How to check if a polynomial $p(x)=a_nx^n+...+a_0$ is a vector space? Let $V=\{p: \mathbf{R}\rightarrow \mathbf{R} \ |\  p(x)=a_nx^n+...+a_0 $ where $a_i\in \mathbf{R}$ and $a_n\ne 0\}$ (the set of polynomial functions of degree exactly $n$).
So to check this, I think we need to check the axioms, but I am not sure if $p$ represents a constant. Anyone helps?

The set of all polynomials of degree exactly $n$ can not be a vector space because you can subtract two polynomials of degree $n$ to get a polynomial of smaller degree. For example:
$$(x^3-x+1)-(x^3-2x-1)=x+2$$
$$(x^3-x^2+3)-(x^3-x+3)=-x^2+x$$
Thus, while the set of all polynomials of degree less than or equal to $n$ would be a vector space, the polynomials of degree exactly $n$ is not.
