Understand this:
https://www.math.iupui.edu/~ccowen/OldCoursePages/Math300F07/300F07Quiz4.pdf
My condensed version:
You know you can factor (x-1) out of (x^n-1) because x=1 is guaranteed to be a root of the polynomial (x^n-1) (because whether n is even or odd we will always get zero when x=1)...