Finite Differences proving

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

[tex]
\Delta^n p(x) = a_o n!
[/tex]

My proof: By mathematical induction

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

hence, S(1) is true

This is where I have a problem. I assume that [tex]\Delta^n p(x) = a_o n! [/tex] 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

Science Advisor
Homework Helper
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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

Homework Helper
2,417
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First show that n=1,2,...
[tex]\Delta x^{n}=n x^{n-1} +P_{n-2}[/tex]
where P_{n} means some polynomial of degree at most n
and thus P_0=0
and
[tex]\Delta P_{0}=0[/tex]
then show by induction that for k=0,...,n
[tex]\Delta^k x^n=\frac{n!}{(n-k)!}x^{n-k}+P_{n-1-k}[/tex]
Take your result as a special case
 

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