Question

Show that each polynomial $f\left(x\right)$is irreducible in $\mathbb{Q}\left[x\right]$by finding a prime $p$such that $f\left(x\right)$is irreducible in ${\mathbb{Z}}_p\left[x\right]$

(a) $7x^3+6x^2+4x+6$

Short answer

It is proved that$7x^3+6x^2+4x+6$ is irreducible in$\mathbb{Q}\left[x\right]$

Step-by-step solution

Rewrite the equation

Rewrite the given equation in ${\mathbb{Z}}_5$as follows:

$7x^3+6x^2+4x+6=2x^3+x^2+4x+1$

Observe the equation and prove

From the obtained equation and direct computation, we can say that equation

$2x^3+x^2+4x+1$is a cubic equation such that it has no root in ${\mathbb{Z}}_5$This implies it is irreducible.

Hence, it is proved$7x^3+6x^2+4x+6$is irreducible in$\mathbb{Q}\left[x\right]$