the model from part (b) in the same viewing window as the scatter plot of the data. (d) Use the graph of the model from part (c) to predict the average value of tuition and fees in 2008 .

Write a rational function \(f\) that has the specified characteristics. (There are many correct answers.) Vertical asymptotes: \(x=-2, x=1\) Horizontal asymptote: None

Graphic Reasoning Solve \(x^{4}-\underline{5 x^{2}}+4=0\). Then use a graphing utility to graph \(y=x^{4}-5 x^{2}+4\). What is the connection between the solutions you found and the intercepts of the graph?

Solve the quadratic equation. $$x^{2}-2 x+2=0$$

Cost of Dental Care The amount that \(\$ 100\) worth of dental care at \(1982-1984\) prices would cost in a different year is given by a CPI (Consumer Price Index). The CPIs for dental care in the United States for the years 1996 to 2005 are shown in the table. (Source: U.S. Bureau of Labor Statistics) $$ \begin{array}{|c|c|} \hline \text { Year } & \text { CPI } \\ \hline 1996 & 216.5 \\ \hline 1997 & 226.6 \\ \hline 1998 & 236.2 \\ \hline 1999 & 247.2 \\ \hline 2000 & 258.5 \\ \hline \end{array} $$ $$ \begin{array}{|c|c|} \hline \text { Year } & \text { CPI } \\ \hline 2001 & 269.0 \\ \hline 2002 & 281.0 \\ \hline 2003 & 292.5 \\ \hline 2004 & 306.9 \\ \hline 2005 & 324.0 \\ \hline \end{array} $$ (a) Use a spreadsheet software program to create a scatter plot of the data. Let \(t\) represent the year, with \(t=6\) corresponding to \(1996 .\) (b) Use the regression feature of a spreadsheet software program to find a linear model, a quadratic model, a cubic model, and a quartic model for the data. (c) Use each model to predict the year in which the CPI for dental care will be about \(\$ 400\). Then discuss the appropriateness of each model for predicting future values.

Modeling Polynomials Sketch the graph of a polynomial function that is of fifth degree, has a zero of multiplicity 2 , and has a negative leading coefficient. Sketch another graph under the same conditions but with a positive leading coefficient.

Liver Transplants The table shows the numbers \(L\) of liver transplant procedures performed in the United States in the years 1995 to \(2005 .\) (Source: U.S. Department of Health and Human Services) $$ \begin{aligned} &\begin{array}{|l|l|l|l|l|} \hline \text { Year } & 1995 & 1996 & 1997 & 1998 \\ \hline \text { Transplants, } L & 3818 & 3918 & 4005 & 4356 \\ \hline \end{array}\\\ &\begin{array}{|l|l|l|l|l|} \hline \text { Year } & 1999 & 2000 & 2001 & 2002 \\ \hline \text { Transplants, } L & 4586 & 4816 & 5177 & 5326 \\ \hline \end{array}\\\ &\begin{array}{|l|l|l|l|} \hline \text { Year } & 2003 & 2004 & 2005 \\ \hline \text { Transplants, } L & 5671 & 6168 & 6444 \\ \hline \end{array} \end{aligned} $$ (a) Use a graphing utility to create a scatter plot of the data. Let \(t\) represent the year, with \(t=5\) corresponding to \(1995 .\) (b) Use the regression feature of a graphing utility to find a quadratic model for the data. (c) Use a graphing utility to graph the model from part (b) in the same viewing window as the scatter plot of the data. (d) Use the graph of the model from part (c) to predict the number of liver transplant procedures performed in \(2008 .\)

Write a rational function \(f\) that has the specified characteristics. (There are many correct answers.) Vertical asymptote: \(x=3\) Horizontal asymptote: \(x\) -axis

Graphical Reasoning Solve \(x^{4}+5 x^{2}+4=0\). Then use a graphing utility to graph \(y=x^{4}+5 x^{2}+4\) What is the connection between the solutions you found and the intercepts of the graph?

Solve the quadratic equation. $$x^{2}+6 x+10=0$$

Solar Energy Photovoltaic cells convert light energy into electricity. The photovoltaic cell and module domestic shipments \(S\) (in peak kilowatts) for the years 1996 to 2005 are shown in the table. (Source: Energy Information Administration) $$ \begin{array}{|c|c|} \hline \text { Year } & \text { Shipments, } S \\ \hline 1996 & 13,016 \\ \hline 1997 & 12,561 \\ \hline 1998 & 15,069 \\ \hline 1999 & 21,225 \\ \hline 2000 & 19,838 \\ \hline \end{array} $$ $$ \begin{array}{|c|c|} \hline \text { Year } & \text { Shipments, } S \\ \hline 2001 & 36,310 \\ \hline 2002 & 45,313 \\ \hline 2003 & 48,664 \\ \hline 2004 & 78,346 \\ \hline 2005 & 134,465 \\ \hline \end{array} $$ (a) Use a spreadsheet software program to create a scatter plot of the data. Let \(t\) represent the year, with \(t=6\) corresponding to 1996 . (b) Use the regression feature of a spreadsheet software program to find a cubic model and a quartic model for the data. (c) Use each model to predict the year in which the shipments will be about \(1,000,000\) peak kilowatts. Then discuss the appropriateness of each model for predicting future values.

