The question I am working on starts of with:
Find the five fifth roots of unity and hence solve the following problems
I have done that and solved several questions using this, however when I came to the last question (the one in the title) I got stumped.
The five fifth root of unity are $z= \cos({2\pi k \over 5})+i\sin({2\pi k \over 5}), k\in \mathbb{Z}$ or more simply put $w=\cos({2\pi \over 5})+i\sin({2\pi \over 5})$ and the roots are $z=1, w, w^2, w^3, w^4$.
Now, using this information we are supposed to find the roots of $(z+1)^5=(z-1)^5$. However having tried some different approaches I don't know how to proceed. I tried simplifying it to $5z^4+10z^2+1=0$ from which I suppose I can use the quadratic equation, but it does not utilize the five fifth roots of unity and so I wont get my answer in terms of $w$ (which is the requirement).