While reading about interpolation I came across the following equation in Norlund. It involves determinants and I don't understand it in full yet. I do know how Lagrange and Newton follow by using the Laplace expansion.
$$ P_n=-\det \begin{bmatrix} 1 & x_0 & \dots & x_0^n & y_0\\ 1 & x_1 & \dots & x_1^n & y_1\\ \vdots & \vdots & \ddots & \vdots & \vdots\\ 1 & x_n & \dots & x_n^n & y_n\\ 1& x & \dots & x^n & 0\\ \end{bmatrix} : \det \begin{bmatrix} 1 & x_0 & \dots & x_0^n\\ 1 & x_1 & \dots & x_1^n\\ \vdots & \vdots & \ddots & \vdots\\ 1 & x_n & \dots & x_n^n\\ \end{bmatrix} $$
Where $P_n$ is the interpolation polynomial. Let's define the following for the interpolation polynomial $P_n(x_i)=y_i$ for $i\in[0,1,...,n]$. But this implies, after converting this equation into the determinant of a single matrix $A$, the following:
$$ \det(A)=\det \begin{bmatrix} 1 & x_0 & \dots & x_0^n & y_0\\ 1 & x_1 & \dots & x_1^n & y_1\\ \vdots & \vdots & \ddots & \vdots & \vdots\\ 1 & x_n & \dots & x_n^n & y_n\\ 1& x & \dots & x^n & P_n\\ \end{bmatrix} =0\quad\quad\quad\quad\quad\quad\quad\quad $$
(Question) What is the geometric interpretation of this statement? My best guess is something along the lines of infinite solution because $x\in\mathbb{R}$.
Thanks in advance.