Chapter 4

Find integers that are upper and lower bounds for the real zeros of the polynomial. $$ P(x)=x^{3}-3 x^{2}+4 $$

Polynomials of Odd Degree The Conjugate Zeros Theorem says that the complex zeros of a polynomial with real coefficients occur in complex conjugate pairs. Explain how this fact proves that a polynomial with real coefficients and odd degree has at least one real zero.

Graph the rational function and find all vertical asymptotes, \(x\) - and \(y\) -intercepts, and local extrema, correct to the nearest decimal. Then use long division to find a polynomial that has the same end behavior as the rational function, and graph both functions in a sufficiently large viewing rectangle to verify that the end behaviors of the polynomial and the rational function are the same. \(y=\frac{2 x^{2}-5 x}{2 x+3}\)

Graph the family of polynomials in the same viewing rectangle, using the given values of \(c .\) Explain how changing the value of \(c\) affects the graph. $$ P(x)=x^{4}-c x ; \quad c=0,1,8,27 $$

Find integers that are upper and lower bounds for the real zeros of the polynomial. $$ P(x)=2 x^{3}-3 x^{2}-8 x+12 $$

Roots of Unity There are two square roots of \(1,\) namely 1 and \(-1 .\) These are the solutions of \(x^{2}=1 .\) The fourth roots of 1 are the solutions of the equation \(x^{4}=1\) or \(x^{4}-1=0 .\) How many fourth roots of 1 are there? Find them. The cube roots of 1 are the solutions of the equation \(x^{3}=1\) or \(x^{3}-1=0\) How many cube roots of 1 are there? Find them. How would you find the sixth roots of 1\(?\) How many are there? Make a conjecture about the number of \(n\) th roots of \(1 .\)

Graph the rational function and find all vertical asymptotes, \(x\) - and \(y\) -intercepts, and local extrema, correct to the nearest decimal. Then use long division to find a polynomial that has the same end behavior as the rational function, and graph both functions in a sufficiently large viewing rectangle to verify that the end behaviors of the polynomial and the rational function are the same. \(y=\frac{x^{4}-3 x^{3}+x^{2}-3 x+3}{x^{2}-3 x}\)

Graph the rational function and find all vertical asymptotes, \(x\) - and \(y\) -intercepts, and local extrema, correct to the nearest decimal. Then use long division to find a polynomial that has the same end behavior as the rational function, and graph both functions in a sufficiently large viewing rectangle to verify that the end behaviors of the polynomial and the rational function are the same. \(y=\frac{x^{5}}{x^{3}-1}\)

(a) On the same coordinate axes, sketch graphs (as accu- rately as possible) of the functions \(y=x^{3}-2 x^{2}-x+2 \quad\) and \(\quad y=-x^{2}+5 x+2\) (b) Based on your sketch in part (a), at how many points do the two graphs appear to intersect? (c) Find the coordinates of all intersection points.

Find integers that are upper and lower bounds for the real zeros of the polynomial. $$ P(x)=x^{5}-x^{4}+1 $$

Graph the rational function and find all vertical asymptotes, \(x\) - and \(y\) -intercepts, and local extrema, correct to the nearest decimal. Then use long division to find a polynomial that has the same end behavior as the rational function, and graph both functions in a sufficiently large viewing rectangle to verify that the end behaviors of the polynomial and the rational function are the same. \(y=\frac{x^{4}}{x^{2}-2}\)

Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes’ Rule of Signs, the quadratic formula, or other factoring techniques. $$ P(x)=2 x^{4}+3 x^{3}-4 x^{2}-3 x+2 $$

Graph the rational function and find all vertical asymptotes, \(x\) - and \(y\) -intercepts, and local extrema, correct to the nearest decimal. Then use long division to find a polynomial that has the same end behavior as the rational function, and graph both functions in a sufficiently large viewing rectangle to verify that the end behaviors of the polynomial and the rational function are the same. \(r(x)=\frac{x^{4}-3 x^{3}+6}{x-3}\)

