Question

Question: Prove that$30x^n-91$(where$n\in \mathrm{\mathbb{Z}},n>1$) has no roots in$\mathrm{\mathbb{Q}}$.

Short answer

It is proved that $30x^n-91$has no root in$\mathrm{\mathbb{Q}}$ .

Step-by-step solution

Step-by-Step Solution Step 1: Eisenstein’s Criterion

Prove the result 

Consider the prime factors of 30 and 91 as:

$30=2·3·5$

$91=7·13$

Then, we have:

$7\cancel{|}30$,$7|91$AND $7^2\cancel{|}91$

By Eisenstein’s Criterion, we can say that $30x^n-91$is irreducible in $\mathrm{\mathbb{Q}}\left[x\right]$for$p=7$.