Rational root theorem

For a polynomial with integer coefficients, every rational root written in lowest terms has a numerator dividing the constant term and a denominator dividing the leading coefficient.

Polynomial: An expression formed from constants and variables using addition, subtraction, and multiplication with nonnegative integer exponents. The theorem applies to polynomials whose coefficients are integers.
Rational root test: A procedure that lists possible rational roots using divisors of a polynomial’s constant and leading coefficients. It applies the theorem by checking a finite set of candidates.
Irrational root: A real or complex number that cannot be expressed as a ratio of integers. The theorem places no restrictions on roots that are not rational.
Divisibility: An integer a divides an integer b when b equals a times an integer. The theorem states that the root’s numerator and denominator divide specific coefficients.
Factor theorem: A polynomial has root r exactly when it is divisible by x − r. Clearing a rational root’s denominator turns its factor into an integer linear factor.
Synthetic division: A compact method for dividing a polynomial by a linear factor x − r. After a candidate root works, synthetic division reduces the polynomial’s degree.
Fundamental theorem of algebra: Every nonconstant polynomial with complex coefficients has a complex root, counting multiplicity across its full factorization. It guarantees complex roots exist, while the rational root theorem only constrains rational ones.
Leading coefficient: The coefficient of the highest-power term in a polynomial. It restricts the denominator of any rational root in lowest terms.
Euclid's lemma: If a prime divides a product of integers, it divides at least one factor. Prime-factor reasoning establishes the theorem’s numerator and denominator divisibility conditions.
Polynomial factorization: The decomposition of a polynomial into a product of simpler polynomials. Finding rational roots supplies linear factors for a factorization.

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