Errors of Interpolating Polynomials

Theorem

Suppose $x_0, x_1, \ldots, x_n$ are distinct numbers in the interval $[a, b]$ and $f \in C^{n+1}[a, b]$. Then, for each $x$ in $[a, b]$, a number $\xi(x)$ (generally unknown) between $x_0, x_1, \ldots, x_n$, and hence in $(a, b)$, exists with:

where $P_n(x)$ is the interpolating polynomial. $R_n(x)$ is the error term. The above equation can only be used for known and differentiable functions, which is usually not the case.