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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
is given and a factor of  is known, we can do the following:
First, we can use  since  is a solution if  .
Carry down the first coefficient:
Multiply  by  :
Add up  and  :
Multiply  by  :
This means that, by factoring the equation by  , we will get another factor as  .
Synthetic is simply an variation of the long division -- a version that's manipulated and simplified to shorten the calculations.
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