Question

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

role="math" localid="1649237593303" $f\left(x\right)$ is irreducible in ${\mathbb{Z}}_p\left[x\right]$

(a) $9x^4+4x^3-3x+7$

Short answer

It is proved$9x^4+4x^3-3x+7$ is irreducible in$\mathbb{Q}\left[x\right]$

Step-by-step solution

Rewrite the equation

Rewrite the given equation in ${\mathbb{Z}}_2$as follows:

$9x^4+4x^3-3x+7=x^4+x+1$

Observe the equation and prove

From the obtained equation and direct computation, we can say that equation $x^4+x+1$has no root in ${\mathbb{Z}}_2$, that is, $x^2+x+1$is not a factor of $x^4+x+1$

This implies it is irreducible.

Hence, it is proved that $9x^4+4x^3-3x+7$is irreducible in $\mathbb{Q}\left[x\right]$