Solution: Factor the polynomial as $\fracz^8 - 1z^2 - 1 = 0$ (excluding roots of $z^2 = 1$). The roots are the 8th roots of unity except $\pm 1$. The roots with maximum imaginary part are $e^i\pi/4$ and $e^i3\pi/4$, with imaginary part $\frac\sqrt22 = \cos(\pi/4)$. - go-checkin.com