The equation $z^6+z^3+1$ has one complex root with argument $ heta$ between $90^{\circ}$ and $180^{

Question

The equation $z^6+z^3+1$ has one complex root with argument $	heta$ between $90^{\circ}$ and $180^{\circ}$ in the complex plane. Determine the degree measure of $	heta$.

Professor PC Thomas Classes · Answer

(160) Let $w=z^3$. Then the given equation reduces to
$$w^2+w+1=0,
$$whose solutions are $(-1+i \sqrt{3}) / 2$ and $(-1-i \sqrt{3}) / 2$, with arguments of $120^{\circ}$ and $240^{\circ}$, respectively. From these, one finds the following six values for the argument of $z$ :
$$
\frac{120^{\circ}}{3}, \frac{120^{\circ}+360^{\circ}}{3}, \frac{120^{\circ}+720^{\circ}}{3}, \frac{240^{\circ}}{3}, \frac{240^{\circ}+360^{\circ}}{3}, \frac{240^{\circ}+720^{\circ}}{3} .
$$Clearly, only the second one of these, $160^{\circ}$, is between $90^{\circ}$ and $180^{\circ}$.