Lagrange interpolation polynomial

Given $n+1$ points $(x_0, y_0), (x_1, y_1), \ldots, (x_n, y_n)$ with all $x_i$ different, there is a unique polynomial of degree at most $n$ that passes through all these points. The Lagrange interpolation polynomial is a way to write this polynomial explicitly.

Important properties

• The Lagrange interpolation polynomial passes through all the given points.

• It is constructed as a sum of terms, each term being zero at all but one of the $x_i$.

• The formula uses products to 'zero out' the polynomial at the other points.