Let z be a complex number such that \(z + \frac{1}{z} = 1\). Find z^3.

I've already tried expanding z. I've also tried just getting z onto one side, but I ended up getting \(z=z-z^2 \rightarrow 0=-z^2\)

z + 1/z = 1

z^2 + 1 = z

z^2 - z + 1 = 0

Complete the square on z

z^2 - z + 1/4 = -1 + 1/4

(z - 1/2)^2 = -3/4

z - 1/2 = ± i √3 / 2

z = 1/2 + (√3/2) i or z = 1/2 - ( √3/2) i

In each case

z^3 = - 1