Polynomial Equations with no Additional Information

So far every technique in §3.7 needed some extra fact about the roots. This section covers the two main tools for when the equation is given completely bare, with nothing else known.

Trial and error, formalised. Sometimes a root can simply be guessed and checked by direct substitution. E.g. for $4x^3-8x^2-x+2=0$: writing $P(x)$ for the left side, $P(2)=32-32-2+2=0$, so $x-2$ is a factor — dividing it out reduces the cubic to an easy quadratic. Guessing blindly is impractical for larger equations, though — the **Rational Root Th …