Shilov on calculating Vandermonde determinant

Question

From Shilov's Linear Algebra, Pg 15 , he shows how one would calculate the Vandermonde determinant by treating is as a polynomial. My question which theorem do he mean by mentioning "by a familiar theorem of elementary algebra"? I am aware of the remainder theorem, but I think this is not what he is getting at. Can someone tell me (1)which theorem is he referring to (2) and explain to me how is he using it to calculate the determinant? Thanks in advance to all!

Answer 1

The theorem of elementary algebra in question is that if a polynomial function $f(x)$ vanishes at $a$, so that $f(a)=0$, then $f$ is divisible by $x-a$. The author is using this to show that the determinant is divisible by a certain polynomial.