= Solution
A polynomial in $\mathbb Z[x]$ is a <primitive polynomial> when the greatest common divisor of its coefficients is one. The <Gauss lemma for polynomials> states that a product of primitive integer polynomials is primitive, and that a primitive positive-degree integer polynomial is reducible over $\mathbb Q$ exactly when it factors into positive-degree integer polynomials. More generally, polynomial content is multiplicative up to sign.
The <Eisenstein criterion> states that a primitive polynomial $a_nx^n+\cdots+a_0\in\mathbb Z[x]$ is irreducible over $\mathbb Q$ if there is a prime $p$ such that $p\nmid a_n$, $p\mid a_j$ for every $j<n$, and $p^2\nmid a_0$. These criteria permit integer coefficient tests for rational irreducibility; the following parts apply them with their exact divisibility hypotheses.
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