I am sure this is a simple standard textbook problem but I seem to have forgotten the basic steps.
Problem: Given a polynomial $f(x)$ of degree $n$ that passes through $n+1$ points and an additional point $P(x_{n+2}, y_{n+2})$, we want to construct the degree-$n+1$ polynomial $g(x)$ such that it passes through all the $n+1$ points of $f(x)$ and also through $P$.
I recall there is a simple construction for this in the lines of Lagrange Interpolation (when the points are specified), but don't remember the details of the interpolation when the polynomial is given and we just want to add more points. Can anyone help?