Regression Problem Let \(x\) be the number of units (in tens of thousands) that a computer company produces and let \(p(x)\) be the profit (in hundreds of thousands of dollars). The table shows the profits for different levels of production. $$ \begin{aligned} &\begin{array}{|l|l|l|l|l|l|} \hline \text { Units, } x & 2 & 4 & 6 & 8 & 10 \\ \hline \text { Profit, } p(x) & 270.5 & 307.8 & 320.1 & 329.2 & 325.0 \\ \hline \end{array}\\\ &\begin{array}{|l|l|l|l|l|l|} \hline \text { Units, } x & 12 & 14 & 16 & 18 & 20 \\ \hline \text { Profit, } p(x) & 311.2 & 287.8 & 254.8 & 212.2 & 160.0 \\ \hline \end{array} \end{aligned} $$ (a) Use a graphing utility to create a scatter plot of the data. (b) Use the regression feature of a graphing utility to find a quadratic model for \(p(x)\). (c) Use a graphing utility to graph your model for \(p(x)\) with the scatter plot of the data. (d) Find the vertex of the graph of the model from part (c). Interpret its meaning in the context of the problem. (e) With these data and this model, the profit begins to decrease. Discuss how it is possible for production to increase and profit to decrease.

Graphical Analysis Find a fourth-degree polynomial function that has (a) four real zeros, (b) two real zeros, and (c) no real zeros. Use a graphing utility to graph the functions and describe the similarities and differences among them.

Solve the quadratic equation. $$4 x^{2}+16 x+17=0$$

Advertising Cost A company that produces video games estimates that the profit \(P\) (in dollars) from selling a new game is given by \(P=-82 x^{3}+7250 x^{2}-450,000, \quad 0 \leq x \leq 80\) where \(x\) is the advertising expense (in tens of thousands of dollars). Using this model, how much should the company spend on advertising to obtain a profit of $$\$ 5,900,000 ?$$

Regression Problem Let \(x\) be the angle (in degrees) at which a baseball is hit with no spin at an initial speed of 40 meters per second and let \(d(x)\) be the distance (in meters) the ball travels. The table shows the distances for the different angles at which the ball is hit. (Source: The Physics of Sports) $$ \begin{aligned} &\begin{array}{|l|l|l|l|l|l|} \hline \text { Angle, } x & 10 & 15 & 30 & 36 & 42 \\ \hline \text { Distance, } d(x) & 58.3 & 79.7 & 126.9 & 136.6 & 140.6 \\ \hline \end{array}\\\ &\begin{array}{|l|l|l|l|l|l|} \hline \text { Angle, } x & 44 & 45 & 48 & 54 & 60 \\ \hline \text { Distance, } d(x) & 140.9 & 140.9 & 139.3 & 132.5 & 120.5 \\ \hline \end{array} \end{aligned} $$ (a) Use a graphing utility to create a scatter plot of the data. (b) Use the regression feature of a graphing utility to find a quadratic model for \(d(x)\). (c) Use a graphing utility to graph your model for \(d(x)\) with the scatter plot of the data. (d) Find the vertex of the graph of the model from part (c). Interpret its meaning in the context of the problem.

Graphical Analysis Find a sixth-degree polynomial function that has (a) six real zeros, (b) four real zeros, (c) two real zeros, and (d) no real zeros. Use a graphing utility to graph the functions and describe the similarities and differences among them.

Solve the quadratic equation. $$9 x^{2}-6 x+37=0$$

Advertising Cost A company that manufactures hydroponic gardening systems estimates that the profit \(P\) (in dollars) from selling a new system is given by \(P=-35 x^{3}+2700 x^{2}-300,000, \quad 0 \leq x \leq 70\) where \(x\) is the advertising expense (in tens of thousands of dollars). Using this model, how much should the company spend on advertising to obtain a profit of $$\$ 1,800,000$$ ?

