If $x, y$ and $z$ are the cube roots of unity, then what is the value of $xy + yz + zx$?

Let $z_1$ and $z_2$ be two complex numbers such that $\left|\frac{z_1 + z_2}{z_1 - z_2}\right| = 1$, then what is $\operatorname{Re} \left(\frac{z_1}{z_2}\right) + 1$ equal to?

If $\omega \neq 1$ is a cube root of unity, then what are the solutions of $(z-100)^3 + 1000 = 0$?

What is $(1 + i)^4 + (1 - i)^4$ equal to, where $i=\sqrt{-1}$?