91Ó°ÊÓ

\(53-56\) . Use the Factor Theorem to show that \(X-c\) is a factor of \(P(x)\) for the given value(s) of \(c .\) $$ P(x)=x^{4}+3 x^{3}-16 x^{2}-27 x+63, \quad c=3,-3 $$

\(51-58\) Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer rounded to two decimal places. $$ y=x^{4}-18 x^{2}+32,[-5,5] \text { by }[-100,100] $$

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

Find the local maximum and minimum values of the function and the value of x at which each occurs. State each answer correct to two decimal places. $$ f(x)=x^{3}-x $$

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. $$ r(x)=\frac{4 x^{2}}{x^{2}-2 x-3} $$

Find all zeros of the polynomial. \(P(x)=x^{5}+x^{3}+8 x^{2}+8 \quad[\text { Hint: Factor by grouping. }]\)

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

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 and state the domain and range. Use a graphing device to confirm your answer. $$ r(x)=\frac{2 x^{2}+10 x-12}{x^{2}+x-6} $$

\(57-58\) . Show that the given value(s) of \(c\) are zeros of \(P(x),\) and find all other zeros of \(P(x) .\) $$ P(x)=x^{3}-x^{2}-11 x+15, \quad c=3 $$