Rational Root Theorem

Intuition and core definition

The rational root theorem lists possible rational zeros of an integer-coefficient polynomial. If $p/q$ in lowest terms is a root of $a_nx^n+\cdots+a_0$, then $p$ divides the constant $a_0$ and $q$ divides the leading coefficient $a_n$. The theorem supplies candidates, not guaranteed roots.

Notation, language, and conditions

Candidates are $\pm p/q$, reduced to avoid duplicates. The theorem requires integer coefficients; rational coefficients can first be cleared. A root $r$ corresponds to factor $x-r$, and evaluation or synthetic division determines whether a candidate is genuine.

Why this idea matters

The rational root theorem narrows possible rational zeros to a finite factor list, turning an open-ended search into structured testing.

A dependable method

1. Write the polynomial in standard form with integer coefficients. If the constant term is zero, factor out the greatest available power of $x$ and record zero as a root first.
2. For the remaining polynomial with nonzero constant term, list positive and negative factors $p$ of that constant term.
3. List factors $q$ of the leading coefficient.
4. Form and reduce every distinct $\pm p/q$ candidate.
5. Test strategically by substitution or synthetic division and factor further when a zero remainder occurs.

Worked example

Common mistakes