Vector subspace clarification We need to check if the following sets are vector subspaces: 
$$ S_1=\{(0,0),(1,2),(2,1)  \};V=(\Bbb Z_3)^2 \\
 S_2=\{P\mid\exists x \in \Bbb R :P_{(x)}=0  \};V=\Bbb R_n [x]$$
For $S_1$ I just add: $u+v=(1,2)+(2,1)=(3,3)=(1,1) \notin S_1$ so this is not a vector subspace.
For $S_2$, P is a polynomial. I don't understand this notation: $V=\Bbb R_n [x]$ and as far as I understand, the set means that there are some $x$ that gives the zero solution to the polynomial but I have no idea how to apply it.
Any help would be appreciated. 

For $(1)$, note that we are working with $($ the integers under addition modulo $3)^2$. So $$u+v=(1,2)+(2,1)=(3,3)=(0,0) \in S_1$$
For the second, $\Bbb R_n[x]$ is the space of all polynomials of degree less than or equal to $n$. In effect, $S_2$ consists of all polynomials with at least one real root.
Hint for $(2)$. We have $$P_1 = (x + 1)^2 = x^2 + 2x + 1 \in S_2,\quad P_2 = 1-x \in S_2$$
But $$P_1 + P_2 = x^2 + x + 2 \notin S_2, \;\text{since} \;\not\exists x \in \mathbb R \;\text{such that }\; x^2 + 2x + 1 = 0$$