Write the quadratic function \(f(x)=a x^{2}+b x+c\) in standard form to verify that the vertex occurs at \(\left(-\frac{b}{2 a}, f\left(-\frac{b}{2 a}\right)\right)\)

Profit The demand and cost equations for a stethoscope are given by \(p=140-0.0001 x\) and \(C=80 x+150,000\) where \(p\) is the unit price (in dollars), \(C\) is the total cost (in dollars), and \(x\) is the number of units. The total profit \(P\) (in dollars) obtained by producing and selling \(x\) units is given by \(P=R-C=x p-C\) Try to determine a price \(p\) that would yield a profit of \(\$ 9\) million, and then use a graphing utility to explain why this is not possible.

Solve the quadratic equation. $$4 x^{2}+16 x+15=0$$

MAKE A DECISION: DEMAND FUNCTION A company that produces cell phones estimates that the demand \(D\) for a new model of phone is given by \(D=-x^{3}+54 x^{2}-140 x-3000, \quad 10 \leq x \leq 50\) where \(x\) is the price of the phone (in dollars). (a) Use a graphing utility to graph \(D\). Use the trace feature to determine the values of \(x\) for which the demand is 14,400 phones. (b) You may also determine the values of \(x\) for which the demand is 14,400 phones by setting \(D\) equal to 14,400 and solving for \(x\) with a graphing utility. Discuss this alternative solution method. Of the solutions that lie within the given interval, what price would you recommend the company charge for the phones?

Credit Cards The numbers of active American Express cards \(C\) (in millions) in the years 1997 to 2006 are shown in the table. (Sourze: American Express) $$ \begin{aligned} &\begin{array}{|l|l|l|l|l|l|} \hline \text { Year } & 1997 & 1998 & 1999 & 2000 & 2001 \\ \hline \text { Cards, C } & 42.7 & 42.7 & 46.0 & 51.7 & 55.2 \\ \hline \end{array}\\\ &\begin{array}{|l|l|l|l|l|l|} \hline \text { Year } & 2002 & 2003 & 2004 & 2005 & 2006 \\ \hline \text { Cards, C } & 57.3 & 60.5 & 65.4 & 71.0 & 78.0 \\ \hline \end{array} \end{aligned} $$ (a) Use a graphing utility to create a scatter plot of the data. Let \(t\) represent the year, with \(t=7\) corresponding to \(1997 .\) (b) Use what you know about end behavior and the scatter plot from part (a) to predict the sign of the leading coefficient of a quartic model for \(C\). (c) Use the regression feature of a graphing utility to find a quartic model for \(C\). Does your model agree with your answer from part (b)? (d) Use a graphing utility to graph the model from part (c). Use the graph to predict the year in which the number of active American Express cards would be about 92 million. Is your prediction reasonable?

Revenue The demand equation for a stethoscope is given by \(p=140-0.0001 x\) where \(p\) is the unit price (in dollars) and \(x\) is the number of units sold. The total revenue \(R\) obtained by producing and selling \(x\) units is given by \(R=x p\) Try to determine a price \(p\) that would yield a revenue of \(\$ 50\) million, and then use a graphing utility to explain why this is not possible.

Solve the quadratic equation. $$9 x^{2}-6 x+35=0$$

MAKE A DECISION: DEMAND FUNCTION A company that produces hand-held organizers estimates that the demand \(D\) for a new model of organizer is given by \(D=-0.005 x^{3}+2.65 x^{2}-70 x-2500,50 \leq x \leq 500\) where \(x\) is the price of the organizer (in dollars). (a) Use a graphing utility to graph \(D\). Use the trace feature to determine the values of \(x\) for which the demand will be 80,000 organizers. (b) You may also determine the values of \(x\) for which the demand will be 80,000 organizers by setting \(D\) equal to 80,000 and solving for \(x\) with a graphing utility. Discuss this alternative solution method. Of the solutions that lie within the given interval, what price would you recommend the company charge for the new organizers?

Population The immigrant population \(P\) (in millions) living in the United States at the beginning of each decade from 1900 to 2000 is shown in the table. (Source: Center of Immigration Studies) $$ \begin{aligned} &\begin{array}{|l|c|c|c|c|} \hline \text { Year } & 1900 & 1910 & 1920 & 1930 \\ \hline \text { Population, } P & 10.3 & 13.5 & 13.9 & 14.2 \\ \hline \end{array}\\\ &\begin{array}{|l|l|l|l|l|} \hline \text { Year } & 1940 & 1950 & 1960 & 1970 \\ \hline \text { Population, } P & 11.6 & 10.3 & 9.7 & 9.6 \\ \hline \end{array}\\\ &\begin{array}{|l|l|l|l|} \hline \text { Year } & 1980 & 1990 & 2000 \\ \hline \text { Population, } P & 14.1 & 19.8 & 30.0 \\ \hline \end{array} \end{aligned} $$ (a) Use a graphing utility to create a scatter plot of the data. Let \(t=0\) correspond to 1900 . (b) Use what you know about end behavior and the scatter plot from part (a) to predict the sign of the leading coefficient of a cubic model for \(P\). (c) Use the regression feature of a graphing utility to find a cubic model for \(P\). Does your model agree with your answer from part (b)? (d) Use a graphing utility to graph the model from part (c). Use the graph to predict the year in which the immigrant population will be about 45 million. Is your prediction reasonable?

