Chapter 4

Pharmaceuticals When a certain drug is taken orally, the concentration of the drug in the patient's bloodstream after \(t\) minutes is given by \(C(t)=0.06 t-0.0002 t^{2},\) where \(0 \leq t \leq 240\) and the concentration is measured in mg/L. When is the maximum serum concentration reached, and what is that maximum concentration?

A polynomial P is given. (a) Factor P into linear and irreducible quadratic factors with real coefficients. (b) Factor P completely into linear factors with complex coefficients. \(P(x)=x^{5}-16 x\)

Use Descartes’ Rule of Signs to determine how many positive and how many negative real zeros the polynomial can have. Then determine the possible total number of real zeros. $$ P(x)=x^{5}+4 x^{3}-x^{2}+6 x $$

Find the slant asymptote, the vertical asymptotes, and sketch a graph of the function. $$ r(x)=\frac{x^{2}+5 x+4}{x-3} $$

\(69-74\) . 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. $$ P(x)=c x^{3} ; \quad c=1,2,5, \frac{1}{2} $$

Agriculture The number of apples produced by each tree in an apple orchard depends on how densely the trees are planted. If \(n\) trees are planted on an acre of land, then each tree produces \(900-9 n\) apples. So the number of apples produced per acre is $$ A(n)=n(900-9 n) $$ How many trees should be planted per acre to obtain the maximum yield of apples?

By the Zeros Theorem, every nth-degree polynomial equation has exactly n solutions (including possibly some that are repeated). Some of these may be real, and some may be imaginary. Use a graphing device to determine how many real and imaginary solutions each equation has. (a) \(x^{4}-2 x^{3}-11 x^{2}+12 x=0\) (b) \(x^{4}-2 x^{3}-11 x^{2}+12 x-5=0\) (c) \(x^{4}-2 x^{3}-11 x^{2}+12 x+40=0\)

Use Descartes’ Rule of Signs to determine how many positive and how many negative real zeros the polynomial can have. Then determine the possible total number of real zeros. $$ P(x)=x^{8}-x^{5}+x^{4}-x^{3}+x^{2}-x+1 $$

Find the slant asymptote, the vertical asymptotes, and sketch a graph of the function. $$ r(x)=\frac{x^{3}+4}{2 x^{2}+x-1} $$

\(69-74\) . 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. $$ P(x)=(x-c)^{4} ; \quad c=-1,0,1,2 $$

Nested Form of a Polynomial Expand \(Q\) to prove that the polynomials \(P\) and \(Q\) are the same. $$ \begin{array}{l}{P(x)=3 x^{4}-5 x^{3}+x^{2}-3 x+5} \\ {Q(x)=(((3 x-5) x+1) x-3) x+5}\end{array} $$ Try to evaluate \(P(2)\) and \(Q(2)\) in your head, using the forms given. Which is easier? Now write the polynomial \(R(x)=x^{5}-2 x^{4}+3 x^{3}-2 x^{2}+3 x+4\) in "nested" form, like the polynomial \(Q\) . Use the nested form to find \(R(3)\) in your head. Do you see how calculating with the nested form follows the same arithmetic steps as calculating the value of a polynomial using synthetic division?

Agriculture At a certain vineyard it is found that each grape vine produces about 10 pounds of grapes in a season when about 700 vines are planted per acre. For each additional vine that is planted, the production of each vine decreases by about 1 percent. So the number of pounds of grapes produced per acre is modeled by $$ A(n)=(700+n)(10-0.01 n) $$ where \(n\) is the number of additional vines planted. Find the number of vines that should be planted to maximize grape production.

So far, we have worked only with polynomials that have real coefficients. These exercises involve polynomials with real and imaginary coefficients. Find all solutions of the equation. (a) \(2 x+4 i=1\) (b) \(x^{2}-i x=0\) (c) \(x^{2}+2 i x-1=0\) (d) \(i x^{2}-2 x+i=0\)

Show that the given values for \(a\) and \(b\) are lower and upper bounds for the real zeros of the polynomial. $$ P(x)=2 x^{3}+5 x^{2}+x-2 ; \quad a=-3, b=1 $$

Find the slant asymptote, the vertical asymptotes, and sketch a graph of the function. $$ r(x)=\frac{x^{3}+x^{2}}{x^{2}-4} $$

\(69-74\) . 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. $$ P(x)=x^{4}+c, \quad c=-1,0,1,2 $$

So far, we have worked only with polynomials that have real coefficients. These exercises involve polynomials with real and imaginary coefficients. (a) Show that 2\(i\) and \(1-i\) are both solutions of the equation $$x^{2}-(1+i) x+(2+2 i)=0$$ but that their complex conjugates \(-2 i\) and \(1+i\) are not. (b) Explain why the result of part (a) does not violate the Conjugate Zeros Theorem.

