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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 1=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 1=an = 1.

The theorem is used to find all rational roots of a polynomial, if any. It gives a finite number of possible fractions which can be checked to see if they are roots. If a rational root 1=x = r is found, a linear polynomial (x – r) can be factored out of the polynomial using polynomial long division, resulting in a polynomial of lower degree whose roots are also roots of the original polynomial.

Should there be a nontrivial factor dividing all the coefficients of the polynomial, then one can divide by the greatest common divisor of the coefficients so as to obtain a primitive polynomial in the sense of Gauss's lemma; this does not alter the set of rational roots and only strengthens the divisibility conditions. That lemma says that if the polynomial factors in 'Q'[X], then it also factors in 'Z'[X] as a product of primitive polynomials. Now any rational root p/q corresponds to a factor of degree 1 in 'Q'[X] of the polynomial, and its primitive representative is then qx − p, assuming that p and q are coprime. But any multiple in 'Z'[X] of qx − p has leading term divisible by q and constant term divisible by p, which proves the statement. This argument shows that more generally, any irreducible factor of P can be supposed to have integer coefficients, and leading and constant coefficients dividing the corresponding coefficients of P.