TITLE: Are polynomials infinitely many times differentiable?
QUESTION [19 upvotes]: Are polynomials infinitely many times differentiable?
If so, does it only mean that at some point we reach 0 and then we keep on getting 0?
Thank you!

REPLY [10 votes]: The other answers probably answered your question, but in case you wanted to learn more, the key word to look up is smoothness of the function (see here for instance: https://en.wikipedia.org/wiki/Smoothness). 
So another, more fancy way of asking your question is whether or not polynomials are smooth functions.