How to show this polynomial is irreducible

Question

Is the polynomial $2x^{10}+25x^3+10x^2-30$ irreducible over $\mathbb Q$?

Answer 1

This polynomial is primitive so you can use Eisenstein to verify if it is irreducible. Take the prime $5$ and notice that $5$ doesn't divides $2$, $5^2$ doesn't divides $30$, but $5$ divides $25$ and $10$, therefore, this polynomial is irredducible over $\mathbb{Z}$ and over $\mathbb{Q}$.

Answer 2

Hint: $p(x)\in \mathbb{Z}[x]$ is irreducible $\mathbb{Z}[x]$ implies that it is irreducible in $ \mathbb{Q}[x]$. Now verify that $2x^{10}+25x^3+10x^2-30\,$ is irreducible in $\mathbb{Z}[x]$. Do this by using Eisenstein's criterion with $p=5$