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 $1$.