<s>[INST] Do not understand formula... How does;
$x^{n} -y^{n}=(x-y)(x^{n-1} + x^{n-2}y+...+x y^{n-2}+ y^{n-1} )$
work on $x^{2} - y^{2}$ 
When I attempt to apply the formula on $x^{2} - y^{2}$ 
I get the following
$x^{2} - y^{2} =(x-y)( x^{1} + x^{0}y+...+x y^{0} + y^{1} )$
$x^{2} - y^{2} =(x-y)(2x+2y)$
which is obviously false. What is the correct way to use the formula?
 [/INST] Note how the last term in the factorization of $x^n-y^n$ has $x$ raised to the $0$th power. That means you have to stop when $x$ is raised to the power of $0$.
$$x^2-y^2=(x-y)(x^1y^0+x^0y^1)$$
$$\boxed{x^2-y^2=(x-y)(x+y)}$$
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