Question 2
Let
$$
p(x) = a_{n} x^{n} + a_{n-1} x^{n-1} + \cdots + a_{1} x + a_{0}
$$
and
$$
q(x) = b_{m} x^{m} + b_{m-1} x^{m-1} + \cdots + b_{1} x + b_{0}
$$
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.
$$
p(x) = a_{n} x^{n} + a_{n-1} x^{n-1} + \cdots + a_{1} x + a_{0}
$$
and
$$
q(x) = b_{m} x^{m} + b_{m-1} x^{m-1} + \cdots + b_{1} x + b_{0}
$$
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.