Lagrange polynomial

Summary

Take $n + 1$ points in the plane with distinct first coordinates. Their Lagrange polynomial is the unique polynomial function of degree at most $n$ whose graph passes through all of them. It is written as a sum of basis polynomials, each equal to $1$ at one node and $0$ at the others. It gives an explicit solution of polynomial interpolation and is the starting point of numerical differentiation and quadrature rules. The definition gives the formula.