Show that the given values for \(a\) and \(b\) are lower and upper bounds for the real zeros of the polynomial. $$ P(x)=x^{4}-2 x^{3}-9 x^{2}+2 x+8 ; \quad a=-3, b=5 $$

Find the slant asymptote, the vertical asymptotes, and sketch a graph of the function. $$ r(x)=\frac{2 x^{3}+2 x}{x^{2}-1} $$

So far, we have worked only with polynomials that have real coefficients. These exercises involve polynomials with real and imaginary coefficients. (a) Find the polynomial with real coefficients of the smallest possible degree for which \(i\) and \(1+i\) are zeros and in which the coefficient of the highest power is \(1 .\) (b) Find the polynomial with complex coefficients of the smallest possible degree for which \(i\) and \(1+i\) are zeros and in which the coefficient of the highest power is \(1 .\)

Show that the given values for \(a\) and \(b\) are lower and upper bounds for the real zeros of the polynomial. $$ P(x)=8 x^{3}+10 x^{2}-39 x+9 ; \quad a=-3, b=2 $$

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 \(f,\) and determine all vertical asymptotes from your graph. Then graph \(f\) and \(g\) in a sufficiently large viewing rectangle to show that they have the same end behavior. $$ f(x)=\frac{2 x^{2}+6 x+6}{x+3}, g(x)=2 x $$

\(69-74\) . 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. $$ P(x)=x^{4}-c x, \quad c=0,1,8,27 $$

Show that the given values for \(a\) and \(b\) are lower and upper bounds for the real zeros of the polynomial. $$ P(x)=3 x^{4}-17 x^{3}+24 x^{2}-9 x+1 ; \quad a=0, b=6 $$

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 \(f,\) and determine all vertical asymptotes from your graph. Then graph \(f\) and \(g\) in a sufficiently large viewing rectangle to show that they have the same end behavior. $$ f(x)=\frac{-x^{3}+6 x^{2}-5}{x^{2}-2 x}, g(x)=-x+4 $$

\(69-74\) . 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. $$ P(x)=x^{c} ; \quad c=1,3,5,7 $$

Graph the rational function \(f,\) and determine all vertical asymptotes from your graph. Then graph \(f\) and \(g\) in a sufficiently large viewing rectangle to show that they have the same end behavior. $$ f(x)=\frac{x^{3}-2 x^{2}+16}{x-2}, g(x)=x^{2} $$

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

Fencing a Horse Corral Carol has 2400 ft of fencing to fence in a rectangular horse corral. (a) Find a function that models the area of the corral in terms of the width x of the corral. (b) Find the dimensions of the rectangle that maximize the area of the corral.

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

Graph the rational function \(f,\) and determine all vertical asymptotes from your graph. Then graph \(f\) and \(g\) in a sufficiently large viewing rectangle to show that they have the same end behavior. $$ f(x)=\frac{-x^{4}+2 x^{3}-2 x}{(x-1)^{2}}, \quad g(x)=1-x^{2} $$

Find integers that are upper and lower bounds for the real zeros of the polynomial. $$ P(x)=x^{4}-2 x^{3}+x^{2}-9 x+2 $$

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. $$ y=\frac{2 x^{2}-5 x}{2 x+3} $$

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

Stadium Revenue A baseball team plays in a stadium that holds 55,000 spectators. With the ticket price at $10, the average attendance at recent games has been 27,000. A market survey indicates that for every dollar the ticket price is lowered, attendance increases by 3000. (a) Find a function that models the revenue in terms of ticket price. (b) Find the price that maximizes revenue from ticket sales. (c) What ticket price is so high that no revenue is generated?

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. $$ y=\frac{x^{4}-3 x^{3}+x^{2}-3 x+3}{x^{2}-3 x} $$

(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.

Maximizing Profit A community bird-watching society makes and sells simple bird feeders to raise money for its conservation activities. The materials for each feeder cost \(6, and the society sells an average of 20 per week at a price of \)10 each. The society has been considering raising the price, so it conducts a survey and finds that for every dollar increase, it loses 2 sales per week. (a) Find a function that models weekly profit in terms of price per feeder. (b) What price should the society charge for each feeder to maximize profits? What is the maximum weekly profit?

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' Zule of Signs, the Uuadratic 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. $$ y=\frac{x^{5}}{x^{3}-1} $$

(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' Zule of Signs, the Uuadratic 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. $$ y=\frac{x^{4}}{x^{2}-2} $$

(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 \(x\) -intercepts and 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.

Maximum of a Fourth-Degree Polynomial Find the maximum value of the function $$ f(x)=3+4 x^{2}-x^{4} $$ [Hint: Let \(t=x^{2} . ]\)

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' Zule of Signs, the Uuadratic formula, or other factoring techniques. $$ P(x)=4 x^{4}-21 x^{2}+5 $$

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. $$ r(x)=\frac{x^{4}-3 x^{3}+6}{x-3} $$