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.

Title synthetic division SyntheticDivision 2013-03-22 14:07:27 2013-03-22 14:07:27 mathwizard (128) mathwizard (128) 7 mathwizard (128) Definition msc 12D05