Chapter 3

3\. Reasoning Let \(f\) be a fourth-degree polynomial function with real coefficients. Three of the zeros of \(f\) are \(-1,2\), and \(3+2 i\) What is the fourth zero? Explain

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 \(y=-x^{2}+3 x-5\) Equation $$-x^{2}+3 x-5=0$$

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

Simplify the rational expression. $$\frac{x^{3}+15 x^{2}+68 x+96}{x+4}$$

Comparing Graphs Use a graphing utility to graph the functions given by \(f(x)=x^{3}, g(x)=x^{5}\), and \(h(x)=x^{7} .\) Do the three functions have a common shape? Are their graphs identical? Why or why not?

Population of Deer The Game Commission introduces 100 deer into newly acquired state game lands. The population \(N\) of the herd is given by \(N=\frac{25(4+2 t)}{1+0.02 t}, \quad t \geq 0\) where \(t\) is time (in years). (a) Find the populations when \(t\) is 5,10 , and 25 . (b) What is the limiting size of the herd as time progresses?

Reasoning Let \(f\) be a third-degree polynomial function with real coefficients. Explain how you know that \(f\) must have at least one zero that is 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 \(y=x^{2}+3 x-5\) Equation $$x^{2}+3 x-5=0$$

Simplify the rational expression. $$\frac{6 x^{3}+x^{2}-21 x-10}{2 x+1}$$

Population of Elk The Game Commission introduces 40 elk into newly acquired state game lands. The population \(N\) of the herd is given by \(N=\frac{10(4+2 t)}{1+0.03 t}, \quad t \geq 0\) where \(t\) is time (in years). (a) Find the populations when \(t\) is 5,10, and 25 . (b) What is the limiting size of the herd as time progresses?

Reasoning Let \(f\) be a fifth-degree polynomial function with real coefficients. Explain how you know that \(f\) must have at least one zero that is 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 \(y=-x^{2}-3 x+4\) Equation $$-x^{2}-3 x+4=0$$

Simplify the rational expression. $$\frac{3 x^{3}-5 x^{2}-34 x+24}{3 x-2}$$

Defense The table shows the national defense outlays \(D\) (in billions of dollars) from 1997 to \(2005 .\) The data can be modeled by \(D=\frac{1.493 t^{2}-39.06 t+273.5}{0.0051 t^{2}-0.1398 t+1}, \quad 7 \leq t \leq 15\) where \(t\) is the year, with \(t=7\) corresponding to 1997 . (Source: U.S. Office of Management and Budget) $$ \begin{array}{|c|c|} \hline \text { Year } & \begin{array}{c} \text { Defense } \\ \text { outlays } \end{array} \\ \hline 1997 & 270.5 \\ \hline 1998 & 268.5 \\ \hline 1999 & 274.9 \\ \hline 2000 & 294.5 \\ \hline 2001 & 305.5 \\ \hline \end{array} $$ $$ \begin{array}{|c|c|} \hline \text { Year } & \begin{array}{c} \text { Defense } \\ \text { outlays } \end{array} \\ \hline 2002 & 348.6 \\ \hline 2003 & 404.9 \\ \hline 2004 & 455.9 \\ \hline 2005 & 465.9 \\ \hline \end{array} $$ (a) Use a graphing utility to plot the data and graph the model in the same viewing window. How well does the model represent the data? (b) Use the model to predict the national defense outlays for the years 2010,2015, and 2020 . Are the predictions reasonable? (c) Determine the horizontal asymptote of the graph of the model. What does it represent in the context of the situation?

Think About It A student claims that a third-degree polynomial function with real coefficients can have three complex zeros. Describe how you could use a graphing utility and the Leading Coefficient Test (Section 3.2) to convince the student otherwise.

Simplify the rational expression. $$\frac{x^{4}-5 x^{3}+14 x^{2}-120 x}{x^{2}+x+20}$$

Average Cost The cost \(C\) (in dollars) of producing \(\underline{x}\) basketballs is \(C=375,000+4 x\). The average cost \(\bar{C}\) per basketball is \(\bar{C}=\frac{C}{x}=\frac{375,000+4 x}{x}, x>0\) (a) Sketch the graph of the average cost function. (b) Find the average costs of producing \(1000,10,000\), and 100,000 basketballs. (c) Find the horizontal asymptote and explain its meaning in the context of the problem.

