interpolating polynomial

The one polynomial of degree at most n that passes exactly through n+1 given data points.

Given $n+1$ distinct nodes $x_0,\ldots,x_n$ and values $y_0,\ldots,y_n$ , the interpolating polynomial is the unique polynomial $P$ of degree at most $n$ with $P(x_i)=y_i$ for every $i$ . Existence and uniqueness are guaranteed by the interpolation existence-uniqueness theorem; the Lagrange and Newton divided-difference forms are two different ways of writing down this same, single polynomial.

Examples

Given 3 points, the interpolating polynomial has degree at most 2 -- a parabola, whether it's written in Lagrange form or Newton's divided-difference form, is the exact same curve either way.