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

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