Decide if f(X) is irreducible in the following rings

Question

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!

Answer 1

Every cubic polynomial with real coefficients has at least one real root and therefore it is reducible in $\mathbb{R}[x]$.

It is also reducible in $\mathbb{Z}[x]$, since it is equal to $2\times(39x^3+87x^2-58)$.

Answer 2

One (other) way to show that it's irreducible over $\mathbb{R}$ is to notice that $f(0) < 0$ and $f(1) >0$. Hence, there must be a root between in $(0,1)$