Rational Root Theorem

The Rational Root Theorem


        If p(x) = aₙxⁿ + … + a₀ has integer coefficients,

        then every rational root = ± (factor of a₀) / (factor of aₙ)

The theorem gives you a finite list of candidates to check. If p(x) has a rational root r = p/q (in lowest terms), then p must divide the constant term a₀ and q must divide the leading coefficient aₙ.

Not every candidate IS a root — they are just the only fractions that could be rational roots. You still need to test each one by evaluating p(candidate) and checking whether the result is 0.

Common ways to test a candidate k: substitute directly, or use synthetic division (remainder = 0 means k is a root).

The theorem only guarantees covering all rational roots. A polynomial can also have irrational roots (like √3) or complex roots that will never appear in the ±p/q list.

Finding All Zeros — Strategy

Once you find a rational root r, you can reduce the problem:

Step 1 — List candidates using the Rational Root Theorem (this calculator does this automatically).
Step 2 — Test candidates by synthetic division. The first root r you confirm gives you a factor (x − r).
Step 3 — Reduce degree: divide p(x) by (x − r) to get a polynomial of degree n−1.
Step 4 — Repeat: apply the theorem again to the reduced polynomial, or use the quadratic formula if degree 2.

A degree-n polynomial has exactly n roots (counting multiplicity) over the complex numbers. The Rational Root Theorem helps you find the rational ones first.

Polynomial zeros still feel overwhelming?