Chapter 3

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. $$P(x)=x^{4}+8 x^{2}+16$$

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. $$P(x)=2 x^{3}-x^{2}+4 x-7$$

\(A\) soft-drink vendor at a popular beach analyzes his sales records and finds that if he sells \(x\) cans of soda pop in one day, his profit (in dollars) is given by $$P(x)=-0.001 x^{2}+3 x-1800$$ What is his maximum profit per day, and how many cans must he sell for maximum profit?

Graph the polynomial and determine how many local maxima and minima it has. $$y=\left(x^{2}-2\right)^{3}$$

Find all solutions of the equation and express them in the form \(a+b i\) $$t+3+\frac{3}{t}=0$$

Find the slant asymptote, the vertical asymptotes, and sketch a graph of the function. $$r(x)=\frac{x^{2}-2 x-8}{x}$$

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. $$P(x)=x^{6}-64$$

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. $$P(x)=2 x^{6}+5 x^{4}-x^{3}-5 x-1$$

The effectiveness of a television commercial depends on how many times a viewer watches it. After some experiments an advertising agency found that if the effectiveness \(E\) is measured on a scale of 0 to \(10,\) then $$E(n)=\frac{2}{3} n-\frac{1}{90} n^{2}$$ where \(n\) is the number of times a viewer watches a given commercial. For a commercial to have maximum effectiveness, how many times should a viewer watch it?

Graph the polynomial and determine how many local maxima and minima it has. $$y=x^{8}-3 x^{4}+x$$

Find all solutions of the equation and express them in the form \(a+b i\) $$z+4+\frac{12}{z}=0$$

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

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. $$P(x)=x^{3}-16 x$$

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. $$P(x)=x^{4}+x^{3}+x^{2}+x+12$$

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 \(\mathrm{mg} / \mathrm{L} .\) When is the maximum serum concentration reached, and what is that maximum concentration?

Graph the polynomial and determine how many local maxima and minima it has. $$y=\frac{1}{3} x^{7}-17 x^{2}+7$$

Find all solutions of the equation and express them in the form \(a+b i\) $$z+4+\frac{12}{z}=0$$

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

By the Zeros Theorem, every \(n\) th-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. $$P(x)=x^{5}+4 x^{3}-x^{2}+6 x$$

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?

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

Find all solutions of the equation and express them in the form \(a+b i\) $$4 x^{2}-16 x+19=0$$

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

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) \(i x^{2}-2 x+i=0\)

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. $$P(x)=x^{8}-x^{5}+x^{4}-x^{3}+x^{2}-x+1$$

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.

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

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?

Find all solutions of the equation and express them in the form \(a+b i\) $$\frac{1}{2} x^{2}-x+5=0$$

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

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. $$P(x)=2 x^{3}+5 x^{2}+x-2 ; \quad a=-3, b=1$$

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

Find all solutions of the equation and express them in the form \(a+b i\) $$x^{2}+\frac{1}{2} x+1=0$$

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. $$P(x)=x^{4}-2 x^{3}-9 x^{2}+2 x+8 ; \quad a=-3, b=5$$

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. $$P(x)=x^{3}+c x ; \quad c=2,0,-2,-4$$

Recall that the symbol \(\bar{z}\) represents the complex conjugate of \(z .\) If \(z=a+b i\) and \(w=c+d i,\) prove each statement. $$\bar{z}+\bar{w}=\overline{z+w}$$

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. $$f(x)=\frac{2 x^{2}+6 x+6}{x+3}, g(x)=2 x$$

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.

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

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

Recall that the symbol \(\bar{z}\) represents the complex conjugate of \(z .\) If \(z=a+b i\) and \(w=c+d i,\) prove each statement. $$\overline{z w}=\bar{z} \cdot \bar{w}$$

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. $$f(x)=\frac{-x^{3}+6 x^{2}-5}{x^{2}-2 x}, \quad g(x)=-x+4$$

There are two square roots of 1, namely, 1 and \(-1 .\) These are the solutions of \(x^{2}=1 .\) The fourth roots of I 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 \(1 .\)

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

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

Recall that the symbol \(\bar{z}\) represents the complex conjugate of \(z .\) If \(z=a+b i\) and \(w=c+d i,\) prove each statement. $$(\bar{z})^{2}=\overline{z^{2}}$$