College Algebra, Chapter 4, Chapter Review, Section Review, Problem 60

Find all rational, irrational and complex zeros (and state their multiplicities) of the polynomial function $P(x) = x^4 + 15x^2 - 54$. Use Descartes' Rule of signs, the Upper and Lower Bounds Theorem, the Quadratic Formula or other factoring techniques.

To find the zeros, we first factor $P$ to obtain


$
\begin{equation}
\begin{aligned}

P(x) =& x^4 + 15x^2 - 54
\\
\\
=& (x^2 + 18)(x^2 + 3)

\end{aligned}
\end{equation}
$



To get the zeros, we set $x^2 + 18 = 0$ and $x^2 + 3 = 0$ so,


$
\begin{equation}
\begin{aligned}

x^2 =& -18
&&\text{and}& x^2 =& -3
\\
\\
x =& \pm \sqrt{-18}
&&& x^2 =& \pm \sqrt{-3}
\\
\\
x =& \pm 3 \sqrt{2} i
&&& x =& \pm \sqrt{3} i

\end{aligned}
\end{equation}
$



Thus, the complex zeros are $3 \sqrt{2}i, -3 \sqrt{2}i, \sqrt{3}i$ and $- \sqrt{3}i$. All the zeros have multiplicity of $1$.