An algebra problem by neelesh vij

Note that $z \times \overline{z} = |z|^2$ .

Observe that $z = 0$ is not a solution (the second term is not defined). Now multiply both sides by $z^2$ to obtain:

$\begin{aligned} z^5 + \frac{3 z^2 \overline{z}^2}{|z|} &= 0 \\ z^5 + \frac{3 |z|^4}{|z|} &= 0 \\ z^5 + 3|z|^3 &= 0 \\ z^5 &= -3|z|^3 \end{aligned}$

Taking the modulus of each side, we have $|z|^5 = 3 |z|^3$ , or $|z|^2 = 3$ (because $|z| \neq 0$ ). Since $|z| > 0$ , we have $|z| = \sqrt{3}$ . Thus we obtain $z^5 = -3^{5/2}$ , giving $\boxed{5}$ solutions for $z$ : $-3^{1/2}$ times each fifth root of unity.