Answer
$P(x)=x(x-\displaystyle \frac{1}{2}-\frac{\sqrt{3}}{2}i)(x-\frac{1}{2}+\frac{\sqrt{3}}{2}i)$
Zeros: 0, $\displaystyle \frac{1}{2}-\frac{\sqrt{3}}{2}i,\ \displaystyle \frac{1}{2}+\frac{\sqrt{3}}{2}i$
(all with multiplicity 1)
Work Step by Step
Factoring x out of all three terms,
$P(x)=x(x^{2}-x+1)$
Zeros of $x^{2}-x+1$ , (quadratic formula)
$x=\displaystyle \frac{-b\pm\sqrt{b^{2}-4ac}}{2a},\quad a=1, b=-1, c=1$
$x=\displaystyle \frac{1\pm\sqrt{1-4(1)(1)}}{2(1)}=\frac{1\pm\sqrt{-3}}{2}$
$x=\displaystyle \frac{1}{2}\pm\frac{\sqrt{3}}{2}i$
$P(x)=x[x-(\displaystyle \frac{1}{2}+\frac{\sqrt{3}}{2}i)][x-(\frac{1}{2}-\frac{\sqrt{3}}{2}i)]$
$P(x)=x(x-\displaystyle \frac{1}{2}-\frac{\sqrt{3}}{2}i)(x-\frac{1}{2}+\frac{\sqrt{3}}{2}i)$
Zeros: 0, $\displaystyle \frac{1}{2}-\frac{\sqrt{3}}{2}i,\ \displaystyle \frac{1}{2}+\frac{\sqrt{3}}{2}i$
(all with multiplicity 1)
