Question on Wiki proof of Vandermonde matrix

Question

On this page, https://proofwiki.org/wiki/Vandermonde_Determinant#Proof_3, I understand how they are creating the $n-1$ degree polynomial $P(x)$ by calculating the determinant based on the final row. I understand how they find the $n-1$ zeroes of $P(x)$, and I understand how they got that $P(x) = C(x-x_1)\ldots(x-x_{n-1})$ but I don't understand how they found that $C=V_{n-1}$. It's not very clear on that step.

Even if the coefficient of the $x^{n-1}$ is $V_{n-1}$, how are we sure that the other coefficients will match up? If the above is true, does that mean that $$\begin{vmatrix} x_1 & x_1^2 & \ldots & x_{1}^{n-1} \\ x_2 & x_2^2 & \ldots & x_{2}^{n-1} \\ \vdots & & \ddots & \vdots \\ x_{n-1} & x_{n-1}^2 & \ddots & x_{n-1}^{n-1} \end{vmatrix}$$ is equal to $x_1x_2x_3\ldots x_{n-1}$ because they are both the constant term of $P(x)$? If so, are these identities useful in any way?

Thanks for any help!

Answer 1

$C$ is the leading coefficient of $P$. Earlier in the proof, they wrote down the coefficients of $P$ as determinants of sub-matrices. In particular, the coefficient of the leading monomial $x^{n-1}$ is the determinant of a smaller Vandermonde matrix involving $x_1, \ldots, x_{n-1}$.

Response to your edit: because you have found all roots of $P$, it is necessarily of the form $C(x-x_1) \cdots (x-x_{n-1})$, so yes all the other coefficients must match up. Note that determinant written at the end of your post is necessarily $V_{n-1} x_1 \cdots x_n$ (not $x_1 \cdots x_n$). One way to see this is to divide the $i$th row by $x_i$ which gives you $x_1 \cdots x_n$ times the determinant of the smaller Vandermonde matrix which is $V_{n-1}$.