show that $p(x)=x^3-x^2-4x+5$ is irreducible in $\mathbb{Q}[x]$

Question

show that $p(x)=x^3-x^2-4x+5$ is irreducible in $\mathbb{Q}[x]$

How do we decide if a polynomial $p (x)$ in $\mathbb{Q}[x]$ is irreducible?

Answer 1

For a non-constant polynomial $f(x)$ of degree $n\le 3$ over a field we know that $f(x)$ is irreducible if and only if it has no root. This can be decided by the rational root test. The divisors of $5$ are $\pm 1$ and $\pm 5$, and so it is easy to see that $f(x)$ is irreducible.

Answer 2

There isn't one single all-encompassing way to show that a polynomial is irreducible. There are a lot of theorems that apply in specific cases, though, and those we try to use whenever possible to see whether we get any results.

In this case, we see that the polynomial is of degree three, which means that if it is reducible over $\Bbb Q$, then it must have a root in $\Bbb Q$. However, the rational root theorem says that such a root can only be $\pm 1$ or $\pm 5$ (the leading coefficient is $1$, and all coefficients are integers, so any roots must be divisors of the constant term). We can easily check that none of those are roots, so we are done.