Lagrange basis polynomials

Summary

The Lagrange basis polynomials of distinct nodes $x_0, \dots, x_n$ are $n + 1$ polynomials of degree $n$. Each one equals $1$ at its own node and $0$ at the other nodes. They form a basis of the polynomials of degree at most $n$. In this basis the coordinates of a polynomial are its values at the nodes. So the Lagrange polynomial of given data is simply the sum of the basis polynomials weighted by the data values.