Decomposition of $x^n-y^n$

Question

As a part of textbook assignment I was asked to prove that $x^2-y^2=(x+y)(x-y)$, and I did so as follows: $$x^2-y^2=x^2-y^2+xy-xy=x(x+y)-y(x+y)=(x+y)(x-y)$$ Later, I used similar method to decompose $x^3-y^3$, with different "cofactors": $$x^3-y^3=x^3-y^3+x^2y+xy^2-x^2y-xy^2=...$$ However, when I needed to do this for $x^n-y^n$ I was slightly disconcerted, because up to this point there was no particular reasoning behind adding fitting "cofactors", I just made them up to suit my equation, and it became a problem, because now it was not as evident. The solution I resorted to looked like this: $$x^n-y^n+(x^{n-1}y+x^{n-2}y^2+...x^2y^{n-2}+xy^{n-1})-(x^{n-1}y+x^{n-2}y^2+...x^2y^{n-2}+xy^{n-1})$$ But this time I too picked the terms to cancel each other out without any reasoning. Is there even one, to justify them appearing out of nowhere? And what interests me is whether or not this can be regarded as satisfactory proof, and are there other, nicer ways to do it?

Answer 1

I would go this way $$ \eqalign{ & x^{\,n} - y^{\,n} = x^{\,n} \left( {1 - \left( {y/x} \right)^{\,n} } \right) = x^{\,n} \left( {1 - z^{\,n} } \right) = x^{\,n} \left( {1 - z} \right)\left( {{{1 - z^{\,n} } \over {1 - z}}} \right) = \cr & = x^{\,n} \left( {1 - z} \right)\left( {1 + z + \cdots z^{\,n - 1} } \right) = x\left( {1 - z} \right)x^{\,n - 1} \left( {1 + z + \cdots + z^{\,n - 1} } \right) = \cr & = \left( {x - y} \right)\left( {x^{\,n - 1} + x^{\,n - 2} y + \cdots + y^{\,n - 1} } \right) \cr} $$