Math Problem

I choose at random a polynomial (call it a function, f (x)) with only non-negative integer coefficients. You get to choose a number, a, and I will tell you what f (a) is. You then get to choose another number, b, and I will tell you what f (b) is. Devise a method of choosing a and b such that after hearing what f (b) is, you can tell me what the polynomial is (i.e., what the coefficients are). Propose your answer in the Comments. You may not participate if you knew the answer before reading this post. I assure you no post-high school math skills are required to solve this problem.