Crazy Polynomials

Here is the set up for this problem. I have a polynomial with positive integer coefficients. You are allowed to ask me the value of this polynomial at any TWO points you choose. I claim that if you ask for the value of the polynomial at cleverly chosen points, then you can completely determine the polynomial from this information. How is this done?! Remember, you initially start out with no information about the polynomial other than the fact that its coefficients are positive integers.

Hareder Version of Problem (Due to Kabe Moen and Jermey Martin from the University of Kansas)

Find a way to solve the above problem by only determining the value of the polynomial at only ONE point.