Recall that a function \(f\) is odd if \(f(-x)=-f(x)\) or even if \(f(-x)=f(x)\) for all real \(x .\) $$\begin{array}{l}{\text { (a) Show that a polynomial } P(x) \text { that contains only odd }} \\ {\text { powers of } x \text { is an odd function. }} \\\ {\text { (b) Show that a polynomial } P(x) \text { that contains only even }} \\ {\text { powers of } x \text { is an even function. }} \\ {\text { (c) Show that if a polynomial } P(x) \text { contains both odd and }} \\ {\text { even powers of } x, \text { then it is neither an odd nor an even }} \\\ {\text { function. }}\end{array}$$ $$ \begin{array}{l}{\text { (d) Express the function }} \\ {\qquad P(x)=x^{5}+6 x^{3}-x^{2}-2 x+5} \\ {\text { as the sum of an odd function and an even function. }}\end{array} $$

Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes’ Rule of Signs, the quadratic formula, or other factoring techniques. $$ P(x)=2 x^{4}+15 x^{3}+31 x^{2}+20 x+4 $$

Graph the rational function and find all vertical asymptotes, \(x\) - and \(y\) -intercepts, and local extrema, correct to the nearest decimal. Then use long division to find a polynomial that has the same end behavior as the rational function, and graph both functions in a sufficiently large viewing rectangle to verify that the end behaviors of the polynomial and the rational function are the same. \(r(x)=\frac{4+x^{2}-x^{4}}{x^{2}-1}\)

(a) Graph the function \(P(x)=(x-1)(x-3)(x-4)\) and find all local extrema, correct to the nearest tenth. (b) Graph the function $$Q(x)=(x-1)(x-3)(x-4)+5$$ and use your answers to part (a) to find all local extrema, correct to the nearest tenth.

Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes’ Rule of Signs, the quadratic formula, or other factoring techniques. $$ P(x)=4 x^{4}-21 x^{2}+5 $$

Suppose that the rabbit population on Mr. Jenkins' farm follows the formula $$p(t)=\frac{3000 t}{t+1}$$ where \(t \geq 0\) is the time (in months) since the beginning of the year. (a) Draw a graph of the rabbit population. (b) What eventually happens to the rabbit population?

(a) Graph the function \(P(x)=(x-2)(x-4)(x-5)\) and determine how many local extrema it has. (b) If \(a

Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes’ Rule of Signs, the quadratic formula, or other factoring techniques. $$ P(x)=6 x^{4}-7 x^{3}-8 x^{2}+5 x $$

After a certain drug is injected into a patient, the concentration \(c\) of the drug in the bloodstream is monitored. At time \(t \geq 0\) (in minutes since the injection), the concentration (in \(\mathrm{mg} / \mathrm{L}\) ) is given by $$c(t)=\frac{30 t}{t^{2}+2}$$ (a) Draw a graph of the drug concentration. (b) What eventually happens to the concentration of drug in the bloodstream?

(a) How many \(x\) -intercepts and how many local extrema does the polynomial \(P(x)=x^{3}-4 x\) have? (b) How many \(x\) -intercepts and how many local extrema does the polynomial \(Q(x)=x^{3}+4 x\) have? (c) If \(a>0,\) how many \(Q(x)=x^{3}+4 x\) how many local extrema does each of the polynomials \(P(x)=x^{3}-a x\) and \(Q(x)=x^{3}+a x\) have? Explain your answer.

Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes’ Rule of Signs, the quadratic formula, or other factoring techniques. $$ P(x)=x^{5}-7 x^{4}+9 x^{3}+23 x^{2}-50 x+24 $$

A drug is administered to a patient and the concentration of the drug in the bloodstream is monitored. At time \(t \geq 0\) (in hours since giving the drug), the concentration (in \(\mathrm{mg} / \mathrm{L}\) ) is given by $$c(t)=\frac{5 t}{t^{2}+1}$$ Graph the function \(c\) with a graphing device. (a) What is the highest concentration of drug that is reached in the patient's bloodstream? (b) What happens to the drug concentration after a long period of time? (c) How long does it take for the concentration to drop below 0.3 \(\mathrm{mg} / \mathrm{L} ?\)

Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes’ Rule of Signs, the quadratic formula, or other factoring techniques. $$ P(x)=8 x^{5}-14 x^{4}-22 x^{3}+57 x^{2}-35 x+6 $$

