Suppose that $\displaystyle f^{(n+1)}$ exists on $\displaystyle (a, b)$ and $\displaystyle x_0, x_1, ..., x_n \in (a, b)$ and p is the polynomail of degree $\displaystyle \leq n$ such that $\displaystyle p(x_i)=f(x_i)$. I want to show that for $\displaystyle x\in (a, b)$:

$\displaystyle f(x)=p(x)+\frac{f^{(n+1)}(c)}{(n+1)!}(x-x_0)(x-x_1)...(x-x_n)$

for $\displaystyle c_x \in(a, b)$

We are just starting Taylor's Theorem but I'm not sure how this works. Any advice?