Factoring $a^{k} - b^{k}$

Question

I am a bit lost how to factor $a^{k} - b^{k}$. I know it links to the binomial theorem but I can't remember how to do it. Could anyone explain?

Answer 1

It has nothing to see with the binomial theorem, but with the geometric series. Actually it's a high-school formula: $$1+x+x^2+\dots+x^{k-1}=\frac{1-x^k}{1-x},$$ which also reads as $$1-x^k=(1-x)(1+x+x^2+\dots+x^{k-1})$$ whence, settink $x=\dfrac ba$, and multiplying both sides by $a^k$, one obtains: $$a^k-b^k=(a-b)(a^{k-1}+a^{k-2}b+a^{k-3}b^2+\dots+b^{k-1}).$$

Answer 2

We have

$$\begin{align}a^k-b^k&=(a-b)\left(a^{k-1}+a^{k-2}b+\cdots+ab^{k-2}+b^{k-1}\right)\end{align}$$