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