rational root theorem

Consider the polynomial

p(x)\;=\;a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0}

where all the coefficients $a_{i}$ are integers.

If $p(x)$ has a rational zero $u/v$ where $\gcd(u,\,v)=1$ , then $u\mid a_{0}$ and $v\mid a_{n}$ . Thus, for finding all rational zeros of $p(x)$ , it suffices to perform a finite number of tests.

The theorem is related to the result about monic polynomials whose coefficients belong to a unique factorization domain. Such theorem then states that any root in the fraction field is also in the base domain.

Title	rational root theorem
Canonical name	RationalRootTheorem
Date of creation	2013-03-22 11:46:18
Last modified on	2013-03-22 11:46:18
Owner	drini (3)
Last modified by	drini (3)
Numerical id	13
Author	drini (3)
Entry type	Theorem
Classification	msc 12D10
Classification	msc 12D05
Classification	msc 26A99
Classification	msc 26A24
Classification	msc 26A09
Classification	msc 26A06
Classification	msc 26-01
Classification	msc 11-00
Related topic	FactorTheorem