Is it possible for a rational function to have all three types of asymptotes (vertical, horizontal, and slant)? Why or why not?

Reasoning The imaginary number \(2 i\) is a zero of \(f(x)=x^{3}-2 i x^{2}-4 x+8 i\) but the complex conjugate of \(2 i\) is not a zero of \(f(x)\). Is this a contradiction of the conjugate pairs statement on page 317 ? Explain.

Solve the quadratic equation. $$16 t^{2}-4 t+3=0$$

Height of a Baseball A baseball is launched upward from ground level with an initial velocity of 48 feet per second, and its height \(h\) (in feet) is \(h(t)=-16 t^{2}+48 t, \quad 0 \leq t \leq 3\) where \(t\) is the time (in seconds). You are told the ball reaches a height of 64 feet. Is this possible?

Modeling Polynomials A third-degree polynomial function \(f\) has real zeros \(-1,2\), and \(\frac{10}{3} .\) Find two different polynomial functions, one with a positive leading coefficient and one with a negative leading coefficient, that could be \(f\). How many different polynomial functions are possible for \(f ?\)

Is it possible for a rational function to have more than one horizontal asymptote? Why or why not?

Reasoning The imaginary number \(1-2 i\) is a zero of \(f(x)=x^{3}-(1-2 i) x^{2}-9 x+9(1-2 i)\) but \(1+2 i\) is not a zero of \(f(x)\). Is this a contradiction of the conjugate pairs statement on page 317 ? Explain.

Solve the quadratic equation. $$5 s^{2}+6 s+3=0$$

Exploration Use a graphing utility to graph the function \(f(x)=x^{4}-4 x^{2}+k\) for different values of \(k\). Find the values of \(k\) such that the zeros of \(f\) satisfy the specified characteristics. (Some parts do not have unique answers.) (a) Four real zeros (b) Two real zeros and two complex roots

Modeling Polynomials A fourth-degree polynomial function \(g\) has real zeros \(-2,0,1\), and \(5 .\) Find two different polynomial functions, one with a positive leading coefficient and one with a negative leading coefficient, that could be \(g .\) How many different polynomial functions are possible for \(g\) ?

MAKE A DECISION: SEIZURE OF ILLEGAL DRUGS The cost \(C\) (in millions of dollars) for the federal government to seize \(p\) percent of an illegal drug as it enters the country is \(C=\frac{528 p}{100-p}, \quad 0 \leq p<100\) (a) Find the cost of seizing \(25 \%\) of the drug. (b) Find the cost of seizing \(50 \%\) of the drug. (c) Find the cost of seizing \(75 \%\) of the drug. (d) According to this model, would it be possible to seize \(100 \%\) of the drug? Explain.

Reasoning Let \(f\) be a fourth-degree polynomial function with real coefficients. Three of the zeros of \(f\) are \(3,1+i\), and \(1-i\) Explain how you know that the fourth zero must be a real number.

Solve the quadratic equation and then use a graphing utility to graph the related quadratic function in the standard viewing window. Discuss how the graph of the quadratic function relates to the solutions of the quadratic equation. Function \(y=x^{2}+x+2\) Equation $$x^{2}+x+2=0$$

Reasoning Is it possible that a second-degree polynomial function with integer coefficients has one rational zero and one irrational zero? If so, give an example.

Simplify the rational expression. $$\frac{x^{3}-10 x^{2}+31 x-30}{x-3}$$

Comparing Graphs Use a graphing utility to graph the functions given by \(f(x)=x^{2}, g(x)=x^{4}\), and \(h(x)=x^{6}\). Do the three functions have a common shape? Are their graphs identical? Why or why not?

MAKE A DECISION: WATER POLLUTION The cost \(\bar{C}\) (in millions of dollars) of removing \(p\) percent of the industrial and municipal pollutants discharged into a river is \(C=\frac{255 p}{100-p}, \quad 0 \leq p<100\) (a) Find the cost of removing \(15 \%\) of the pollutants. (b) Find the cost of removing \(50 \%\) of the pollutants. (c) Find the cost of removing \(80 \%\) of the pollutants. (d) According to the model, would it be possible to remove \(100 \%\) of the pollutants? Explain.