# Rational Root Theorem The rational root theorem postulates that [[Rational Numbers|rational]] roots of a [[Polynomials|polynomial]] with integer coefficients must be in the form $\frac{p}{q}$ where $p$ is an integer factor of the constant term and $q$ is an integer factor of the leading coefficient. This theorem only applies to those polynomials with nonzero constant terms. Additionally, all of the polynomial’s coefficients must be integers. ## Proof Any polynomial of order $n$ can be expressed as: $(q_1x-p_1)(q_2x-p_2)\ldots(q_nx-p_n)=0$ Any and all roots of this polynomial are then in the form $\frac{p_n}{q_n}$. If we were to multiply all of the terms, the factor of the first term would be all of the $q$ terms, while the factor of the constant term would be all $p$ terms.