College Algebra, Chapter 4, 4.5, Section 4.5, Problem 22

Factor the polynomial $P(x) = x^3 - x^2 + x$, and find all its zeros. State the multiplicity of each zero.

To find the zeros of $P$, we set $x^3 - x^2 + x = 0$, so $x(x^2 - x + 1) = 0$ by using quadratic formula


$
\begin{equation}
\begin{aligned}

x =& \frac{-(-1) \pm \sqrt{(-1)^2 - 4 (1)(1)}}{2(1)}
\\
\\
=& \frac{1 \pm \sqrt{-3}}{2}
\\
\\
=& \frac{1 \pm \sqrt{3} i}{2}

\end{aligned}
\end{equation}
$


By factorization,


$
\begin{equation}
\begin{aligned}

P(x) =& x \left[ x - \left( \frac{1 + \sqrt{3} i}{2} \right) \right] \left[ x - \left( \frac{1 - \sqrt{3} i}{2} \right) \right]

\end{aligned}
\end{equation}
$


The zeros of $P$ are $\displaystyle 0, \frac{1 + \sqrt{3} i}{2}$ and $\displaystyle \frac{1 - \sqrt{3} i}{2}$. Each zeros has multiplicity of $1$.