Question

Use Eisenstein’s Criterion to show that each polynomial is irreducible in $\mathrm{\mathbb{Q}}\left[\mathbf{x}\right]$:

1. ${\mathbf{x}}^5-4\mathbf{x}\mathbf{+}\mathbf{22}$
   
2. $10-15\mathbf{x}+25{\mathbf{x}}^2-7{\mathbf{x}}^4$
   
3. $5{\mathbf{x}}^{\mathbf{11}}-6{\mathbf{x}}^4+12{\mathbf{x}}^3+36\mathbf{x}-6$
   

Short answer

1. It is proved $x^5-4x+22$is irreducible in $\mathrm{\mathbb{Q}}\left[x\right]$.

Step-by-step solution

Eisenstein’s criterion

Consider $f\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0$as a nonconstant polynomial with integer coefficients. When thereis a prime $p$, in which $p$divides each of $a_0,a_1,\cdots ,a_{n-1}$, $p$does not divide $a_n$,$p^2$,and $a_0$, then $f\left(x\right)$ will beirreducible in$\mathrm{\mathbb{Q}}\left[x\right]$ .

Show that the polynomial is irreducible inℚx 

a)

There is a $21,\left.2\right|-4,\left.2\right|22,$androle="math" localid="1648642521423" $2^2/22$ . As a result,$x^5-4x+22$ is irreducible.

Hence, it is proved$x^5-4x+22$ is irreducible in $\mathrm{\mathbb{Q}}\left[x\right]$.