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Homerational root theorem

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# 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

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