Finite Differences proving

[relinquished™]

I am to prove by mathematical induction that for a polynomial of degree n p(x) with leading coefiicient a_0,

$$\Delta^n p(x) = a_o n!$$

My proof: By mathematical induction

$$\Delta^1 p(x) = [a_0(x+1) + a_1] - [a_0x + a_1]$$
$$\Delta^1 p(x) = [a_0x + a_0 + a_1] - [a_0x + a_1]$$
$$\Delta^1 p(x) = a_0$$
$$\Delta^1 p(x) = a_0 \cdot 1!$$

hence, S(1) is true

This is where I have a problem. I assume that $$\Delta^n p(x) = a_o n!$$ is true... how do i show that S(n+1) is also true? The degree of the polynomial becomes n+1 and my S(n) becomes inapplicable already...

 

[HallsofIvy]

What is the first difference of a polynomial of degree n+1? Isn't it a polynomial of degree n? What is its leading coefficient?

 

[lurflurf]

First show that n=1,2,...
$$\Delta x^{n}=n x^{n-1} +P_{n-2}$$
where P_{n} means some polynomial of degree at most n
and thus P_0=0
and
$$\Delta P_{0}=0$$
then show by induction that for k=0,...,n
$$\Delta^k x^n=\frac{n!}{(n-k)!}x^{n-k}+P_{n-1-k}$$
Take your result as a special case