College Algebra, Chapter 4, Chapter Review, Section Review, Problem 56

Find all rational, irrational and complex zeros (and state their multiplicities) of the polynomial function $P(x) = x^4 - 81$. Use Descartes' Rule of signs, the Upper and Lower Bounds Theorem, the Quadratic Formula or other factoring techniques.

To determine the zeros, we first factor $P$.


$
\begin{equation}
\begin{aligned}

P(x) =& x^4 - 81
&&
\\
\\
=& (x^2 - 9)(x^2 + 9)
&& \text{Difference of squares}
\\
\\
=& (x + 3)(x - 3)(x^2 + 9)
&& \text{Difference of squares}

\end{aligned}
\end{equation}
$


Then to find the remaining zeros of $P$, we set


$
\begin{equation}
\begin{aligned}

x^2 + 9 =& 0
\\
\\
x^2 =& -9
\\
\\
x =& \pm 3i

\end{aligned}
\end{equation}
$



Therefore, the zeros of $P$ are $-3, 3, 3i$ and $- 3i$. Each zeros have multiplicity of $1$.