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

Question

$\displaystyle W=\{p(x) \in V: p(x)=p(1-x)\} , x\in R$.}

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?

Answer 1

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.

Answer 2

It is sufficient to verify that, for $p,q \in W$ and $ \alpha \in \mathbb{R}$ we have $ \alpha p+q \in W$. And this is true because we have:

$$ (\alpha p+q)(1+x)=\alpha p(1+x)+q(1+x)= \alpha p(x)+q(x)=(\alpha p +q)(x) $$