Is the set of polynomials with rational roots a vector space?
The polynomial $p_i(x)=(x-{{a_i}/{b_i}})(q_i(x))$ is from that given set.
If $p_1(x) + p_2(x) = (x-{{a_1}/{b_1}})(q_1(x)) + (x-{{a_2}/{b_2}})(q_2(x))$ is the polynomial from that set, it is ok. Where $b_i$ can't be 0.
If $rp_1(x) = r(x-{{a_1}/{b_1}})(q_1(x))$ is the polynomial from that set, it is ok. Where $r$ is a scalar.
We are restricted that if those polynomials form a subspace, all of them must have rational roots.
Any suggestions how to find a contradiction? Thank you and sorry for my English.