Given distinct points

$$(x_0,y_0),\ldots,(x_n,y_n),$$

there is exactly one polynomial of degree at most $n$ that passes through all of them.

One explicit form is the Lagrange interpolating polynomial

$$p(x)=\sum_{i=0}^n y_i L_i(x),$$

where

$$L_i(x)=\prod_{j\ne i}\frac{x-x_j}{x_i-x_j}.$$

Each basis polynomial satisfies $L_i(x_i)=1$ and vanishes at the other data points.

Interpolation matches the supplied values exactly, but exact matching does not imply good behavior between or beyond them. High-degree polynomial interpolation can oscillate strongly, and the choice and spacing of interpolation nodes affects the error.

Polynomial interpolation is a building block for numerical differentiation, quadrature and local approximation.