Relationship between the rational roots of a polynomial and its extreme coefficients
In algebra, the rational root theorem (or rational root test, rational zero theorem, rational zero test or p/q theorem) states a constraint on rational solutions of a polynomial equation
with integer coefficients and . Solutions of the equation are also called roots or zeros of the polynomial on the left side.
The theorem states that each rational solution x = p⁄q, written in lowest terms so that p and q are relatively prime, satisfies:
p is an integer factor of the constant term a0, and
q is an integer factor of the leading coefficient an.
The rational root theorem is a special case (for a single linear factor) of Gauss's lemma on the factorization of polynomials. The integral root theorem is the special case of the rational root theorem when the leading coefficient is an = 1.
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