Is this generalized factorization of a difference of powers correct?

Question

Is this factorization true?

$$(x^n - y^n) = (x+y)^{n-1}(x-y)^{n-1}$$

I am trying to use it in my computation of the determinant of a Vandermonde matrix.

Thanks,

Answer 1

An alternative (true) identity is

$$x^n - y^n = (x-y)(x^{n-1} + x^{n-2}y + x^{n-3}y^2 + \cdots + xy^{n-2} + y^{n-1})$$

Answer 2

You can't have equality in your equation since the LHS is degree $n$ while the RHS is degree $2n-2$. These are only equal for $n=2$. The correct factorization is:

$$\begin{align}x^n-y^n & =(x-y)(x^{n-1}+x^{n-2}y+\cdots+xy^{n-2}+y^{n-1})\\ & =(x-y)\sum_{k=0}^{n-1}x^{n-1-k}y^k. \end{align}$$