62nd Ukrainian National Mathematical Olympiad

Question

We call a polynomial *mixed* if it has both positive and negative coefficients (we do not consider zero to be either positive or negative). Is it true that the product of two mixed polynomials is always a mixed polynomial?

Accepted Answer

Here is a counterexample: f(x) = x^5 + x^4 - x^3 - x^2 + x + 1 = (x^4 - x^2 + 1)(x + 1) g(x) = x^4 - x^3 + 2x^2 - x + 1 = (x^2 - x + 1)(x^2 + 1). aligned f(x)g(x) &= &= (x^4 - x^2 + 1)(x + 1)(x^2 - x + 1)(x^2 + 1) = &= (x^6 + 1)(x^3 + 1) = x^9 + x^6 + x^3 + 1. aligned $$ Both of these polynomials are mixed, but their product is not mixed, since all coefficients are positive.