Interpolating polynomials and divided differences

We shall take it as known that a polynomial of degree n has at most n distinct zeros (a proof is given in Lemma 1 below). Given n+1 distinct real numbers xj and any numbers αj (0 ≤ j ≤ n), there is a unique polynomial p of degree at most n satisfying p(xj) = αj (0 ≤ j ≤ n). The polynomial is unique, since if p1 and p2 were two such polynomials, then p1−p2 would be zero at each xj: since it has degree at most n, it can only be zero. Existence can be deduced from the fact that the matrix with entries xj (0 ≤ j ≤ n, 0 ≤ k ≤ n) is non-singular, but it is easy to describe an explicit construction, as follows.