 Interpolation polynomial

An interpolation polynomial for distinct nodes $x_0,\dots ,x_n$ and prescribed values $y_0,\dots ,y_n$ is a polynomial $p$ satisfying $p(x_i)=y_i$. There is a unique such polynomial of degree at most $n$: the Lagrange interpolation polynomial constructs it explicitly, and uniqueness follows because a nonzero degree-at-most-$n$ difference cannot have $n+1$ distinct roots. Newton interpolation polynomials express the same object using divided differences.