# rational root theorem

Consider the polynomial^{}

$$p(x)={a}_{n}{x}^{n}+{a}_{n-1}{x}^{n-1}+\mathrm{\cdots}+{a}_{1}x+{a}_{0}$$ |

where all the coefficients ${a}_{i}$ are integers.

If $p(x)$ has a rational zero $u/v$ where $\mathrm{gcd}(u,v)=1$, 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 |