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:

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

Carry down the first coefficient:

Multiply $2$ by $-3$:

Add up $10$ and $-6$:

Multiply $4$ by $-3$:

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.