# factor theorem

If $f(x)$ is a polynomial^{} over a ring with identity, then $x-a$ is a factor if and only if $a$ is a root (that is, $f(a)=0$).

This theorem is of great help for finding factorizations of higher degree polynomials. As example, let us think that we need to factor the polynomial $p(x)={x}^{3}+3{x}^{2}-33x-35$. With some help of the rational root theorem we can find that $x=-1$ is a root (that is, $p(-1)=0$), so we know $(x+1)$ must be a factor of the polynomial. We can write then

$$p(x)=(x+1)q(x)$$ |

where the polynomial $q(x)$ can be found using long or synthetic division^{} of $p(x)$ between $x-1$. In our case $q(x)={x}^{2}+2x-35$ which can be easily factored as $(x-5)(x+7)$. We conclude that

$$p(x)=(x+1)(x-5)(x+7).$$ |

Title | factor theorem |

Canonical name | FactorTheorem |

Date of creation | 2013-03-22 12:17:24 |

Last modified on | 2013-03-22 12:17:24 |

Owner | drini (3) |

Last modified by | drini (3) |

Numerical id | 10 |

Author | drini (3) |

Entry type | Theorem |

Classification | msc 12D10 |

Classification | msc 12D05 |

Synonym | root theorem |

Related topic | Polynomial |

Related topic | RationalRootTheorem |

Related topic | Root |

Related topic | APolynomialOfDegreeNOverAFieldHasAtMostNRoots |