Think About It A student claims that the polynomial \(x^{4}-7 x^{2}+12\) may be factored over the rational numbers as \((x-\sqrt{3})(x+\sqrt{3})(x-2)(x+2)\) Do you agree with this claim? Explain your answer.

Plot the complex number. $$i$$

Simplify the rational expression. $$\frac{x^{4}+x^{3}+3 x^{2}+10 x}{x^{2}-x+5}$$

Human Memory Model Psychologists have developed mathematical models to predict memory performance as a function of the number of trials \(n\) of a certain task. Consider the learning curve modeled by \(P=\frac{0.6+0.95(n-1)}{1+0.95(n-1)}, \quad n>0\) where \(P\) is the percent of correct responses (in decimal form) after \(n\) trials. (a) Complete the table. $$ \begin{array}{|l|l|l|l|l|l|l|l|l|l|l|} \hline n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline P & & & & & & & & & & \\ \hline \end{array} $$ (b) According to this model, what is the limiting percent of correct responses as \(n\) increases?

Plot the complex number. $$-2+i$$

Simplify the rational expression. $$\frac{x^{4}+4 x^{3}-6 x^{2}-36 x-27}{x^{2}-9}$$

Human Memory Model Consider the learning curve modeled by \(P=\frac{0.55+0.87(n-1)}{1+0.87(n-1)}, \quad n>0\) where \(P\) is the percent of correct responses (in decimal form) after \(n\) trials. (a) Complete the table. $$ \begin{array}{|l|l|l|l|l|l|l|l|l|l|l|} \hline n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline P & & & & & & & & & & \\ \hline \end{array} $$ (b) According to this model, what is the limiting percent of correct responses as \(n\) increases?

Plot the complex number. $$-2-3 i$$

Simplify the rational expression. $$\frac{x^{4}+x^{3}-13 x^{2}-x+12}{x^{2}+x-12}$$

Average Recycling Cost The cost \(C\) (in dollars) of recycling a waste product is \(C=450,000+6 x, \quad x>0\) where \(x\) is the number of pounds of waste. The average recycling cost \(\bar{C}\) per pound is \(\bar{C}=\frac{C}{x}=\frac{450,000+6 x}{x}, x>0\) (a) Use a graphing utility to graph \(\bar{C}\). (b) Find the average costs of recycling \(10,000,100,000\), \(1,000,000\), and \(10,000,000\) pounds of waste. What can you conclude?

Plot the complex number. $$1-2 i$$

Examination Room A rectangular examination room in a veterinary clinic has a volume of \(x^{3}+11 x^{2}+34 x+24\) cubic feet. The height of the room is \(x+1\) feet (see figure). Find the number of square feet of floor space in the examination room.

Drug Concentration The concentration \(C\) of a medication in the bloodstream \(t\) minutes after sublingual (under the tongue) application is given by \(C(t)=\frac{3 t-1}{2 t^{2}+5}, \quad t>0\) (a) Use a graphing utility to graph the function. Estimate when the concentration is greatest. (b) Does this function have a horizontal asymptote? If so, discuss the meaning of the asymptote in terms of the concentration of the medication.

Plot the complex number. $$-2 i$$

Veterinary Clinic A rectangular veterinary clinic has a volume of \(x^{3}+55 x^{2}+650 x+2000\) cubic feet (the space in the attic is not counted). The height of the clinic is \(x+5\) feet (see figure). Find the number of square feet of floor space on the first floor of the clinic.

