Question

Factor x^6-1 completely and find real and complex zeros.

Answer 1

\[(-1+x) (1+x) (1-x+x^2) (1+x+x^2)\]

The first two roots are trivial. -1, and 1. The latter two require the quadratic formula.

Answer 2

rewrite as
(x^3)2 - 1^2
that's
((x^3 -1)(x^3+ 1)
that's sum of cubes and difference of cubes (x^3 and 1^3)

(x-1)(x^2+x +1)(x+1)(x^2 - x +1)
so your final answer is

(x^2 - 1)(x^2+ x + 1)(x^2 - x +1)