 Solution

Write $\omega (t)={\prod }_{i=0}^n(t-x_i)$. If $x$ is an interpolation node, the polynomial interpolation error is zero. Otherwise setThe function $F$ has $n+2$ distinct zeros: the $n+1$ interpolation nodes and $x$. Repeated Rolle's theorem therefore gives a point $\xi$ strictly between the smallest and largest of these points with $F^{(n+1)}(\xi )=0$. Since $p^{(n+1)}=0$ and ${\omega }^{(n+1)}=(n+1)!$, this yields the polynomial interpolation errorAll differentiations are justified by $f\in C^{n+1}[a,b]$. At an interpolation node the same displayed equality holds with any $\xi \in [a,b]$, because the product is zero.