Complex Numbers – Complex Roots of z^3 + z? = 0

Question

I'm trying to find the complex roots of $z^3 + \bar{z} = 0$ using De Moivre.

Some suggested multiplying both sides by z first, but that seems wrong to me as it would add a root ( and I wouldn't know which root was the extra ).

I noticed that $z=a+bi$ and there exists $\theta$ such that the trigonometric representation of $z$ is $\left ( \sqrt{a^2+b^2} \right )\left ( \cos \theta + i \sin \theta \right )$ .

It seems that $-\bar{z} = -\left ( \sqrt{a^2+b^2} \right )\left ( \cos (-\theta) + i \sin (-\theta) \right )$

However, my trig is pretty rusty and I'm not quite sure where to go from here.

Answer 1

Hint: First try to narrow down the value of $|z|$