How to show that the following is satisfied for all vector space axiom?

Question

Let $V=\{a_2x^2+a_1x+a_0|a_1, a_2, a_3\in \mathbb{R}, a_2\ne 0\}$ with operation defined by $$(a_2x^2+a_1x+a_0)+(b_2x^2+b_1x+b_0)=(a_2+b_2)x^2+(a_1+b_1)x+(a_0+b_0)$$

$$c(a_2x^2+a_1x+a_0)=ca_2x^2+ca_1x+ca_0$$

Here I am wondering if the identity is $(0x^2+0x+0)$ and the additive inverse is $-(a_2x^2+a_1x+a_0)$.

For other axioms, I don't see any that is not satisfied, could someone point out?

Answer 1

If $V$ is defined the way you say with the constraint on $a_2$, how can $(0x^2+0x+0)$ be an element of $V$?

So it is not a vector space.

If you eliminate the constraint $a_2 \ne 0$ then you get a vector space: it is the space of polinomials of degree less than or equal to 2.