Suppose $x_{0}, x_{1}, \ldots, x_{n} \in[a, b] \subset \mathbf{R}$ are pointwise distinct and $f(x)$ is continuous on $[a, b]$. For $k=1,2, \ldots, n$ define

$$I_{0, k}(x)=\frac{f\left(x_{0}\right)\left(x_{k}-x\right)-f\left(x_{k}\right)\left(x_{0}-x\right)}{x_{k}-x_{0}}$$

and for $k=2,3, \ldots, n$

$$I_{0,1, \ldots, k-2, k-1, k}(x)=\frac{I_{0,1, \ldots, k-2, k-1}(x)\left(x_{k}-x\right)-I_{0,1, \ldots, k-2, k}(x)\left(x_{k-1}-x\right)}{x_{k}-x_{k-1}}$$

Show that $I_{0,1}, \ldots, k-2, k-1, k(x)$ is a polynomial of order $k$ which interpolates $f(x)$ at $x_{0}, x_{1}, \ldots, x_{k}$.

Given $x_{k}=\{-1,0,2,5\}$ and $f\left(x_{k}\right)=\{33,5,9,1335\}$, determine the interpolating polynomial.