RationalNullstellenSatz

RationalNullstellenSatz, commonly known in English as the Rational Root Theorem, is a standard result in algebra about rational zeros of polynomials with integer coefficients. It provides a finite list of possible rational roots and explains how their numerators and denominators relate to the coefficients of the polynomial.

Statement: Let f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 be a polynomial with integer coefficients. If r = p/q ∈ Q is a root of f in lowest terms (gcd(p,q) = 1), then p divides the constant term a_0 and q divides the leading coefficient a_n. Equivalently, any rational root must be of the form p/q where p is a divisor of a_0 and q is a divisor of a_n.

Consequences: The theorem restricts the search for rational roots to a finite set, namely all fractions p/q with p | a_0 and q | a_n. This is especially useful for factoring polynomials with integer coefficients and for quick root tests in algebra homework and computer algebra systems. It does not say anything about irrational or complex roots.

Example: Consider f(x) = x^3 - 6x^2 + 11x - 6. Here a_n = 1 and a_0 = -6. Possible rational roots are ±1, ±2, ±3, ±6. Indeed, the roots are 1, 2, and 3.

Remarks: The Rational Root Theorem is often presented together with Gauss’s lemma on primitive polynomials and is a foundational tool in polynomial factorization algorithms. It is sometimes referred to as the Rational Zeros Test in English-language texts.