Rational Root Theorem

Higher-degree polynomials do not factor by inspection, but you can narrow the search to a finite list.

Any rational root p/q must have p dividing the constant term and q dividing the leading coefficient.

For 2x³ − 3x² − 8x + 12: constants dividing 12 are ±1, 2, 3, 4, 6, 12; leading coefficient factors are ±1, 2. Candidates are those over 1 and over 2.

Test candidates by substitution or synthetic division. When one gives zero, divide it out and continue with a smaller polynomial.

Two caveats: the theorem only finds *rational* roots, and a candidate list is not a promise that any of them work.

If P(x) = a_n xⁿ + … + a₀ has integer coefficients and a rational root p/q in lowest terms, then p | a₀ and q | a_n.

The list is a superset of the actual rational roots and can be long, so it is usually pruned before testing: a rough graph or a sign check narrows the region, and Descartes' Rule of Signs bounds the number of positive and negative real roots.

Once a root is found, synthetic division depresses the degree, and the process repeats. This is the standard route to fully factoring a cubic or quartic by hand.

The theorem says nothing about irrational or complex roots, which is why x³ − 2 has no rational root despite ∛2 being real. When no candidate works, the polynomial is irreducible over ℚ and other methods — numerical approximation or the quadratic formula on a depressed factor — are required.