Connection between polynomials and Pascal's triangle

Main Question or Discussion Point

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!
 

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Stephen Tashi
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