Identify a polynomial with two questions

Question

I am thinking of a polynomial p whose coefficients are all non-negative whole numbers. I will not tell you its degree. You may name a number x and I will tell you p(x), and you may do this twice.

Determine the polynomial exactly. Why do two questions suffice, and why does one not?

Accepted Answer

Ask for p(1). Call the answer S. Then ask for p(S + 1), and write that value in base S + 1: its digits are the coefficients. Why. p(1) is the sum of all the coefficients, so every coefficient is at most S. Now evaluate at x = S + 1: p(S + 1) = c0 + c1 (S + 1) + c2 (S + 1) squared + ..., and since every coefficient is between 0 and S, this is exactly the base-(S + 1) representation of the number p(S + 1), with the coefficients as its digits. Reading the digits from the right gives c0, c1, c2, ... and the degree is the position of the last non-zero digit. Example. Suppose p(1) = 7. Ask for p(8)…

Identify a polynomial with two questions

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