Let $$f(X) = 78X^3 + 174X^2 − 116 ∈ Z[X]$$
My question is to decide if $f(X)$ is irreducible in $Z[X], Q[X] and R[X]$
I have tried finding a prime number 29, and to fulfil the Eisenstein's Irreducibility Criterion
1) $29$ is a common factor of $-116$ and $174$ in $Z$;
2) $29^2 = 841$ is not a factor of $-116$ in $Z$
3) $29$ is not a factor of $78$.
So that its enough to show $f(X)$ is irreducible in $Q[X]$.
And for $R[X]$ there should be no root as a integers so its should be irreducible but are there any way i can prove there really not integers root? And i also getting stuck in how to check with $Z[X]$, are there anything needed from the above calculation or is it something new?
Thanks a lot!