Calculating DFT of a specific polynomial
You haven't explained what you mean by the DFT of a polynomial, so I'm assuming you mean the polynomial evaluated at all $n$th roots of unity, that is,
$$ v_i = P(\omega^i), $$
where $\omega = e^{2\pi i/n}$ is a primitive $n$th root of unity. Let now $Q(x) = P(x^2)$, and let $\varpi = e^{\pi i/n}$ be a primitive $2n$th root of unity, $\varpi^2 = \omega$. Let us examine its DFT $w_1,\ldots,w_{2n}$:
$$
w_k = Q(\varpi^k) = P(\varpi^{2k}) = P(\omega^k) = v_{k \mod{n}}.
$$
We conclude that the DFT of $Q$ is
$$ v_1,\ldots,v_n,v_1,\ldots,v_n. $$