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If $a_0+a_1 x+ \ldots + a_ n x^ n$ is irreducible over $\mathbb Q$ then $a_n+ a_{n-1} x +\ldots+a_0x^{n}$ is irreducible over $\mathbb Q$.

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  • $\begingroup$ In such cases uses Gauss's lemma. Choose a prime to do the job. $\endgroup$ – Haha Oct 8 '14 at 9:02
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Assume $f(x)=g(x)h(x)$ with $\deg g=k$, $\deg h=m$. Then the other polynomial is $x^nf(1/x)$ and we have $x^nf(1/x)=x^kg(1/x)\cdot x^mh(1/x)$.

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