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:

$$	$2$	$10$	$12$

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