Decide whether a given cubic polynomial is irreducible over the rationals

Decide whether the following two polynomials are irreducible over \mathbb{Q}: p(x) = x^3 - x + 2 and q(x) = x^3 - 12x^2 + 44x - 52.


Note that if a cubic polynomial in \mathbb{Z}[x] is reducible over \mathbb{Q}, then it must have a linear factor- that is, it has a root in \mathbb{Q}. By the rational root theorem, there are finitely many candidates for this root. If none of these is a root, then the polynomial is irreducible in \mathbb{Q}[x].

By the rational root theorem, if p(x) has a root in \mathbb{Q} it is in the set \{ \pm 2/\pm 1, \pm 1/\pm 1 \} = \{ 2, -2, 1, -1\}. Since p(1) = p(-1) = 2, p(2) = 8, and p(-2) = -4, p(x) is irreducible in \mathbb{Q}[x].

Again by the rational root theorem, if q(x) has a root in \mathbb{Q} then it is in the set \{ \pm 1, \pm 2, \pm 4, \pm 13, \pm 26, \pm 52 \}. It is easy but rather tedious to see that q(1) = -19, q(-1) = -109, q(2) = -4, q(-2) = -196, q(4) = -4, q(-4) = -484, q(13) = 689, q(-13) = -4849, q(26) = 10556, q(-26) = -26884, q(52) = 110396, and q(-52) = -175396. Thus q(x) is irreducible in \mathbb{Q}[x].

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