Finding an interpolation polynomial of degree greater than n+1

Question

The degree of an interpolation polynomial of n data points is n-1. However, this is THE interpolation polynomial. If we have already found it, how can we build an interpolation polynomial for the same data points of degree n or n+1? Any help please?

Answer 1

You can add any expression $$q(x)\prod_{k=1}^n(x-x_k),$$ $q$ being some polynomial, to the interpolation polynomial and get another polynomial that takes the same values at the sample points $x_k$.