I am trying to prove whether a specific form of polynomials form a subspace of $P_3$. I know that any set $W$ forms a subspace of a vector space $V$ if the set $W$ is closed under addition and scalar multiplication (is this right? please clarify).
Based on this, are polynomials of form $a_0 + a_1x + a_2x^2 + a_3x^3$ where $a_0,a_1,a_2,a_3$ are rationals, a subspace of $P_3$ ? I am not sure how any polynomials of this form actually violates the closure properties.
Thanks in advance