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

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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.


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

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

Thus from Factors of Polynomial with Integer Coefficients have Integer Coefficients we have:

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

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


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

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.