Let $n\in\mathbb{N}\cup\{0\}$, let $x_0,\ldots,x_n\in\mathbb{R}$ be pairwise distinct, and set $X_n:=\{x_0,\ldots,x_n\}$. Let $I\subset\mathbb{R}$ be an interval such that $X_n\subset I$, let $f:I\to\mathbb{R}$ be a function, and let $t\in I\setminus X_n$ be such that the [divided difference](/page/Divided%20Difference) $f[x_0,\ldots,x_n,t]$ is defined. If $p\in\mathcal{P}_n$ satisfies

\begin{align*} p(x_m)=f(x_m) \end{align*}

for every $m\in\{0,\ldots,n\}$, then

\begin{align*} f(t)-p(t)=f[x_0,\ldots,x_n,t]\prod_{m=0}^{n}(t-x_m). \end{align*}