Finding the Simplest Polynomial for a Series

[PEZenfuego]

I cannot for the life of me remember where I learned to do this nor can I remember why this works, but I know a method for finding the simplest polynomial for a series. I had something half typed up and then I found this site: http://www.johansens.us/sane/technotes/formula.htm which explains a method more thoroughly. While this is not identical to the way I do it, it is certainly similar enough.

So why does this work?

 

[HallsofIvy]

That is "Newton's divided difference formula": http://en.wikipedia.org/wiki/Newton_polynomial