College Algebra, Chapter 4, 4.5, Section 4.5, Problem 24

Factor the polynomial $P(x) = x^4 - 625$, and find all its zeros. State the multiplicity of each zero.

To find the zeros of $P$, we set $x^4 - 625 = 0$, by factoring


$
\begin{equation}
\begin{aligned}

x^4 - 625 =& 0
\\
\\
(x^2 - 25)(x^2 + 25) =& 0

\end{aligned}
\end{equation}
$


Thus, the zeros are $x = \pm 5$ and $x = \pm 5i$.

By factorization,


$
\begin{equation}
\begin{aligned}

P(x) =& (x - 5) \left[ x - (-5) \right] (x - 5i) \left[ x - (-5i) \right]
\\
\\
=& (x - 5)(x + 5)(x - 5i)(x + 5i)

\end{aligned}
\end{equation}
$


We can say that each factors have multiplicity of $1$.