Rational Root Theorem

In algebra, the rational root theorem (or rational root test) states a constraint on rational solutions (or roots) of the polynomial equation

with integer coefficients.

If a₀ and a_n are nonzero, then each rational solution x, when written as a fraction x = p/q in lowest terms (i.e., the greatest common divisor of p and q is 1), satisfies

• p is an integer factor of the constant term a₀, 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 formula .

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 a special case of the rational root theorem if the leading coefficient a_n = 1.