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# Rational root theorem

### From Wikipedia, the free encyclopedia

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

*a*+_{n}x^{n}*a*_{n−1}*x*^{n −1}+ ... +*a*_{1}*x*+*a*_{0}= 0

with integer
coefficients. Let *a*_{n} 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*a*_{0}, and*q*is an integer factor of the leading coefficient*a*_{n}.

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

- 3
*x*^{3}− 5*x*^{2}+ 5*x*− 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
*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 on the factorization of polynomials.