Domestic Demand The U.S. domestic demand \(D\) (in millions of barrels) for refined oil products from 1995 to 2005 can be modeled by \(D=100.9708 t+6083.999,5 \leq t \leq 15\) where \(t\) represents the year, with \(t=5\) corresponding to 1995\. The population \(P\) (in millions) of the United States from 1995 to 2005 can be modeled by \(P=3.0195 t+251.817, \quad 5 \leq t \leq 15\) where \(t\) represents the year, with \(t=5\) corresponding to 1995\. (Sources: U.S. Energy Information Administration and the U.S. Census Bureau) (a) Construct a rational function \(B\) to describe the per capita demand for refined oil products. (b) Use a graphing utility to graph the rational function \(B\). (c) Use the model to predict the per capita demand for refined oil products in \(2010 .\)

Decide whether the number is in the Mandelbrot Set. Explain your reasoning. $$c=0$$

Health Care Spending The total health care spending \(H\) (in millions of dollars) in the United States from 1995 to 2005 can be modeled by \(H=6136.36 t^{2}-22,172.7 t+979,909, \quad 5 \leq t \leq 15\) where \(t\) represents the year, with \(t=5\) corresponding to 1995\. The population \(P\) (in millions) of the United States from 1995 to 2005 can be modeled by \(P=3.0195 t+251.817, \quad 5 \leq t \leq 15\) where \(t\) represents the year, with \(t=5\) corresponding to 1995\. (Sources: U.S. Centers for Medicare and Medicaid Services and the U.S. Census Bureau) (a) Construct a rational function \(S\) to describe the per capita health spending. (b) Use a graphing utility to graph the rational function \(S\). (c) Use the model to predict the per capita health care spending in 2010 .

Decide whether the number is in the Mandelbrot Set. Explain your reasoning. $$c=2$$

Profit A company that produces calculators estimated that the profit \(P\) (in dollars) from selling a particular model of calculator was \(P=-152 x^{3}+7545 x^{2}-169,625, \quad 0 \leq x \leq 45\) where \(x\) was the advertising expense (in tens of thousands of dollars). For this model of calculator, the advertising expense was \(\$ 400,000(x=40)\) and the profit was \(\$ 2,174,375\). (a) Use a graphing utility to graph the profit function. (b) Could the company have obtained the same profit by spending less on advertising? Explain your reasoning.

100-Meter Freestyle The winning times for the men's 100-meter freestyle swim at the Olympics from 1952 to 2004 can be approximated by the quadratic model \(y=86.24-0.752 t+0.0037 t^{2}, \quad 52 \leq t \leq 104\) where \(y\) is the winning time (in seconds) and \(t\) represents the year, with \(t=52\) corresponding to \(1952 .\) (Sources: The World Almanac and Book of Facts 2005 ) (a) Use a graphing utility to graph the model. (b) Use the model to predict the winning times in 2008 and \(2012 .\) (c) Does this model have a horizontal asymptote? Do you think that a model for this type of data should have a horizontal asymptote?

Writing Briefly explain what it means for a divisor to divide evenly into a dividend.

3000-Meter Speed Skating The winning times for the women's 3000 -meter speed skating race at the Olympics from 1960 to 2006 can be approximated by the quadratic model \(y=0.0202 t^{2}-5.066 t+550.24, \quad 60 \leq t \leq 106\) where \(y\) is the winning time (in seconds) and \(t\) represents the year, with \(t=60\) corresponding to \(1960 .\) (Sources: World Almanac and Book of Facts 2005 and NBC) (a) Use a graphing utility to graph the model. (b) Use the model to predict the winning times in 2010 and \(2014 .\) (c) Does this model have a horizontal asymptote? Do you think that a model for this type of data should have a horizontal asymptote?

Decide whether the number is in the Mandelbrot Set. Explain your reasoning. $$c=-1$$

Writing Briefly explain how to check polynomial division, and justify your answer. Give an example.

Find the constant c such that the denominator will divide evenly into the numerator. $$\frac{x^{3}+4 x^{2}-3 x+c}{x-5}$$

Decide whether the number is in the Mandelbrot Set. Explain your reasoning. $$c=-i$$

Find the constant c such that the denominator will divide evenly into the numerator. $$\frac{x^{5}-2 x^{2}+x+c}{x+2}$$

Determine whether the statement is true or false. Explain. The conjugate of the sum of two complex numbers is equal to the sum of the conjugates of the two complex numbers.