Answer
The zeros of the function are: $\{0,\sqrt 3i,-\sqrt 3i\}$
The complete factorization of P is:
$P(x)=x[(x-\sqrt 3i)(x+\sqrt 3i)]^{2}$
$x =0$ with multiplicity 1
$x =\sqrt 3i$ with multiplicity 2
$x =-\sqrt 3i$ with multiplicity 2
Work Step by Step
Factor the polynomial completely to obtain:
$P(x)=x^{5}+6x^{3}+9x$
$P(x)=x(x^{4}+6x^{2}+9)$
$P(x)=x(x^{2}+3)^{2}$
$P(x)=x[(x+\sqrt 3i)(x-\sqrt 3i)]^{2}$
Equate each unique factor to zero then solve each equation to obtain:
$x=0 $
$x+\sqrt 3i=0 \rightarrow x=-\sqrt 3i$
$x-\sqrt 3i=0 \rightarrow x=\sqrt 3i$
The zeros of the function are: $\{0,\sqrt 3i,-\sqrt 3i\}$
The complete factorization of P is:
$P(x)=x[(x-\sqrt 3i)(x-(-\sqrt 3i))]^{2}$
$P(x)=x[(x-\sqrt 3i)(x+\sqrt 3i)]^{2}$
$x =0$ with multiplicity 1
$x =\sqrt 3i$ with multiplicity 2
$x =-\sqrt 3i$ with multiplicity 2
