Show that the following polynomial is irreducible over $\mathbb Q$

Question

Show that the following polynomial is irreducible over $\mathbb Q$:

$x^7 + 28x^2 + 32x + 6$

I know how to use the Eisenstein's Criterion method and testing with roots.
The problem is that I can't use the methods here

For Eisenstein's Criterion I need a leading polynomial of degree higher than $1$

For the root method I need a polynomial of degree $2$ or $3$.

How can I solve the problem and what is the name of the method I should use?

Answer 1

Apply Eisenstein's Criterion with $p = 2$ to the polynomial $$\color{red}{1} x^7 + \color{blue}{28}x^2 + \color{blue}{32}x + \color{green}6$$

We need to check that

• $2 \not \mid \color{red}1$
• $2 \mid \color{blue}{28}$
• $2 \mid \color{blue}{32}$
• $2 \mid \color{green}{6}$ and $2^2 \not\mid \color{green}{6}$