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Rational root theorem

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

an xn + an−1 xn −1 + ... + a1 x + a0 = 0

with integer coefficients. Let an be nonzero. Then each rational 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 a0, and
  • q is an integer factor of the leading coefficient an.

Thus, a list of possible rational roots of the equation can be derived using the formulae x = ± p/q.

For example, every rational solution of the equation

3x3 − 5x2 + 5x − 2 = 0

must be among the numbers

1/3, −1/3, 2/3, −2/3, 1, −1, 2, −2.

These root candidates can be tested using the Horner scheme. If a root r1 is found, the Horner scheme will also yield a polynomial of degree n − 1 whose roots, together with r1, 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 a0, 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 on the factorization of polynomials.

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