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In algebra, the rational root theorem states that for any
polynomial equation
- an xn + an-1 xn
-1 + ... + a1 x + a0 = 0
with integer coefficients (and an nonzero), every
rational solution x (also called "root") is of the form
p/q, where p is an integer factor of the constant term
a0 and q is a integer factor of the leading coefficient an.
For example, every rational solution of the equation 3x3-5x2+5x-2 = 0 must be among the numbers
1/3, 2/3, -1/3, -2/3, 1, -1.
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. When a polynomial is brought
down to a quadratic equation, the quadratic formula may be used.
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 the Gauss
lemma on factorization of polynomials.
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