## You are here

Homerational root theorem

## Primary tabs

# 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.

Keywords:

polynomial

Related:

FactorTheorem

Type of Math Object:

Theorem

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

12D10*no label found*12D05

*no label found*26A99

*no label found*26A24

*no label found*26A09

*no label found*26A06

*no label found*26-01

*no label found*11-00

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

Oct 21

new question: Prime numbers out of sequence by Rubens373

Oct 7

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag

new question: Prime numbers out of sequence by Rubens373

Oct 7

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag