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