91Ó°ÊÓ

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. $$P(x)=x^{3}-5 x^{2}+4 x-20$$

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

Graph the polynomial and determine how many local maxima and minima it has. $$y=x^{4}-5 x^{2}+4$$

Find a polynomial of the specified degree that has the given zeros. Degree \(4 ; \quad\) zeros \(-1,1,3,5\)

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

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

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

Find a polynomial of the specified degree that has the given zeros. Degree \(5 ; \quad\) zeros \(-2,-1,0,1,2\)

Graph the polynomial and determine how many local maxima and minima it has. $$y=1.2 x^{5}+3.75 x^{4}-7 x^{3}-15 x^{2}+18 x$$

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