Answer
The zeros of the function are: $\{-2,1+\sqrt 3i,1-\sqrt 3i\}$
The complete factorization of P is:
$P(x)=[(x+2)(x-(1+\sqrt 3i))(x-(1-\sqrt 3i))]^{2}$
$x =-2$ with multiplicity 2
$x =1+\sqrt 3i$ with multiplicity 2
$x =1-\sqrt 3i$ with multiplicity 2
Work Step by Step
Factor the polynomial completely to obtain:
$P(x)=x^{6}+16x^{3}+64$
$P(x)=(x^{3}+8)^{2}$
$P(x)=[(x+2)(x^{2}-2x+4)]^{2}$
$P(x)=[(x+2)(x-(1+\sqrt 3i))(x-(1-\sqrt 3i))]^{2}$
Equate each unique factor to zero then solve each equation to obtain:
$x+2=0 \rightarrow x=-2$
$x-(1+\sqrt 3i)=0 \rightarrow x=1+\sqrt 3i$
$x-(1-\sqrt 3i)=0 \rightarrow x=1-\sqrt 3i$
The zeros of the function are: $\{-2,1+\sqrt 3i,1-\sqrt 3i\}$
The complete factorization of P is:
$P(x)=[(x+2)(x-(1+\sqrt 3i))(x-(1-\sqrt 3i))]^{2}$
$x =-2$ with multiplicity 2
$x =1+\sqrt 3i$ with multiplicity 2
$x =1-\sqrt 3i$ with multiplicity 2
