Can the set L, of all even polynomials be a subspace of F[X]?

Question

So I have the question Let F be a field and let L be the set of all polynomials f(x) element of F[X] satisfying the condition that deg(f) is even. Is L a subspace of F[X]?

I would say that L is not a subspace of F[X].

Taking the condition f(-a)=-f(a)

if we took f(x)=x^2 -2, which has an even degree, then

f(-a)=(-a)^2 -2 = a^2 -2 which does not equal -(x^2 -2)

Is my approach correct?

Answer 1

Even degree polynomial is NOT the same as even function. For example: $x^2+x$ is an even degree polynomial but not an even function. So if your set $L$ has even degree polynomials then it is NOT a subspace because $f(x)=x^2+x$ and $g(x)=-x^2$ are both in $L$ but $f+g \not\in L$.