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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 $x+3=0$
Carry down the first coefficient:
Multiply $2$ by $-3$
Add up $10$ and $-6$
| $-3$ |
|
$2$ |
$10$ |
$12$ |
| |
|
$2$ |
$4$ |
|
Multiply $4$ by $-3$
| $-3$ |
|
$2$ |
$10$ |
$12$ |
| |
|
$2$ |
$4$ |
$0$ |
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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