Question

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

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

(b)role="math" localid="1649757360240" $\mathbf{10}\mathbf{-}\mathbf{15}\mathbf{x}\mathbf{+}\mathbf{25}{\mathbf{x}}^{\mathbf{2}}\mathbf{-}\mathbf{7}{\mathbf{x}}^{\mathbf{4}}$

(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

(a) It is proved is $5x^{11}-6x^4+12x^3+36x-6$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 $a_n$,role="math" localid="1649757774247" $p^2$,and $a_0$, then$f\left(x\right)$ willbe irreducible in $\mathrm{\mathbb{Q}}\left[x\right]$.

Use Eisenstein’s criterion to show that the polynomial is irreducible in ℚ[x]

(c)

There is a$2\cancel{\mathrm{I}}5,\left.2\right|-6,\left.2\right|12,\left.2\right|36,\left.2\right|-6,$ and $2^2\cancel{\mathrm{I}}-6$. As a result,$5x^{11}-6x^4+12x^3+36x-6$ is irreducible.

Hence, it is that$5x^{11}-6x^4+12x^3+36x-6$ is irreducible in $\mathrm{\mathbb{Q}}\left[x\right]$.