Does $ W $ form a vectors subspace of the vector space of polynomial with real coefficients. $\displaystyle W=\{p(x) \in V: p(x)=p(1-x)\} , x\in R$.}

Does $ W $ form a  vectors subspace  of the vector space  of polynomial with real coefficients?

My attempts:  $p(x) $ = $p(1-x)$   as  $p(x) - p(1-x)  =  p(x+1-x)$

now   $c .p(x+1-x)  =  cp(x) - cp(1-x)$ so my answer is yes.......

Is that correct?
Let $p(x),q(x)\in W$ and $\alpha,\beta\in\mathbb{R}$ then 
$$r(x):=\alpha p(x)+\beta q(x)=\alpha p(1-x)+\beta q(1-x)=r(1-x)$$
hence $r(x)\in W$ so $W$ is a subspace.
Let $p(x),q(x)\in W$ and $\alpha,\beta\in\mathbb{R}$ then 
$$r(x):=\alpha p(x)+\beta q(x)=\alpha p(1-x)+\beta q(1-x)=r(1-x)$$
hence $r(x)\in W$ so $W$ is a subspace.