Chapter 3

Use Descartes' Rule of Signs to determine the number of positive and negative zeros of \(p\). You need not find the zeros. $$p(x)=3 x^{4}-2 x^{3}+3 x^{2}-4 x+1$$

Sketch a graph of the rational function. Indicate any vertical and horizontal asymptote(s) and all intercepts. $$f(x)=\frac{x-2}{2 x^{2}+x-3}$$

This set of exercises will draw on the ideas presented in this section and your general math background. Why can't the numbers \(i, 2 i, 1,\) and 2 be the set of zeros for some fourth- degree polynomial with real coefficients?

Find a function of the form \(y=c x^{k}\) that has the same end behavior as the given function. Confirm your \(\mathrm{re}\) sults with a graphing utility. $$g(x)=-3.6 x^{4}+4 x^{2}+x-20$$

Graph the polynomial function using a graphing utility. Then (a) approximate the \(x\) -intercept(s) of the graph of the function; (b) find the intervals on which the function is positive or negative; (c) approximate the values of \(x\) at which a local maximum or local minimum occurs; and (d) discuss any symmetries. $$f(x)=x^{3}+x^{2}+\frac{1}{2}$$

The concentration \(C(t)\) of a drug in a patient's bloodstream \(t\) hours after administration is given by $$C(t)=\frac{4 t}{3+t^{2}}$$ where \(C(t)\) is in milligrams per liter. During what time interval will the concentration be greater than 1 milligram per liter?

Use Descartes' Rule of Signs to determine the number of positive and negative zeros of \(p\). You need not find the zeros. $$p(x)=x^{5}+3 x^{4}-4 x^{2}+10$$

Sketch a graph of the rational function. Indicate any vertical and horizontal asymptote(s) and all intercepts. $$f(x)=\frac{x^{2}+x-6}{x^{2}-1}$$

For each polynomial function, (a) find a function of the form \(y=c x^{2}\) that has the same end behavior. (b) find the \(x\) - and \(y\) -intercept(s) of the graph. (c) find the interval(s) on which the value of the function is positive. (d) find the interval(s) on which the value of the function is negative. (e) use the information in parts ( \(a\) ) \(-\) (d) to sketch a graph of the function. $$f(x)=-2 x^{3}+8 x$$

Graph the polynomial function using a graphing utility. Then (a) approximate the \(x\) -intercept(s) of the graph of the function; (b) find the intervals on which the function is positive or negative; (c) approximate the values of \(x\) at which a local maximum or local minimum occurs; and (d) discuss any symmetries. $$f(x)=x^{4}+2 x^{3}-1$$

To print booklets, it costs 400 dollars plus an additional 0.50 dollars per booklet. What is the minimum number of booklets that must be printed so that the average cost per booklet is less than 0.55 dollars?

Use Descartes' Rule of Signs to determine the number of positive and negative zeros of \(p\). You need not find the zeros. $$p(x)=2 x^{5}-6 x^{3}+7 x^{2}-8$$

Sketch a graph of the rational function. Indicate any vertical and horizontal asymptote(s) and all intercepts. $$f(x)=\frac{x^{2}+3 x+2}{x^{2}-9}$$

For each polynomial function, (a) find a function of the form \(y=c x^{2}\) that has the same end behavior. (b) find the \(x\) - and \(y\) -intercept(s) of the graph. (c) find the interval(s) on which the value of the function is positive. (d) find the interval(s) on which the value of the function is negative. (e) use the information in parts ( \(a\) ) \(-\) (d) to sketch a graph of the function. $$f(x)=3 x^{3}-27 x$$

Graph the polynomial function using a graphing utility. Then (a) approximate the \(x\) -intercept(s) of the graph of the function; (b) find the intervals on which the function is positive or negative; (c) approximate the values of \(x\) at which a local maximum or local minimum occurs; and (d) discuss any symmetries. $$f(x)=-x^{4}+3 x-1$$

Sketch a graph of the rational function. Indicate any vertical and horizontal asymptote(s) and all intercepts. $$h(x)=\frac{1}{x^{2}+1}$$

To solve the inequality \(x(x+1)(x-1)<2,\) a student starts by setting up the following inequalities. $$x<2 ; \quad x+1<2 ; \quad x-1<2$$ Why is this the wrong way to start the problem? What is the correct way to start this problem?

