Intermediate Algebra

Squaring the equation z + 1/z = 1 gives z^2 + 2 + 1/z^2 = 1. Then multiplying by z^2 gives z^4 + 2z^2 + 1 = z^2. Rearranging, we get z^4 + 2z^2 + 1 - z^2 = 0, which factors as (z^2 + 1)(z^2 + 1) = 0. Since z^2 + 1 = 0 has no solutions, z^2 = -1. Taking the square root of both sides, we find z = i or z = -i.

If z = i, then z^3 = i^3 = -i.

Alternatively, if z = -i, then z^3 is also equal to -i.

Therefore,  in either case, z^3 = -i.

Yeah, I managed to solve it 30 minutes later. (Sorry, the correct answer was -1.)

Anyways, thanks for the answer. What I did was take

$z + \frac{1}{z}=1$ and multiply it by $z^2$, getting $z^3-z^2+z=0$. However, when you multiply $z+ \frac{1}{z}=1$ by $z$, you will get

$z^2-z=1$

From which you can replace the $-z^2+z$ with $-1$ (The negative of the equation above) to get

$z^3-1=0$

From which you can easily get $z^3=-1$