Sorry, do you mean, solve ?
EDIT:// I checked your work and there seems nothing wrong with your method. It is quite slick to think of
You could always use polynomial expansion, foiling out if you like, but that wouldn't seem to be worth the effort
You are dead on with your explanation (that -2 is the other answer). I believe this is the step where the oversight was:
If you are dealing with equations over the complex numbers, things are different. By the fundamental theorem of algebra, there roots for a polynomial of degree . So has solutions for . So can picture the solutions as vertices of a polygon in the complex plane (as seen here), with one vertex at the principal root. For even powers you have the symmetry of flipping around the vertical axis which is not present in odd powers.
I'm sorry for double posting
however, on this post, i asked a question that dealt with the algebra part of the problem, it was an algebra concept that I couldn't understand, on the other post, it was a calculus concept.
Should I put it both in one thread next time?