It is not a vector space.
Work Step by Step
The set of all fifth-degree polynomials with the standard operationss is not a vector space because it is not closed under addition. For example, $f(x)=x^5$ and $g(x)=-x^5+x^2+1$. Now, $f(x)+g(x)=x^2+1$ which is a polynomial of degree $2$.