Suppose a rocket is fired upward from the surface of the earth with an initial velocity \(v\) (measured in \(\mathrm{m} / \mathrm{s}\) ). Then the maximum height \(h\) (in meters) reached by the rocket is given by the function $$h(v)=\frac{R v^{2}}{2 g R-v^{2}}$$ where \(R=6.4 \times 10^{6} \mathrm{m}\) is the radius of the earth and \(g=9.8 \mathrm{m} / \mathrm{s}^{2}\) is the acceleration due to gravity. Use a graphing device to draw a graph of the function \(h .\) Note that \(h\) and \(v\) must both be positive, so the viewing rectangle need not contain negative values.) What does the vertical asymptote represent physically?

Population Change The rabbit population on a small is- land is observed to be given by the function $$P(t)=120 t-0.4 t^{4}+1000$$ where \(t\) is the time (in months) since observations of the island began. (a) When is the maximum population attained, and what is that maximum population? (b) When does the rabbit population disappear from the island?

Show that the polynomial does not have any rational zeros. $$ P(x)=x^{3}-x-2 $$

As a train moves toward an observer (see the figure), the pitch of its whistle sounds higher to the observer than it would if the train were at rest, because the crests of the sound waves are compressed closer together. This phenomenon is called the Doppler effect. The observed pitch \(P\) is a function of the speed \(v\) of the train and is given by $$P(v)=P_{0}\left(\frac{s_{0}}{s_{0}-v}\right)$$ where \(P_{0}\) is the actual pitch of the whistle at the source and \(s_{0}=332 \mathrm{m} / \mathrm{s}\) is the speed of sound in air. Suppose that a train has a whistle pitched at \(P_{0}=440 \mathrm{Hz}\) . Graph the function \(y=P(v)\) using a graphing device. How can the vertical asymptote of this function be interpreted physically?

Volume of a Box An open box is to be constructed from a piece of cardboard 20 cm by 40 cm by cutting squares of side length x from each corner and folding up the sides, as shown in the figure. $$ \begin{array}{l}{\text { (a) Express the volume } V \text { of the box as a function of } x .} \\ {\text { (b) What is the domain of } V ? \text { (Use the fact that length and }} \\ {\text { volume must be positive.) }} \\\ {\text { (c) Draw a graph of the function } V \text { and use it to estimate }} \\ {\text { the maximum volume for such a box. }}\end{array} $$

Show that the polynomial does not have any rational zeros. $$ P(x)=2 x^{4}-x^{3}+x+2 $$

For a camera with a lens of fixed focal length \(F\) to focus on an object located a distance \(x\) from the lens, the film must be placed a distance \(y\) behind the lens, where \(F, x,\) and \(y\) are related by $$\frac{1}{x}+\frac{1}{y}=\frac{1}{F}$$ (See the figure.) Suppose the camera has a \(55-\mathrm{mm}\) lens \((F=55) .\) (a) Express \(y\) as a function of \(x\) and graph the function. (b) Express happens to the focusing distance \(y\) as the object moves far away from the lens? (c) What happens to the focusing distance \(y\) as the object moves close to the lens?

Show that the polynomial does not have any rational zeros. $$ P(x)=3 x^{3}-x^{2}-6 x+12 $$

Give an example of a rational function that has vertical asymptote \(x=3 .\) Now give an example of one that has vertical asymptote \(x=3\) and horizontal asymptote \(y=2 .\) Now give an example of a rational function with vertical asymptotes \(x=1\) and \(x=-1\) , horizontal asymptote \(y=0,\) and \(x\) -intercept 4 .

Show that the polynomial does not have any rational zeros. $$ P(x)=x^{50}-5 x^{25}+x^{2}-1 $$

Explain how you can tell (without graphing it) that the function $$r(x)=\frac{x^{6}+10}{x^{4}+8 x^{2}+15}$$ has no \(x\) -intercept and no horizontal, vertical, or slant asymptote. What is its end behavior?

