Editing Polynomial long division (section)

{{Short description|Algorithm for division of polynomials}}
{{For|a shorthand version of this method|synthetic division}}
In [[algebra]], '''polynomial long division''' is an [[algorithm]] for dividing a [[polynomial]] by another polynomial of the same or lower [[Degree of a polynomial|degree]], a generalized version of the arithmetic technique called [[long division]]. It can be done easily by hand, because it separates an otherwise complex division problem into smaller ones. Sometimes using a shorthand version called [[synthetic division]] is faster, with less writing and fewer calculations. Another abbreviated method is polynomial [[short division]] (Blomqvist's method).

Polynomial long division is an algorithm that implements the [[Euclidean division of polynomials]], which starting from two polynomials ''A'' (the ''dividend'') and ''B'' (the ''divisor'') produces, if ''B'' is not zero, a ''[[quotient]]'' ''Q'' and a ''remainder'' ''R'' such that
:''A'' = ''BQ'' + ''R'',
and either ''R'' = 0 or the degree of ''R'' is lower than the degree of ''B''. These conditions uniquely define  ''Q'' and ''R'', which means that ''Q'' and ''R'' do not depend on the method used to compute them.

The result ''R'' = 0 occurs [[if and only if]] the polynomial ''A'' has ''B'' as a [[polynomial factorization|factor]]. Thus long division is a means for testing whether one polynomial has another as a factor, and, if it does, for factoring it out. For example, if a [[root of a polynomial|root]] ''r'' of ''A'' is known, it can be factored out by dividing ''A'' by (''x''&nbsp;–&nbsp;''r'').