How to prove $x^3-y^3 = (x-y)(x^2+xy+y^2)$ without expand the right side? I can prove that $x^3-y^3 = (x-y)(x^2+xy+y^2)$ by expanding the right side.


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*$x^3-y^3 = (x-y)x^2 + (x-y)(xy) + (x-y)y^2$ 

*$\implies x^3 - x^2y + x^2y -xy^2 + xy^2 - y^3$

*$\implies x^3 - y^3$


I was wondering what are other ways to prove that $x^3-y^3 = (x-y)(x^2+xy+y^2)$

First, notice that for all $u$,
\begin{align*}(1-u)(1 + u + u^2) &= 1 + u + u^2 - (u - u^2 - u^3)\\
&= 1 + (u-u) + (u^2-u^2) - u^3\\
&= 1 - u^3.\end{align*}
Now, take $u = \frac{y}{x}$ and multiply by $x^3$.
