Editing Synthetic division (section)

{{short description|Algorithm for Euclidean division of polynomials}}
{{For|the album|Synthetic Division{{!}}''Synthetic Division''}}
[[File:Synthdiv.gif|thumb|Animation showing the use of synthetic division to find the quotient of <math>2x^4 + 3x^2 + 5x + 1</math> by <math>x - 2</math>. Note that there is no term in <math>x^3</math>, so the fourth column from the right contains a zero.]]
In [[algebra]], '''synthetic division''' is a method for manually performing [[Euclidean division of polynomials]], with less writing and fewer calculations than [[Polynomial long division|long division]].

It is mostly taught for division by [[linear]] [[Monic polynomial|monic polynomials]] (known as [[Ruffini's rule]]), but the method can be generalized to division by any [[polynomial]].

The advantages of synthetic division are that it allows one to calculate without writing variables, it uses few calculations, and it takes significantly less space on paper than long division. Also, the subtractions in long division are converted to additions by switching the signs at the very beginning, helping to prevent sign errors.