Rational roots theorem

Rational root theorem

In algebra, the rational root theorem (or 'rational root test' to find the zeros) states a constraint on solutions (or roots) to the polynomial equation

a_nx^n+a_{n-1}x^{n-1}+cdots+a_0 = 0,!

with integer coefficients.

Let a0 and an be nonzero. Then each rational solution x can be written in the form x = p/q for p and q satisfying two properties:

Thus, a list of possible rational roots of the equation can be derived using the formula x = pm frac{p}{q}.

For example, every rational solution of the equation

3x^3 - 5x^2 + 5x - 2 = 0,!
must be among the numbers symbolically indicated by

± tfrac{1,2}{1,3},,

which gives the list of possible answers:

1, -1, 2, -2, frac{1}{3}, -frac{1}{3}, frac{2}{3}, -frac{2}{3},.

These root candidates can be tested, for example using the Horner scheme. In this particular case there is exactly one rational root.

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 integer (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.

The integral root theorem is a special case of the rational root theorem if the leading coefficient an=1.

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