Is a polynomial of degree 2 a subspace of the Vector space of all real-valued functions defined on R

Question

I encountered this question: Is this polynomial of degree 2 a subspace of F(-infinity, +infinity)?

My thinking was that is. I looked at the solution for it and it gave a counter example:

U = 2 - x

V = 2 + x

u + v = 4, which is a polynomial of degree 0 and is thus not a subspace.

But I recently saw that P2 is subspace of P3, then shouldn't P0 be a subspace of P2? Making polynomials of degree 2 a subspace of F(-infinity, +infinity)?

Answer 1

It is incorrect to say that the set of all polynomials of degree "n" are a subspace. What is correct is that the set of all polynomial of degree "n or less" is a subspace.