Finding the roots of the sum of complex numbers

Question

Find the roots of $(z-1)^6 +(z+1)^6$. So far we've tried binomial expansion, but where to go now, as it is a non-calculator question?

Answer 1

Hint: Consider the equation $\left(\frac{z-1}{z+1}\right)^6=-1$.

Answer 2

This is equivalent to finding the roots of $$2(z^6 + 15z^4 + 15z^2 + 1) = 0.$$ If we substitute $x = z^2$, we get $x^3 + 15x^2 + 15x + 1 = 0$, which can be factored as $(x + 1)(x^2 + 14x + 1) = 0$. This can now be solved, and now we can use our substitution $x = z^2$ to find the roots of $z$ from the roots of $x$.