# synthetic division

*Synthetic division ^{}* is simply a shorthand for long division,
when one of the factors is linear.
So when an equation such as

$$y=2{x}^{2}+10x+12$$ |

is given and a factor of $x+3$ is known, we can do the following:

$$ | $2$ | $10$ | $12$ |
---|

First, we can use $-3$ since $-3$ is a solution if $x+3=0$.

$-3$ | $2$ | $10$ | $12$ |
---|

Carry down the first coefficient^{}:

$-3$ | $2$ | $10$ | $12$ |
---|---|---|---|

$2$ |

Multiply $2$ by $-3$:

$-3$ | $2$ | $10$ | $12$ |
---|---|---|---|

$-6$ | |||

$2$ |

Add up $10$ and $-6$:

$-3$ | $2$ | $10$ | $12$ |
---|---|---|---|

$-6$ | |||

$2$ | $4$ |

Multiply $4$ by $-3$:

$-3$ | $2$ | $10$ | $12$ |
---|---|---|---|

$-6$ | $-12$ | ||

$2$ | $4$ | $0$ |

This means that, by factoring the equation by $x+3$, we will get another factor as $2x+4$.

Synthetic is simply an variation of the long division — a version that’s manipulated and simplified to shorten the calculations.

Title | synthetic division |
---|---|

Canonical name | SyntheticDivision |

Date of creation | 2013-03-22 14:07:27 |

Last modified on | 2013-03-22 14:07:27 |

Owner | mathwizard (128) |

Last modified by | mathwizard (128) |

Numerical id | 7 |

Author | mathwizard (128) |

Entry type | Definition |

Classification | msc 12D05 |