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Homesynthetic division

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# synthetic division

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

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

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

$\phantom{-3}$ | $2$ | $10$ | $12$ | |

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.

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