Uniformly at random polynomial

Question

We have a polynomial of degree $d$, and multiply it by a polynomial whose coefficients are chosen uniformly at random and its degree is equal to or less than $d$. My question is whether the result is a polynomial with coefficients distributed uniformly?

Answer 1

The result is a polynomial of degree $2d$. Assuming the base field is $\mathbb{Z}_q$, there are $q^{2d+1}$ such polynomials, but you only get $q^{d+1}$ polynomials, so for $d > 0$ the distribution is not going to be uniform.