Rational root theorem

The rational root theorem states that for any polynomial equation

a_n xⁿ + a_n-1 x^{n -1} + ... + a₁ x + a₀ = 0

with integer coefficients (and a_n nonzero), every rational solution x (also called "root") is of the form p/q, where p is a factor of the constant term a₀ and q is a factor of the leading coefficient a_n.

These root candidates can be tested using the Horner scheme. If a root is found, the Horner scheme will also yield a polynomial of degree n - 1 which is then to be investigated. After a polynomial is brought down to a quadratic equation, the quadratic formula may be used.

If the equation lacks a constant term a₀, then 0 is one of the rational roots of the equation.

The theorem is a special case (for a single linear factor) of the Gauss lemma on factorization of polynomials.