Chapter 4

Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer correct to two decimal places. $$ y=-x^{2}+8 x, \quad[-4,12] \text { by }[-50,30] $$

\(41-58=\) Find all zeros of the polynomial. $$ P(x)=2 x^{3}-8 x^{2}+9 x-9 $$

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example 3(a). $$ P(x)=3 x^{3}-5 x^{2}-8 x-2 $$

Find the intercepts and asymptotes, and then sketch a graph of the rational function. Use a graphing device to confirm your answer. \(r(x)=\frac{4 x^{2}}{x^{2}-2 x-3}\)

Use synthetic division and the Remainder Theorem to evaluate \(P(c) .\) \(P(x)=x^{3}-x+1, \quad c=\frac{1}{4}\)

Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer correct to two decimal places. $$ y=x^{3}-3 x^{2}, \quad[-2,5] \text { by }[-10,10] $$

\(41-58=\) Find all zeros of the polynomial. $$ P(x)=x^{4}+x^{3}+7 x^{2}+9 x-18 $$

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example 3(a). $$ P(x)=2 x^{4}+15 x^{3}+17 x^{2}+3 x-1 $$

Find the intercepts and asymptotes, and then sketch a graph of the rational function. Use a graphing device to confirm your answer. \(r(x)=\frac{2 x^{2}+10 x-12}{x^{2}+x-6}\)

Use synthetic division and the Remainder Theorem to evaluate \(P(c) .\) \(P(x)=x^{3}+2 x^{2}-3 x-8, \quad c=0.1\)

Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer correct to two decimal places. $$ y=x^{3}-12 x+9, \quad[-5,5] \text { by }[-30,30] $$

\(41-58=\) Find all zeros of the polynomial. $$ P(x)=x^{4}-2 x^{3}-2 x^{2}-2 x-3 $$

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example 3(a). $$ P(x)=4 x^{5}-18 x^{4}-6 x^{3}+91 x^{2}-60 x+9 $$

Find the intercepts and asymptotes, and then sketch a graph of the rational function. Use a graphing device to confirm your answer. \(r(x)=\frac{2 x^{2}+2 x-4}{x^{2}+x}\)

Let $$\begin{aligned} P(x)=6 x^{7}-40 x^{6}+16 x^{5}-200 x^{4} & \\\\-&-60 x^{3}-69 x^{2}+13 x-139 \end{aligned}$$ Calculate \(P(7)\) by (a) using synthetic division and (b) substituting \(x=7\) into the polynomial and evaluating directly.

Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer correct to two decimal places. $$ y=2 x^{3}-3 x^{2}-12 x-32,[-5,5] \text { by }[-60,30] $$

\(41-58=\) Find all zeros of the polynomial. $$ P(x)=x^{5}-x^{4}+7 x^{3}-7 x^{2}+12 x-12 $$

\(51-58=\) A polynomial \(P\) is given. (a) Find all the real zeros of \(P .\) (b) Sketch the graph of \(P\) . $$ P(x)=x^{3}-3 x^{2}-4 x+12 $$

Find the intercepts and asymptotes, and then sketch a graph of the rational function. Use a graphing device to confirm your answer. \(r(x)=\frac{x^{2}-x-6}{x^{2}+3 x}\)

Use the Factor Theorem to show that \(x-c\) is a factor of \(P(x)\) for the given value(s) of \(c .\) \(P(x)=x^{3}-3 x^{2}+3 x-1, \quad c=1\)

Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer correct to two decimal places. $$ y=x^{4}+4 x^{3}, \quad[-5,5] \text { by }[-30,30] $$

\(51-58=\) A polynomial \(P\) is given. (a) Find all the real zeros of \(P .\) (b) Sketch the graph of \(P\) . $$ P(x)=-x^{3}-2 x^{2}+5 x+6 $$

\(41-58=\) Find all zeros of the polynomial. $$ P(x)=x^{5}+x^{3}+8 x^{2}+8 $$

Find the intercepts and asymptotes, and then sketch a graph of the rational function. Use a graphing device to confirm your answer. \(r(x)=\frac{x^{2}+3 x}{x^{2}-x-6}\)

Use the Factor Theorem to show that \(x-c\) is a factor of \(P(x)\) for the given value(s) of \(c .\) \(P(x)=x^{3}+2 x^{2}-3 x-10, \quad c=2\)

Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer correct to two decimal places. $$ y=x^{4}-18 x^{2}+32, \quad[-5,5] \text { by }[-100,100] $$

\(51-58=\) A polynomial \(P\) is given. (a) Find all the real zeros of \(P .\) (b) Sketch the graph of \(P\) . $$ P(x)=2 x^{3}-7 x^{2}+4 x+4 $$

\(41-58=\) Find all zeros of the polynomial. $$ P(x)=x^{4}-6 x^{3}+13 x^{2}-24 x+36 $$

Find the intercepts and asymptotes, and then sketch a graph of the rational function. Use a graphing device to confirm your answer. \(r(x)=\frac{3 x^{2}+6}{x^{2}-2 x-3}\)

Use the Factor Theorem to show that \(x-c\) is a factor of \(P(x)\) for the given value(s) of \(c .\) \(P(x)=2 x^{3}+7 x^{2}+6 x-5, \quad c=\frac{1}{2}\)

Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer correct to two decimal places. $$ y=3 x^{5}-5 x^{3}+3, \quad[-3,3] \text { by }[-5,10] $$

\(51-58=\) A polynomial \(P\) is given. (a) Find all the real zeros of \(P .\) (b) Sketch the graph of \(P\) . $$ P(x)=3 x^{3}+17 x^{2}+21 x-9 $$

\(41-58=\) Find all zeros of the polynomial. $$ P(x)=x^{4}-x^{2}+2 x+2 $$

Find the intercepts and asymptotes, and then sketch a graph of the rational function. Use a graphing device to confirm your answer. \(r(x)=\frac{5 x^{2}+5}{x^{2}+4 x+4}\)

Use the Factor Theorem to show that \(x-c\) is a factor of \(P(x)\) for the given value(s) of \(c .\) \(P(x)=x^{4}+3 x^{3}-16 x^{2}-27 x+63, \quad c=3,-3\)

Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer correct to two decimal places. $$ y=x^{5}-5 x^{2}+6, \quad[-3,3] \text { by }[-5,10] $$

\(51-58=\) A polynomial \(P\) is given. (a) Find all the real zeros of \(P .\) (b) Sketch the graph of \(P\) . $$ P(x)=x^{4}-5 x^{3}+6 x^{2}+4 x-8 $$

\(41-58=\) Find all zeros of the polynomial. $$ P(x)=4 x^{4}+4 x^{3}+5 x^{2}+4 x+1 $$

Find the intercepts and asymptotes, and then sketch a graph of the rational function. Use a graphing device to confirm your answer. \(s(x)=\frac{x^{2}-2 x+1}{x^{3}-3 x^{2}}\)

Show that the given value(s) of \(c\) are zeros of \(P(x)\), and find all other zeros of \(P(x)\). \(P(x)=x^{3}-x^{2}-11 x+15, \quad c=3\)

Graph the polynomial and determine how many local maxima and minima it has. $$ y=-2 x^{2}+3 x+5 $$

\(41-58=\) Find all zeros of the polynomial. $$ P(x)=4 x^{4}+2 x^{3}-2 x^{2}-3 x-1 $$

\(51-58=\) A polynomial \(P\) is given. (a) Find all the real zeros of \(P .\) (b) Sketch the graph of \(P\) . $$ P(x)=-x^{4}+10 x^{2}+8 x-8 $$

Find the intercepts and asymptotes, and then sketch a graph of the rational function. Use a graphing device to confirm your answer. \(t(x)=\frac{x^{3}-x^{2}}{x^{3}-3 x-2}\)

Show that the given value(s) of \(c\) are zeros of \(P(x)\), and find all other zeros of \(P(x)\). \(P(x)=3 x^{4}-x^{3}-21 x^{2}-11 x+6, \quad c=\frac{1}{3},-2\)

Graph the polynomial and determine how many local maxima and minima it has. $$ y=x^{3}+12 x $$

\(41-58=\) Find all zeros of the polynomial. $$ P(x)=x^{5}-3 x^{4}+12 x^{3}-28 x^{2}+27 x-9 $$

\(51-58=\) A polynomial \(P\) is given. (a) Find all the real zeros of \(P .\) (b) Sketch the graph of \(P\) . $$ P(x)=x^{5}-x^{4}-5 x^{3}+x^{2}+8 x+4 $$

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

Find a polynomial of the specified degree that has the given zeros. Degree \(3 ; \quad\) zeros \(-1,1,3\)