What is the rational root theroem?

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

with integer coefficients.

If a0 and an 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 an.

The rational root theorem is a special case (for a single linear factor) of Gauss's lemmaon the factorization of polynomials. The integral root theorem is a special case of the rational root theorem if the leading coefficient an = 1.