Reward for Solution

If anyone can find a generalization of the following factorizations for square matrices and prove that it holds, I will mail you my deepest gratitude. Seriously.

a.) 1 - x^3 = (1 - x)(1 + x + x^2)

b.) 1 + x + x^2 = (1 - x^3) / (1 - x)

I'm going nucking futs over this. Matrices and I do NOT get along.