Chapter 4

Find a polynomial with integer coefficients that satisfies the given conditions. \(Q\) has degree 3 and zeros \(-3\) and \(1+i\)

\(27-40\) Factor the polynomial and use the factored form to find the zeros. Then sketch the graph. $$ P(x)=x^{6}-2 x^{3}+1 $$

Find the maximum or minimum value of the function. $$ f(x)=-\frac{x^{2}}{3}+2 x+7 $$

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$ P(x)=2 x^{4}-7 x^{3}+3 x^{2}+8 x-4 $$

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

\(41-46\) . Determine the end behavior of \(P\) . Compare the graphs of \(P\) and \(Q\) in large and small viewing rectangles, as in Example 3\((\mathrm{b})\) . $$ P(x)=3 x^{3}-x^{2}+5 x+1 ; \quad Q(x)=3 x^{3} $$

\(39-51\) . Use synthetic division and the Remainder Theorem to evaluate \(P(c) .\) $$ P(x)=x^{3}+3 x^{2}-7 x+6, \quad c=2 $$

Find a polynomial with integer coefficients that satisfies the given conditions. \(R\) has degree 4 and zeros \(1-2 i\) and \(1,\) with 1 a zero of multiplicity \(2 .\)

Find the maximum or minimum value of the function. $$ f(x)=3-x-\frac{1}{2} x^{2} $$

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

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

\(41-46\) . Determine the end behavior of \(P\) . Compare the graphs of \(P\) and \(Q\) in large and small viewing rectangles, as in Example 3\((\mathrm{b})\) . $$ P(x)=-\frac{1}{8} x^{3}+\frac{1}{4} x^{2}+12 x, \quad Q(x)=-\frac{1}{8} x^{3} $$

\(39-51\) . Use synthetic division and the Remainder Theorem to evaluate \(P(c) .\) $$ P(x)=x^{3}-x^{2}+x+5, \quad c=-1 $$

Find a polynomial with integer coefficients that satisfies the given conditions. S has degree 4 and zeros 2\(i\) and 3\(i\).

Find the maximum or minimum value of the function. $$ g(x)=2 x(x-4)+7 $$

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

Find the intercepts and asymptotes, and then sketch a graph of the rational function and state the domain and range. Use a graphing device to confirm your answer. $$ s(x)=\frac{4-3 x}{x+7} $$

\(39-51\) . Use synthetic division and the Remainder Theorem to evaluate \(P(c) .\) $$ P(x)=x^{3}+2 x^{2}-7, \quad c=-2 $$

Find a polynomial with integer coefficients that satisfies the given conditions. Thas degree \(4,\) zeros \(i\) and \(1+i,\) and constant term 12.

Find a function whose graph is a parabola with vertex and that passes through the point . $$ \begin{array}{l}{\text { Find a function whose graph is a parabola with vertex }(1,-2)} \\ {\text { and that passes through the point }(4,16) .}\end{array} $$

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

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

\(39-51\) . Use synthetic division and the Remainder Theorem to evaluate \(P(c) .\) $$ P(x)=2 x^{3}-21 x^{2}+9 x-200, \quad c=11 $$

Find a polynomial with integer coefficients that satisfies the given conditions. \(U\) has degree \(5,\) zeros \(\frac{1}{2},-1,\) and \(-i,\) and leading coefficient \(4 ;\) the zero \(-1\) has multiplicity \(2 .\)

$$ \begin{array}{l}{\text { Find a function whose graph is a parabola with vertex }(3,4)} \\ {\text { and that passes through the point }(1,-8) .}\end{array} $$

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

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

\(39-51\) . Use synthetic division and the Remainder Theorem to evaluate \(P(c) .\) $$ P(x)=5 x^{4}+30 x^{3}-40 x^{2}+36 x+14, \quad c=-7 $$

Find all zeros of the polynomial. \(x^{3}+2 x^{2}+4 x+8\)

Find the domain and range of the function. $$ f(x)=-x^{2}+4 x-3 $$

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

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

\(39-51\) . Use synthetic division and the Remainder Theorem to evaluate \(P(c) .\) $$ P(x)=6 x^{5}+10 x^{3}+x+1, \quad c=-2 $$

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

Find the domain and range of the function. $$ f(x)=x^{2}-2 x-3 $$

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

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

\(47-50\) The graph of a polynomial function is given. From the graph, find (a) the \(x-\) and \(y\) -intercepts, and (b) the coordinates of all local extrema. $$ P(x)=-x^{2}+4 x $$

\(39-51\) . Use synthetic division and the Remainder Theorem to evaluate \(P(c) .\) $$ P(x)=x^{7}-3 x^{2}-1, \quad c=3 $$

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

Find the domain and range of the function. $$ f(x)=2 x^{2}+6 x-7 $$

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example \(3(\mathrm{a}) .\) $$ P(x)=x^{3}-5 x^{2}+2 x+12 $$

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

\(39-51\) . Use synthetic division and the Remainder Theorem to evaluate \(P(c) .\) $$ P(x)=-2 x^{6}+7 x^{5}+40 x^{4}-7 x^{2}+10 x+112, \quad c=-3 $$

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 \(3(\mathrm{a}) .\) $$ P(x)=x^{4}-6 x^{3}+4 x^{2}+15 x+4 $$

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

\(47-50\) The graph of a polynomial function is given. From the graph, find (a) the \(x-\) and \(y\) -intercepts, and (b) the coordinates of all local extrema. $$ P(x)=-\frac{1}{2} x^{3}+\frac{3}{2} x-1 $$

\(39-51\) . Use synthetic division and the Remainder Theorem to evaluate \(P(c) .\) $$ P(x)=3 x^{3}+4 x^{2}-2 x+1, \quad c=\frac{2}{3} $$

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