Polynomial Puzzle and Lill’s Method

I have in mind a polynomial with nonnegative integer coefficients. Can you determine completely the polynomial by just asking two questions?

I forgot where I saw this from but it was interesting enough to spur a few lunch time discussions with colleagues. I finally saw a reference in Dan Kalman’s book “Polynomia and Related Realms.” It was asked by I. B. Keene on the College Mathematics Journal, vol 36 (2005) page 100 and answered on page 159.

2005? I suspect the problem is much older than that.

Kalman’s book is a gem. In the first few pages alone, I learned about the Horner form of a polynomial and the fascinating Lill’s method for geometric visualization of the real roots of a polynomial.

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