# 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