TITLE: How to show this polynomial is irreducible
QUESTION [5 upvotes]: Is the polynomial $2x^{10}+25x^3+10x^2-30$ irreducible over $\mathbb Q$?

REPLY [3 votes]: 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 $\mathbb{Q}$.

REPLY [3 votes]: 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 $p=5$