If you have a polynomial with integer coefficients (and a non-zero constant term) and there’s some prime number $p$ such that
- $p$ goes into every coefficient but not the first (or last) and
- $p^2$ doesn't go into the last (or first, respectively)
then your polynomial doesn’t factor into two polynomials with integer coefficients.
(But the converse isn’t true. That is, if the criterion fails, the polynomial might nonetheless not factor.)
This is true of polynomials of any degree, but is most useful for quadratics, for two reasons:
- Those are the polynomials that are most frequently factored over the integers in high school.
- It’s easy to see why the criterion holds true for quadratics.
Its use is as a quick check for non-factoring, so as not to waste time trying to factor. Thus, for example, a student can tell at a glance that $x^2+9x+24$ doesn't factor (because $3|9$ and $3^2 \nmid 24$), without checking which of the many factors of $24$ might sum to $9$. And can tell that $x^2-10x+60$ doesn't factor without checking factors of $60$.
However, this criterion doesn’t seem to be taught in high school at all, which makes me wonder whether doing so is pedagogically unsound. So my question is: Should one teach this in a high-school algebra class? For quadratics at least? Why or why not?
 It’s especially easy to see for $x^2+bx+c$. There, that $p$ goes into the $b$ means that the $r,s$ in $(x+r)(x+s)$ must both or neither be divisible by $p$; but then $c=rs$ must be divisible by $p^2$ or not by $p$ at all.