Rational Root Theorem

For a polynomial with integer coefficients $a_n z^n + \cdots + a_0$, any rational root $\tfrac{p}{q}$ (written in lowest terms) must have $p$ divide the constant term $a_0$ and $q$ divide the leading coefficient $a_n$.

That gives a short, finite list of candidates to test — instead of guessing blindly.

Example: for $2z^3 - 3z^2 - 11z + 6 = 0$, candidates are $\pm1, \pm2, \pm3, \pm6, \pm\tfrac12, \pm\tfrac32$. Testing $z=3$: $2(27) - 3(9) - 11(3) + 6 = 54-27-33+6 = 0$ — so $z=3$ is a root, and the rest factors out from there.