Factoring and expanding $1-x^8$ It's been a while since I've studied factoring, and I need it for a question. How did this go from $1-x^8$ to $(1-x^4)(1+x^4)$ and then to $(1-x)(1+x+...+x^7)$? I remember studying this a few years go but unfortunately I don't remember.
$$1-x^8=(1-x^4)(1+x^4)=(1-x)(1+x+\dots + x^7)$$

Taking your question, “How did this go from $1-x^8$ to $(1-x^4)(1+x^4)$ and then to...” literally, it involves repeated applications of the identity $1-z^2=(1-z)(1+z)$: $$\begin{align} 1+x^8 &= (1-x^4)(1+x^4) \\ &= (1-x^2)(1+x^2)(1+x^4) \\ &= (1-x)(1+x)(1+x^2)(1+x^4) \\ &= (1-x)(1+x^2+x^3+x^4+x^5+x^6+x^7). \end{align}$$
