Connection between polynomials and Pascal's triangle

[LightningB0LT]

I recently discovered that for a 3rd degree polynomial I was studying, f(5) - 4f(4) + 6f(3) - 4f(2) + f(1) = 0. At first I just though it was coincidental that the coefficients were the 5th row of Pascal's Triangle, but then I tried a 2nd degree polynomial and found that f(4) - 3f(3) + 3f(2) - f(1) = 0, which is the 4th row. The same thing worked for 1st and 4th degree polynomials I tried, using the 3rd and 6th row as coefficients. I've tried to reason through why this might be the case, but without success. Can someone explain this to me? Thanks in advance!

 

[SteamKing]

Check out the Binomial Theorem:

http://en.wikipedia.org/wiki/Binomial_theorem

 

[Stephen Tashi]

I recently discovered that for a 3rd degree polynomial I was studying, f(5) - 4f(4) + 6f(3) - 4f(2) + f(1) = 0.

Another way of looking at the same equation is that expresses f(5) in terms of the value of the polynomial at previous consecutive values: f(5) = 4f(4) - 6f(3) + 4f(2) - f(1) and so this link might be relevant: http://ckrao.wordpress.com/2012/02/28/finite-differences-for-polynomial-extrapolation/