# Polynomial proof Newton

Let $f(x)=a_mx^{m}$ + (lower degree terms) be a polynomial. Show that

$$f[x_0,...,x_n,x] = \begin{cases} {degree}[m-n-1], & n < m-1 & \\ a_m, &n =m-1 \\ 0 & n>m-1\end{cases}$$where $f[x_0,...,x_n,x]$ is Newton divided difference.

I am having difficulty proving this question. I don't know how to prove it. What I know up to this point is that from writing the general polynomial of degree $m$, I see that there are $m+1$ independent parameters $a_0,...,a_m$. From a function $p(x_i)$, $i = 0,...,n$ then it imposes $n+1$ conditions on $p(x)$. From that and the conditions in my question how can I prove this?

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