Is this polynomial irreducible? $f(x)=x^3-117x+53$

Question

I am trying to find if $f(x)=x^3-117x+53$ is irreducible or reducible.

I tried Eisenstein's criterion but this fails as $53$ does not divide $117$

After some thinking, if f(x) has any roots then they will be negative, but I have had no luck finding any.

Would really appreciate some help, thanks

Answer 1

A cubic (or quadratic) with rational coefficients is irreducible over $\mathbb{Q}$ if and only if it has no rational roots.

By the Rational Roots Theorem, the only candidates for rational roots of our cubic are $\pm 1$ and $\pm 53$. None of them works.

We conclude that our cubic is irreducible over the rationals, and as a consequence irreducible over the integers.

Answer 2

Consider the equation modulo $5$.

$x^3-2x+3\pmod{5}$

When $x\equiv 0\pmod{5}$, the above equates to $3\pmod{5}$

When $x\equiv 1\pmod{5}$, the above equates to $2\pmod{5}$

When $x\equiv 2\pmod{5}$, the above equates to $4\pmod{5}$

When $x\equiv 3\pmod{5}$, the above equates to $4\pmod{5}$

When $x\equiv 4\pmod{5}$, the above equates to $4\pmod{5}$

Since the integers can be partitioned into the classes modulo $5$ and we see that no class could possibly be a root, we know there are no integer roots and it is irreducible over $\mathbb{Z}$.

Since a monic polynomial whose coefficients are all integers will be irreducible over $\mathbb{Z}$ if and only if it is also irreducible over $\mathbb{Q}$ by Gauss's lemma, we see that it is irreducible.