Is $f(x)=x^3-9x^2+6x-12$ irreducible over $\mathbb{R}$? I want to show that $f(x)=x^3-9x^2+6x-12$ is reducible over $\mathbb{R}$. Since $\mathbb{R}$ is a field I know that if $f$ has a root in $\mathbb{R}$ then $f$ has a linear factor and hence it is reducible. My way of checking this seems like "cheating" because we don't talk about the notion of continuity in ring theory. One can easily show that $f(0)<0$ and $f(3)>0$, and since polynomials are continuous, the intermediate value theorem guarantees that $f$ has a root in $\mathbb{R}$. How could I show this in another way? Thanks!

Irreducible polynomials over the real numbers have degree $1$ or $2$, see here. So any real cubic polynomial is reducible.
