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.


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


I was wondering what are other ways to prove that $x^3-y^3 = (x-y)(x^2+xy+y^2)$
Divide $x^3-y^3$  by  $x-y$ as polynomials
You can use "Long Division of Polynomials"

$\frac {1-(\frac y x)^3}{1-(\frac y x)}=1+\frac y x+(\frac y x)^2$

Multiply both sides by $x^3$

$\frac {x^3-y^3}{1-(\frac y x)}=x(x^2+xy+y^2)$

Then you have

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