I'm trying to find the fifth difference (f(x+1) - f(x)) of a regular polynomial f(x)=ax^{5}+bx^{4}+cx^{3}+dx^{2}+ex+f but I keep messing up, help please!
Try finding the first difference of a linear function, the second difference of a quadratic, the third difference of a cubic, and you should see a pattern emerge.
When I compute the differences I suggested above, the pattern that emerges suggest the nth difference of an n degree polynomial will be $\displaystyle n!a_n$. Proofs of this can be found online.