I'm having a lot of trouble with this topic. I have no idea on what to do for these questions:

1) Show that the set $\displaystyle S=\{p\in\mathbb{P}_2(0)=1\}$ is NOT a subspace of $\displaystyle \mathbb{P}_2$

The only thing I can think of doing for this question is making $\displaystyle p(x)=ax^2+bx+1$

2) Show that the set:

$\displaystyle S=\{p\in\mathbb{P}_3''(x)=0 \ \ \forall x \ \ \in\mathbb{R}\}$

is a subspace of $\displaystyle \mathbb{P}_3$

Would I do the following:

$\displaystyle p(x)=ax^3+bx^2+cx+d$

$\displaystyle \Longrightarrow p'(x)=3ax^2+2bx+c$

$\displaystyle \Longrightarrow p''(x)=6ax+2b$

$\displaystyle p''(x)=0 \ \mbox{for} \ x=-\dfrac{b}{3a}$