How do you find the imaginary roots of a fourth degree polynomial that cannot be simplified? I started out with $f(x)=16x^6-1$, and I got it down to $64x^4+16x^2+4$ by synthetically dividing by roots $0.5$ and $-0.5$ How should I continue in order to find the other roots?

$16x^6=1$
$(\sqrt[3]{4}x)^6=1$
$u=\sqrt[3]{4}x$
$u^6=1$
These roots are spaced at u=$\pm 1$  and $\pm \frac{1}{2} \pm \frac{i\sqrt{3}}{2}$ ($n^{th}$ roots of unity)
Or $+1,-1, +\frac{1}{2} -\frac{i\sqrt{3}}{2}, +\frac{1}{2}  -\frac{i\sqrt{3}}{2} , - \frac{1}{2} + \frac{i\sqrt{3}}{2}, - \frac{1}{2} - \frac{i\sqrt{3}}{2}$
So substituting x back into the solutions... and solving for x, you get the solutions
