Sketch set in the complex plane re = 0

Question

Problem: Describe and sketch the set $\{Z\in \mathbb C| Re(Z^3)=0\}$.

This is what I began with:

$Z^3=(x+iy)^3 =x^3-3xy^2+(3x^2y-y^3)i$

The set seems to be completely imaginary(?) since $\ Re(Z^3)=0$

$\ Re(Z^3)= x^3-3xy^2=0$

Then I think this is the graph I need to sketch..

Any advice? Am I working in the right direction?

Answer 1

I would suggest that you think geometrically from the start. That's a lot easier. What kind of point lands on the imaginary axis when you cube it?

But now that we're here: $$ x^3-3xy^2=x(x-\sqrt3y)(x+\sqrt3y) $$ is $0$ iff one (or more) of these three factors is zero. Each of the factors are zero along a straight line in the plane ($x=0$, for instance, is the imaginary axis itself). So your set is the union of those three lines. And yes, you are asked to draw it as well. Or at least sketch it.