Characterization of Polynomial In terms of Derivative

Question

In many problem or theorem in which I have to show that given function is polynomial, there is the argument that for some k show that $f^{(n)}(z)=0, \forall n>k $ .
I know that by definition of polynomial above characteristics is ok. But I am not convinced with fact that converse part. that there is no function other than polynomial which satisfies this property.
Where am I missing?
Any help will be appreciated.

Answer 1

If you know $f^{(n)}(z)$ is identically zero, integrate $n$ times, remembering the arbitrary constant at each stage.