Chapter 8.4, Problem 21E

Use Theorem $8.34$ to show that each of the following polynomials is irreducible over the field $\mathbb{Q}$ of rational numbers.

$f\left(x\right)=27x^3-16x^2+3x-25$

$f\left(x\right)=8x^3-2x^2-5x+10$

$f\left(x\right)=12x^3-2x^2+15x-2$

$f\left(x\right)=30x^3+11x^2-2x+8$

$f\left(x\right)=3x^4+9x^3-7x^2+15x+25$

$f\left(x\right)=9x^5-x^4+6x^3+5x^2-x+21$

Theorem $8.34$

Irreducibility of $f\left(x\right)$ in $\mathbb{Q}\left[x\right]$

Suppose $f\left(x\right)=a_0+a_{1}x+\cdot \cdot \cdot +a_n{x}^n$ is a polynomial of positive degree with integral coefficients and $p$ is a prime integer that does not divide $a_n$. Let

$f_p\left(x\right)=\left[a_0\right]+\left[a_1\right]x+\cdot \cdot \cdot +\left[a_n\right]x^n,$

Where $\left[a_i\right]\in {\mathbb{Z}}_p$ for $i=0,1,2,\cdots ,n.$

If $f_p\left(x\right)$ is irreducible in ${\mathbb{Z}}_p\left[x\right],$ then $f\left(x\right)$ is irreducible in $\mathbb{Q}\left[x\right]$.