Transforming the difference of cubes into a product

Question

I want to know what happened at the denominator at line 2... Why did it it become like that, going from $x^3 - y^3$ to $(x-y)(x^2 +xy + y^2)$?

Answer 1

That comes from the identity

$x^n - y^n = (x-y)(x^{n-1}+x^{n-2}y + \cdots + xy^{n-2} + y^{n-1})$.

To see why it is true, simply expand the right hand side. All the terms except the ones containing $x^n$ and $y^n$ will cancel.