For the polynomial function $$ f(x)=x^{5}-x^{4}-9 x^{3}+9 x^{2} $$, find the zeros. Then determine the multiplicity at each zero and state whether the graph displays the behavior of a touch or a cross at each intercept. $$ \begin{array}{l}x=-3, ext { cross; } x=0 ext {, touch; } x=1 ext {, cross; } x= \ 3, ext { cross } \ x=-3 ext {, cross; } x=0, ext { cross; } x=1 ext {, touch; } x= \ 3, ext { touch } \ x=-3, ext { cross; } x=0, ext { cross; } x=1 ext {, cross; } x=3 \ ext { touch } \ x=-3, ext { cross; } x=0, ext { cross; } x=1 ext {, touch; } x= \ 3, ext { cross }\end{array} $$

Question

For the polynomial function $$ f(x)=x^{5}-x^{4}-9 x^{3}+9 x^{2} $$, find the zeros. Then determine the multiplicity at each zero and state whether the graph displays the behavior of a touch or a cross at each intercept. $$ \begin{array}{l}x=-3, 	ext { cross; } x=0 	ext {, touch; } x=1 	ext {, cross; } x= \ 3, 	ext { cross } \ x=-3 	ext {, cross; } x=0, 	ext { cross; } x=1 	ext {, touch; } x= \ 3, 	ext { touch } \ x=-3, 	ext { cross; } x=0, 	ext { cross; } x=1 	ext {, cross; } x=3 \ 	ext { touch } \ x=-3, 	ext { cross; } x=0, 	ext { cross; } x=1 	ext {, touch; } x= \ 3, 	ext { cross }\end{array} $$

Malone Hart · Accepted Answer

The zeros of $$ f(x) $$ are $$ x = -3 $$, $$ x = 0 $$, $$ x = 1 $$, and $$ x = 3 $$. The behavior at each zero is: $$ x = -3 $$ (cross), $$ x = 0 $$ (touch), $$ x = 1 $$ (cross), $$ x = 3 $$ (cross).

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