Rational root theorem

In Algebra, the rational root theorem states a constraint on solutions (also called roots ) to the polynomial equation

: an xn + a n -1 x n -1 + ... + a 1 x + a 0 = 0

with integer coefficients. Let a n be nonzero. Then each rational number solution x can be written in the form x=p / q for p and q satisfying two properties:

p is an integer factor of the constant term a 0, and

q is an integer factor of the leading coefficient a n .

For example, every rational solution of the equation

:3 x 3 − 5 x 2 + 5 x − 2 = 0 must be among the numbers

:1/3, 2/3, -1/3, -2/3, 1, −1, 2, −2.

These root candidates can be tested using the Horner scheme. If a root r 1 is found, the Horner scheme will also yield a polynomial of degree n - 1 whose roots, together with r 1 , are exactly the roots of the original polynomial.

It may also be the case that none of the candidates is a solution; in this case the equation has no rational solution. The fundamental theorem of algebra states that any polynomial with integral (or real, or even complex) coefficients must have at least one root in the set of complex numbers. Any polynomial of odd degree (degree being n in the example above) with real coefficients must have a root in the set of real numbers.

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

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