College Algebra, Chapter 1, 1.5, Section 1.5, Problem 64

Find all solutions, real and complex of the equation $\displaystyle x^4 - 16 = 0$


$\begin{equation} \begin{aligned} x^4 - 16 =& 0 && \text{Given} \\ \\ (x^4 + x^3) + (x^2 + x) =& 0 && \text{Group terms} \\ \\ x^2(x^2 + x) + (x^2 + x) =& 0 && \text{Factor out } x^2 \\ \\ (x^2 + 1)(x^2 + x) =& 0 && \text{Factor out } x^2 + 1 \\ \\ x^2 + 1 =& 0 \text{ and } x^2 + x = 0 && \text{Zero Product Property} \\ \\ x =& \pm \sqrt{-1} \text{ and } x (x + 1) = 0 && \text{Solve for } x \\ \\ x =& \pm \sqrt{i^2} \text{ and } x = 0, x = -1 && \text{Recall that } i^2 = -1 \\ \\ x =& \pm i \text{ and } x = 0, x = -1 && \end{aligned} \end{equation}$