[Math] Set of all polynomials of degree 4 or 6 with usual addition and scalar multiplication is a vector space or not

Question

I understand that since its usual addition, any two polynomials of degree 4 or 6, when added will result in a polynomial of degree 4 or 6 again.
i.e. a+b belong to set V(say).

Again if i add any such polynomial with a polynomial with zero in its coefficient will result in the same polynomial. so this implies that a+0=a(existence of unique zero element in v)
Similarly it satisfies existence of additive inverse and scalar multiplication too.
So, am i wrong somewhere? Or is it not a vector space?

Answer 1

No. More generally, the set of polynomials of degree $d$ is not a subspace of the vector space of polynomials, for two reasons:

1. It does not have a zero element, since $0$ has no degree. In some contexts, on may accept the convention that $0$ has a degree, but this degree is either $-1$ or $-\infty$, not any $d\ge 0$.
2. The sum of two polynomials of degree $d$ is not necessarily of degree $d$: for instance, if $d=3$, $f(x)=x^3$, $g(x)=-x^3+x+1$, then $f(x)+g(x)=x+1$.

However, as is well known, the set of polynomials of degree at most $d$ is a vector space (of dimension $d+1$).