Question

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

Answer 1

Considering right hand side,

$\begin{array}{c}\left(x-y\right)\left(x^2+xy+y^2\right)=x\left(x^2+xy+y^2\right)-y\left(x^2+xy+y^2\right)\\ =\left(x^3+x^{2}y+x{y}^2\right)-\left(x^{2}y+x{y}^2+y^3\right)\\ =x^3+x^{2}y+x{y}^2-x^{2}y-x{y}^2-y^3\\ =x^3-y^3+x^{2}y-x^{2}y+x{y}^2-x{y}^2\\ =x^3-y^3\end{array}$

which is equal to left hand side

Hence proved.