What is the degree of the zero polynomial ?Pet P4 be the set of all polynomials of degree less than four.

Is the set S of polynomials of even degree less than four a subspace of P4?

I think that it is; however my text says it isn't.

Here's why I think that it is.

letvbe an element of the Set described. Thenvis an even degree polynomial of deg less than four. If we take a scaler œ and multiplyvby it, we get another polynomial of the same deg before. So, œvis in S. Satisfying closure one.

Letv,xbe in S, then considerv+x v+xis a polynomial with the highest even degree between them, and this will be less than 4, and it will be even. So,v+x is in S.

What did I do wrong, if anything?