The real solutions of the given equation are rational. List all possible rational roots using the Rational Zeros Theorem, and then graph the polynomial in the given viewing rectangle to determine which values are actually solutions. (All solutions can be seen in the given viewing rectangle.) $$ x^{3}-3 x^{2}-4 x+12=0 ;[-4,4] \text { by }[-15,15] $$

In this chapter we adopted the convention that in rational functions, the numerator and denominator don't share a common factor. In this exercise we consider the graph of a rational function that doesn't satisfy this rule. (a) Show that the graph of $$r(x)=\frac{3 x^{2}-3 x-6}{x-2}$$ is the line \(y=3 x+3\) with the point \((2,9)\) removed. [Hint: Factor. What is the domain of \(r\)?] (b) Graph the rational functions: $$\begin{aligned} s(x) &=\frac{x^{2}+x-20}{x+5} \\ t(x) &=\frac{2 x^{2}-x-1}{x-1} \\ u(x) &=\frac{x-2}{x^{2}-2 x} \end{aligned}$$

Possible Number of Local Extrema Is it possible for a third-degree polynomial to have exactly one local extremum? Can a fourth-degree polynomial have exactly two local extrema? How many local extrema can polynomials of third, fourth, fifth, and sixth degree have? (Think about the end behavior of such polynomials.) Now give an example of a polynomial that has six local extrema.

The real solutions of the given equation are rational. List all possible rational roots using the Rational Zeros Theorem, and then graph the polynomial in the given viewing rectangle to determine which values are actually solutions. (All solutions can be seen in the given viewing rectangle.) $$ x^{3}-3 x^{2}-4 x+12=0 ;[-4,4] \text { by }[-15,15] $$

In Example 2 we saw that some simple rational functions can be graphed by shifting, stretching, or reflecting the graph of \(y=1 / x .\) In this exercise we consider rational functions that can be graphed by transforming the graph of \(y=1 / x^{2},\) shown on the following page. (a) Graph the function $$r(x)=\frac{1}{(x-2)^{2}}$$ by transforming the graph of \(y=1 / x^{2}\) (b) Use long division and factoring to show that the function $$s(x)=\frac{2 x^{2}+4 x+5}{x^{2}+2 x+1}$$ can be written as $$s(x)=2+\frac{3}{(x+1)^{2}}$$ Then graph \(s\) by transforming the graph of \(y=1 / x^{2}\) . (c) One of the following functions can be graphed by transforming the graph of \(y=1 / x^{2} ;\) the other cannot. Use transformations to graph the one that can be, and explain why this method doesn't work for the other one. $$p(x)=\frac{2-3 x^{2}}{x^{2}-4 x+4} \quad q(x)=\frac{12 x-3 x^{2}}{x^{2}-4 x+4}$$

Impossible Situation? Is it possible for a polynomial to have two local maxima and no local minimum? Explain.

The real solutions of the given equation are rational. List all possible rational roots using the Rational Zeros Theorem, and then graph the polynomial in the given viewing rectangle to determine which values are actually solutions. (All solutions can be seen in the given viewing rectangle.) $$ 2 x^{4}-5 x^{3}-14 x^{2}+5 x+12=0 ; \quad[-2,5] \text { by }[-40,40] $$

The real solutions of the given equation are rational. List all possible rational roots using the Rational Zeros Theorem, and then graph the polynomial in the given viewing rectangle to determine which values are actually solutions. (All solutions can be seen in the given viewing rectangle.) $$ 3 x^{3}+8 x^{2}+5 x+2=0 ;[-3,3] \text { by }[-10,10] $$

Use a graphing device to find all real solutions of the equation, correct to two decimal places. $$ x^{4}-x-4=0 $$

Use a graphing device to find all real solutions of the equation, correct to two decimal places. $$ 2 x^{3}-8 x^{2}+9 x-9=0 $$

Use a graphing device to find all real solutions of the equation, correct to two decimal places. $$ 4.00 x^{4}+4.00 x^{3}-10.96 x^{2}-5.88 x+9.09=0 $$

Use a graphing device to find all real solutions of the equation, correct to two decimal places. $$ x^{5}+2.00 x^{4}+0.96 x^{3}+5.00 x^{2}+10.00 x+4.80=0 $$

Show that the equation $$ x^{5}-x^{4}-x^{3}-5 x^{2}-12 x-6=0 $$ has exactly one rational root, and then prove that it must have either two or four irrational roots.