Let $V\subset \mathbb{Q}[x]$ be the set of all polynomials of degree at most $n$, with $n \ge 1.$
Prove that $V$ is a vector space over $\mathbb{Q}[x]$ and that V is not a subring of $\mathbb{Q}[x]$.
Proving that $V$ is a vector space is easy.
However, I came across some trouble when trying to show that $V$ is not a subring of $\mathbb{Q}[x]$. Now, from what I know, $V$ is a subring of $\mathbb{Q}[x]$ if and only if:
$0 \in V$
if $a,b \in V$ then $a+b\in V$ and $a\cdot b \in V$
if $a \in V$ then $-a \in V$
However, these all hold in $V$. I am not really sure how we are supposed to prove that $V$ is not a subring of $\mathbb{Q}[x]$.