MathDB

Let

p(x) = a_n x^n + a_{n - 1} x^{n - 1} + \dots + a_1 x + a_0

and

q(x) = b_m x^m + a_{m - 1} x^{m - 1} + \dots + b_1 x + b_0

be two polynomials with integer coefficients. Suppose that all the coefficients of the product

p(x) \cdot q(x)

are even but not all of them are divisible by 4. Show that one of

p(x)

and

q(x)

has all even coefficients and the other has at least one odd coefficient.

Problem Statement