Given $S = \{ (x_0, a_0), (x_1, a_1), \dots (x_n,a_n) \}$ we know there exist an unique polynomial of degree $\leq n$ such that $p(x_i) = a_i$ for $i = 0, \dots, n$, the Lagrange interpolating polynomial.
I am trying to find a general expression for all the interpolating polynomials (of any degree) of this set. It is clear that if $q$ is a polynomial such that $q(x_i) = 1$ for all $i = 1, \dots, n$ then $pq$ is an interpolating polynomial of those points. Are all interpolating polynomials like this?