Are the polynomials of form $a_0 + a_1x + a_2x^2 +a_3x^3$, with $a_i$ rational, a subspace of $P_3$?

Question

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

Answer 1

Hint: Are all scalar multiples of $1$ ($= 1+0x+0x^2+0x^3$) in the set $$ \{ a_0+a_1x+a_2x^2+a_3x^3 \mid a_0,a_1,a_2,a_3 \in \mathbb{Q}\} $$ ?

Answer 2

The product of an irrational and a rational is irrational. .. this appears to present a problem, and your subset would appear not to be closed under scalar multiplication. ..