Finite Differences and Polynomials
Date: 06/17/99 at 00:22:52 From: Chris Harris Subject: Why are finite differences and polynomials linked? I have read that f(x) is a polynomial in x and of degree n if and only if the finite differences of x go constant at the nth row of differences. I do not doubt the validity of this, but could you please explain why this should be true in general? I can't understand why it is so except for in the case of linear polynomials P(x) = mx+b. Or is this something one can't easily grasp until college?
Date: 06/17/99 at 10:07:59 From: Doctor Anthony Subject: Re: Why are finite differences and polynomials linked? It is easy to show why this happens. If you have say a cubic in x such as f(x) = ax^3 f(1) = a f(2) = 8a f(3) = 27a f(4) = 64a f(5) = 125a n = 1 2 3 4 5 f(n)= a 8a 27a 64a 125a 1st diff. 7a 19a 37a 61a 2nd diff. 12a 18a 24a 3rd diff. 6a 6a With 3rd differences constant. If you wish to write out the rather tedious algebra you could make up a table similar to that shown above with the entries x (x+1) (x+2) (x+3) (x+4) f(x)= ax^3 a(x+1)^3 a(x+2)^3 a(x+3)^3 a(x+4)^3 1st diff a(3x^2+3x+1) a(3x^2+9x+7) a(3x^2+15x+19) a(3x^2+21x+37) 2nd diff a(6x+6) a(6x+12) a(6x+18) 3rd diff 6a 6a again 3rd differences are constant. You can carry out a similar exercise with functions like f(x) = ax^3 + bx^2 + cx + d You will find again that third differences are constant. A theoretical proof follows along the lines below: Define [x]^n = x(x-1)(x-2).....(x-n+1) (there are n factors) so for example [x]^3 = x(x-1)(x-2) and we define [x]^0 = 1 Let D[f(x)] = f(x+1) - f(x) and so D[[x]^n] = [x+1]^n - [x]^n = (x+1)[x]^(n-1) - [x]^(n-1)(x-n+1) = [x]^(n-1) [x + 1 - x + n -1] = n.[x]^(n-1) also D^2[x]^n = D.D[x]^n = D[n.[x]^(n-1)] = n.D[x]^(n-1) = n(n-1)[x]^(n-2) which is of course the familiar result for ordinary differentiation. Similarly D^3[x]^n = n(n-1)(n-2)[x]^(n-3) and if n = 3 this gives D^3[x]^3 = n(n-1)(n-2)[x]^0 = n(n-1)(n-2) = 3 x 2 x 1 = 6 So with a cubic polynomial the third differences are constant. In the general case an nth degree polynomial will have the nth differences constant. - Doctor Anthony, The Math Forum http://mathforum.org/dr.math/
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