Use Descartes' Rule of Signs to determine the number of positive and negative zeros of \(p\). You need not find the zeros. $$p(x)=x^{6}+4 x^{3}-3 x+7$$

For each polynomial function, (a) find a function of the form \(y=c x^{2}\) that has the same end behavior. (b) find the \(x\) - and \(y\) -intercept(s) of the graph. (c) find the interval(s) on which the value of the function is positive. (d) find the interval(s) on which the value of the function is negative. (e) use the information in parts ( \(a\) ) \(-\) (d) to sketch a graph of the function. $$g(x)=(x-3)(x+4)(x-1)$$

In this set of exercises, you will use polynomials to study real-world problems. A rectangular solid has height \(h\) and a square base. One side of the square base is 3 inches greater than the height. (a) Find an expression for the volume of the solid in terms of \(h\) (b) Sketch a graph of the volume function. (c) For what values of \(h\) does the volume function make sense?

Sketch a graph of the rational function. Indicate any vertical and horizontal asymptote(s) and all intercepts. $$h(x)=\frac{2}{x^{2}+4}$$

To solve the inequality \(\frac{x}{x+1} \geq 2,\) a student first "simplifies" the problem by multiplying both sides by \(x+1\) to get $$x \geq 2(x+1)$$ Why is this an incorrect way to start the problem?

Use Descartes' Rule of Signs to determine the number of positive and negative zeros of \(p\). You need not find the zeros. $$p(x)=5 x^{6}-7 x^{5}+4 x^{3}-6$$

For each polynomial function, (a) find a function of the form \(y=c x^{2}\) that has the same end behavior. (b) find the \(x\) - and \(y\) -intercept(s) of the graph. (c) find the interval(s) on which the value of the function is positive. (d) find the interval(s) on which the value of the function is negative. (e) use the information in parts ( \(a\) ) \(-\) (d) to sketch a graph of the function. $$f(x)=(x+1)(x-2)(x+3)$$

In this set of exercises, you will use polynomials to study real-world problems. Manufacturing An open box is to be made by cutting four squares of equal size from a 12 -inch by 12 -inch square piece of cardboard (one at each corner) and then folding up the sides. (a) Let \(x\) be the length of a side of the square cut from each corner. Find an expression for the volume of the box in terms of \(x\) (b) Sketch a graph of the volume function. (c) \(\quad\) Find the value of \(x\) that gives the maximum volume for the box.

Sketch a graph of the rational function and find all intercepts and slant asymptotes. Indicate all asymptotes onthe graph. $$g(x)=\frac{x^{2}}{x+4}$$

Find a polynomial \(p(x)\) such that \(p(x)>0\) has the solution set \((0,1) \cup(3, \infty) .\) There may be more than one correct answer.

Graph the function using a graphing utility, and find its zeros. $$f(x)=x^{3}-3 x^{2}-3 x-4$$

For each polynomial function, (a) find a function of the form \(y=c x^{2}\) that has the same end behavior. (b) find the \(x\) - and \(y\) -intercept(s) of the graph. (c) find the interval(s) on which the value of the function is positive. (d) find the interval(s) on which the value of the function is negative. (e) use the information in parts ( \(a\) ) \(-\) (d) to sketch a graph of the function. $$f(x)=-\frac{1}{2}\left(x^{2}-4\right)\left(x^{2}-1\right)$$

Gross Domestic Product (GDP) is the market value of all final goods and services produced within a country during a given time period. The following fifthdegree polynomial approximates the per capita GDP (the average GDP per person) for the United States for the years 1933 to 1950 \(g(x)=0.294 x^{5}-12.2 x^{4}+169 x^{3}-912 x^{2}+2025 x+4508\) where \(g(x)\) is in 1996 dollars and \(x\) is the number of ycars since \(1933 .\) Note that when dollar amounts are measured over time, they are converted to the dollar value for a specific base year. In this case, the base ycar is \(1996 .\) (Source: Economic History Scrvices) (a) Use this model to calculate the per capita GDP (in 1996 dollars) for the years \(1934,1942,\) and 1949 What do you observe? (b) Explain why this model may not be suitable for predicting the per capita GDP for the year 20024 (c) Use your graphing utility to find the year(s), during the period \(1933-1950\), when the GDP reached a local maximum.

Sketch a graph of the rational function and find all intercepts and slant asymptotes. Indicate all asymptotes onthe graph. $$g(x)=\frac{4 x^{2}}{x+3}$$

Find polynomials \(p(x)\) and \(q(x),\) with \(q(x)\) not a constant function, such that \(\frac{p(x)}{q(x)} \geq 0\) has the solution set \([3, \infty)\) There may be more than one correct answer.

