Irreducibility of Non-monic Quartic Polynomials in Q[x]

Question

Decide whether the following polynomials are reducible or not in the given ring.

$f(x)=5x^4+22x^3+35x^2+47x+15$ in Q$[x]$

$g(x)=2x^4-4x^3-12x^2+14x-2$ in Q$[x]$

I am not sure what method to use to determine irreducibility here. I know, for quadratic and cubic polynomials, the polynomial is irreducible if it has no roots in Q$[x]$. But of course these are quartic polynomials. I cannot use the Eisenstein Criterion because no prime number divides the coefficients of $f(x)$ and the only prime number, 2, that divides the coefficients of $g(x)$ also divides the leading coefficient, which is not acceptable when using the Criterion. I would also want to convert the coefficients of each to modulo 2 so that I could show they are irreducible in F$_2$[x], but this only works for monic functions. Any idea of what I can do?

Thanks.

Answer 1

Your $f$ is not irreducible because $-3$ is a root.

As for $g$, first you should divide it by $2$ because it is not primitive (in $\mathbb{Z}[x]$ it is not irreducible, but of course for $\mathbb{Q}[x]$ this is irrelevant); after dividing by $2$, if you reduce mod $2$ you get $x^4 + x + 1$ which is irreducible in $\mathbb{F}_2[x]$ (as is easily checked by trying all possible factors of degrees $1$ and $2$), so $g$ is irreducible.

Answer 2

The polynomial $g(x)$ can be written as $2h(x)$ with a monic polynomial $h(x)$ which is irreducible. Since it is monic you can try the different methods. $f(x)$ is obviously reducible since $(x+3)$ is a linear factor.