Chapter 4

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph. $$ P(x)=x^{4}-3 x^{2}-4 $$

\(31-40=\) Find a polynomial with integer coefficients that satisfies the given conditions. $$ Q \text { has degree } 3, \text { and zeros }-3 \text { and } 1+i $$

Find all rational zeros of the polynomial. $$ P(x)=6 x^{4}-7 x^{3}-12 x^{2}+3 x+2 $$

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

Find the quotient and remainder using synthetic division. \(\frac{x^{4}-16}{x+2}\)

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph. $$ P(x)=x^{6}-2 x^{3}+1 $$

\(31-40=\) Find a polynomial with integer coefficients that satisfies the given conditions. $$ \begin{array}{l}{R \text { has degree } 4, \text { and zeros } 1-2 i \text { and } 1, \text { with } 1 \text { a zero of }} \\ {\text { multiplicity } 2 .}\end{array} $$

Find all rational zeros of the polynomial. $$ P(x)=x^{5}+3 x^{4}-9 x^{3}-31 x^{2}+36 $$

Find the intercepts and asymptotes, and then sketch a graph of the rational function. Use a graphing device to confirm your answer. \(r(x)=\frac{18}{(x-3)^{2}}\)

Use synthetic division and the Remainder Theorem to evaluate \(P(c) .\) \(P(x)=4 x^{2}+12 x+5, \quad c=-1\)

\(31-40=\) Find a polynomial with integer coefficients that satisfies the given conditions. $$ S \text { has degree } 4, \text { and zeros } 2 i \text { and } 3 i $$

Find all rational zeros of the polynomial. $$ P(x)=x^{5}-4 x^{4}-3 x^{3}+22 x^{2}-4 x-24 $$

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

Use synthetic division and the Remainder Theorem to evaluate \(P(c) .\) \(P(x)=2 x^{2}+9 x+1, \quad c=\frac{1}{2}\)

\(31-40=\) Find a polynomial with integer coefficients that satisfies the given conditions. $$ T \text { has degree } 4, \text { zeros } i \text { and } 1+i, \text { and constant term } 12 $$

Find all rational zeros of the polynomial. $$ P(x)=3 x^{5}-14 x^{4}-14 x^{3}+36 x^{2}+43 x+10 $$

Find the intercepts and asymptotes, and then sketch a graph of the rational function. Use a graphing device to confirm your answer. \(s(x)=\frac{4 x-8}{(x-4)(x+1)}\)

Use synthetic division and the Remainder Theorem to evaluate \(P(c) .\) \(P(x)=x^{3}+3 x^{2}-7 x+6, \quad c=2\)

\(31-40=\) Find a polynomial with integer coefficients that satisfies the given conditions. $$ \begin{array}{l}{U \text { has degree } 5, \text { zeros } \frac{1}{2},-1, \text { and }-i, \text { and leading coefficient }} \\ {4 ; \text { the zero }-1 \text { has multiplicity } 2 .}\end{array} $$

Find all rational zeros of the polynomial. $$ P(x)=2 x^{6}-3 x^{5}-13 x^{4}+29 x^{3}-27 x^{2}+32 x-12 $$

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

Use synthetic division and the Remainder Theorem to evaluate \(P(c) .\) \(P(x)=x^{3}-x^{2}+x+5, \quad c=-1\)

\(41-58=\) Find all zeros of the polynomial. $$ P(x)=x^{3}+2 x^{2}+4 x+8 $$

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

Find the intercepts and asymptotes, and then sketch a graph of the rational function. Use a graphing device to confirm your answer. \(s(x)=\frac{6}{x^{2}-5 x-6}\)

Use synthetic division and the Remainder Theorem to evaluate \(P(c) .\) \(P(x)=x^{3}+2 x^{2}-7, \quad c=-2\)

\(41-58=\) Find all zeros of the polynomial. $$ P(x)=x^{3}-7 x^{2}+17 x-15 $$

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

Use synthetic division and the Remainder Theorem to evaluate \(P(c) .\) \(P(x)=2 x^{3}-21 x^{2}+9 x-200, \quad c=11\)

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

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example 3(a). $$ P(x)=x^{4}-6 x^{3}+4 x^{2}+15 x+4 $$

Find the intercepts and asymptotes, and then sketch a graph of the rational function. Use a graphing device to confirm your answer. \(t(x)=\frac{3 x+6}{x^{2}+2 x-8}\)

Use synthetic division and the Remainder Theorem to evaluate \(P(c) .\) \(P(x)=5 x^{4}+30 x^{3}-40 x^{2}+36 x+14, \quad c=-7\)

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

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

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

Use synthetic division and the Remainder Theorem to evaluate \(P(c) .\) \(P(x)=6 x^{5}+10 x^{3}+x+1, \quad c=-2\)

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

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example 3(a). $$ P(x)=x^{4}-7 x^{3}+14 x^{2}-3 x-9 $$

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

Use synthetic division and the Remainder Theorem to evaluate \(P(c) .\) \(P(x)=x^{7}-3 x^{2}-1, \quad c=3\)

The graph of a polynomial function is given. From the graph, find (a) the \(x\) - and \(y\) -intercepts (b) the coordinates of all local extrema $$ P(x)=-\frac{1}{2} x^{3}+\frac{3}{2} x-1 $$

\(41-58=\) Find all zeros of the polynomial. $$ P(x)=x^{3}-x-6 $$

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example 3(a). $$ P(x)=x^{5}-4 x^{4}-x^{3}+10 x^{2}+2 x-4 $$

Find the intercepts and asymptotes, and then sketch a graph of the rational function. Use a graphing device to confirm your answer. \(r(x)=\frac{2 x(x+2)}{(x-1)(x-4)}\)

Use synthetic division and the Remainder Theorem to evaluate \(P(c) .\) \(P(x)=-2 x^{6}+7 x^{5}+40 x^{4}-7 x^{2}+10 x+112, \quad c=-3\)

\(41-58=\) Find all zeros of the polynomial. $$ P(x)=2 x^{3}+7 x^{2}+12 x+9 $$

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example 3(a). $$ P(x)=4 x^{3}-6 x^{2}+1 $$

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

Use synthetic division and the Remainder Theorem to evaluate \(P(c) .\) \(P(x)=3 x^{3}+4 x^{2}-2 x+1, \quad c=\frac{2}{3}\)