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
+1242 
