# Irreducible Polynomial/Examples/8 x^3 - 6 x - 1 in Rationals

## Examples of Irreducible Polynomials

Consider the polynomial:

$\map P x = 8 x^3 - 6 x - 1$

over the ring of polynomials $\Q \sqbrk X$ over the rational numbers.

Then $\map P x$ is irreducible.

## Proof

Aiming for a contradiction, suppose $\map P x$ has proper factors.

Then one of these has to be of degree $1$.

$8 x^3 - 6 x - 1 = \paren {a x + b} \paren {c^2 + d x + e}$

for some $a, b, c, d, e \in \Z$.

Hence:

 $\ds a c$ $=$ $\ds 8$ $\ds \leadsto \ \$ $\ds a$ $\divides$ $\ds 8$ $\ds b e$ $=$ $\ds -1$ $\ds \leadsto \ \$ $\ds b$ $\divides$ $\ds 1$

The only possible degree $1$ factors with integer coefficients are:

$x \pm 1, 2 x \pm 1, 4 x \pm 1, 8 x \pm 1$

By trying each of these possibilities, it is determined that no integer value of $d$ gives the correct values.

Hence the result.

$\blacksquare$