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 $1$.