Graph the function using a graphing utility, and find its zeros. $$g(x)=2 x^{5}+x^{4}-2 x-1$$

For each polynomial function, (a) find a function of the form \(y=c x^{2}\) that has the same end behavior. (b) find the \(x\) - and \(y\) -intercept(s) of the graph. (c) find the interval(s) on which the value of the function is positive. (d) find the interval(s) on which the value of the function is negative. (e) use the information in parts ( \(a\) ) \(-\) (d) to sketch a graph of the function. $$f(x)=\left(x^{2}-4\right)(x+1)(x-3)$$

In this set of exercises, you will use polynomials to study real-world problems. Coir is a fiber obtained from the husk of a coconut. It is used chiefly in making rope and floor mats. The amount of coir exported from India during the years 1995 to 2001 is summarized in the following table. (Source: Food and Agriculture Organization of the United Nations)$$\begin{array}{cc}\text { Year } & \begin{array}{c}\text { Quantity Exported } \\\\\text { (metric tons) }\end{array} \\\\\hline 1995 & 1,577 \\\1996 & 963 \\\1997 & 1,691 \\\1998 & 3,268 \\\1999 & 4,323 \\\2000 & 5,768 \\\2001 & 11,538\end{array}$$ (a) Make a scatter plot of the data, and find the cubic function of best fit for this data set. Let \(x\) be the number of years since 1995 (b) Use the cubic function to estimate the quantity of Indian coir exported in 2003 (c) Use the cubic function to estimate the quantity of coir exported in \(2001 .\) How close is this value to the actual data valuc? (d) Explain why the cubic function is not adequate for describing the long- term trend in exports of Indian coir.

Sketch a graph of the rational function and find all intercepts and slant asymptotes. Indicate all asymptotes onthe graph. $$g(x)=\frac{-3 x^{2}}{x-5}$$

Graph the function using a graphing utility, and find its zeros. $$h(x)=4 x^{3}-12 x^{2}+5 x+6$$

For each polynomial function, (a) find a function of the form \(y=c x^{2}\) that has the same end behavior. (b) find the \(x\) - and \(y\) -intercept(s) of the graph. (c) find the interval(s) on which the value of the function is positive. (d) find the interval(s) on which the value of the function is negative. (e) use the information in parts ( \(a\) ) \(-\) (d) to sketch a graph of the function. $$f(x)=x^{3}-2 x^{2}-3 x$$

Sketch a graph of the rational function and find all intercepts and slant asymptotes. Indicate all asymptotes onthe graph. $$h(x)=\frac{-x^{2}}{x-3}$$

Graph the function using a graphing utility, and find its zeros. $$p(x)=-x^{4}-x^{3}+18 x^{2}+16 x-32$$

This set of exercises will draw on the ideas presented in this section and your general math background. Sketch the graph of a cubic polynomial function with exactly two real zeros. There can be more than one correct answer.

Sketch a graph of the rational function and find all intercepts and slant asymptotes. Indicate all asymptotes onthe graph. $$h(x)=\frac{4-x^{2}}{x}$$

Graph the function using a graphing utility, and find its zeros. $$p(x)=-2 x^{4}+13 x^{3}-23 x^{2}+3 x+9$$

For each polynomial function, (a) find a function of the form \(y=c x^{2}\) that has the same end behavior. (b) find the \(x\) - and \(y\) -intercept(s) of the graph. (c) find the interval(s) on which the value of the function is positive. (d) find the interval(s) on which the value of the function is negative. (e) use the information in parts ( \(a\) ) \(-\) (d) to sketch a graph of the function. $$f(x)=-x(2 x+1)(x-3)$$

This set of exercises will draw on the ideas presented in this section and your general math background. Find a polynomial function whose zeros are \(x=0,1\) and \(-1 .\) Is your answer the only correct answer? Why or why not? You may confirm your answer with a graphing utility.

Sketch a graph of the rational function and find all intercepts and slant asymptotes. Indicate all asymptotes onthe graph. $$h(x)=\frac{x^{2}-9}{x}$$

Graph the function using a graphing utility, and find its zeros. $$f(x)=x^{3}+x^{2}+x-3.1 x^{2}-2.5 x-4$$

For each polynomial function, (a) find a function of the form \(y=c x^{2}\) that has the same end behavior. (b) find the \(x\) - and \(y\) -intercept(s) of the graph. (c) find the interval(s) on which the value of the function is positive. (d) find the interval(s) on which the value of the function is negative. (e) use the information in parts ( \(a\) ) \(-\) (d) to sketch a graph of the function. $$g(x)=2 x(x-2)(2 x-1)$$

This set of exercises will draw on the ideas presented in this section and your general math background. Find a polynomial function whose graph crosses the \(x\) -axis at (2,0) and \((1,0) .\) Is your answer the only correct answer? Why or why not? You may confirm your answer with a graphing utility.

Sketch a graph of the rational function and find all intercepts and slant asymptotes. Indicate all asymptotes onthe graph. $$h(x)=\frac{x^{2}+x+1}{x-1}$$