Question

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

a)${\mathbf{x}}^{\mathbf{5}}\mathbf{-}\mathbf{4}\mathbf{x}\mathbf{+}\mathbf{22}$

b) $\mathbf{10}\mathbf{-}\mathbf{15}\mathbf{x}\mathbf{+}\mathbf{25}{\mathbf{x}}^{\mathbf{2}}\mathbf{-}\mathbf{7}{\mathbf{x}}^{\mathbf{4}}$

c)$\mathbf{5}{\mathbf{x}}^{\mathbf{11}}\mathbf{-}\mathbf{6}{\mathbf{x}}^{\mathbf{4}}\mathbf{+}\mathbf{12}{\mathbf{x}}^{\mathbf{3}}\mathbf{+}\mathbf{36}\mathbf{x}\mathbf{-}\mathbf{6}$

Short answer

b) It is proved$-7x^4+25x^2-15x+10$ 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 there is a prime $p$, in which $p$divides each of $a_0,a_1,\cdots ,a_{n-1}$, $p$does not divide $a_n$,$p^2$, androle="math" localid="1649756439048" $a_0$, then$f\left(x\right)$ willbeirreducible in $\mathrm{\mathbb{Q}}\left[x\right]$.

Show that the polynomial is irreducible in

b)

There is a $5\cancel{\mathrm{I}}-7,\left.5\right|25,\left.5\right|-15,\left.5\right|10$and $5^2\cancel{\mathrm{I}}10$. As a result,$-7x^4+25x^2-15x+10$ is irreducible.

Hence, it is proved$-7x^4+25x^2-15x+10$ is irreducible in $\mathrm{\mathbb{Q}}\left[x\right]$.