Editing Polynomial (section)

=== Division ===

The division of one polynomial by another is not typically a polynomial.  Instead, such ratios are a more general family of objects, called ''[[rational fraction]]s'', ''rational expressions'',  or ''[[rational function]]s'', depending on context.<ref>{{cite book|last1 = Marecek | first1 = Lynn | last2 = Mathis | first2 = Andrea Honeycutt | title = Intermediate Algebra 2e | date = 6 May 2020 | publisher = [[OpenStax]] <!-- | location = Houston, Texas -->|  url = https://openstax.org/details/books/intermediate-algebra-2e | at = §7.1}}</ref>  This is analogous to the fact that the ratio of two [[integer]]s is a [[rational number]], not necessarily an integer.<ref>{{Cite book|last1=Haylock|first1=Derek|url=https://books.google.com/books?id=hgAr3maZeQUC&q=division+integers+not+closed&pg=PA49|title=Understanding Mathematics for Young Children: A Guide for Foundation Stage and Lower Primary Teachers|last2=Cockburn|first2=Anne D.|date=2008-10-14|publisher=SAGE|isbn=978-1-4462-0497-9|pages=49|language=en|quote=We find that the set of integers is not closed under this operation of division.}}</ref><ref name = openstax>{{harvnb|Marecek|Mathis|2020|loc=§5.4]}}</ref>  For example, the fraction {{math|1/(''x''<sup>2</sup> + 1)}} is not a polynomial, and it cannot be written as a finite sum of powers of the variable {{mvar|x}}.

For polynomials in one variable, there is a notion of [[Euclidean division of polynomials]], generalizing the [[Euclidean division]] of integers.{{efn|This paragraph assumes that the polynomials have coefficients in a [[field (mathematics)|field]].}}  This notion of the division {{math|''a''(''x'')/''b''(''x'')}} results in two polynomials, a ''quotient'' {{math|''q''(''x'')}} and a ''remainder'' {{math|''r''(''x'')}}, such that {{math|''a'' {{=}} ''b'' ''q'' + ''r''}} and {{math|degree(''r'') < degree(''b'')}}.  The quotient and remainder may be computed by any of several algorithms, including [[polynomial long division]] and [[synthetic division]].<ref>{{cite book |first1=Peter H. |last1=Selby |first2=Steve |last2=Slavin |title=Practical Algebra: A Self-Teaching Guide |date=1991 |publisher=Wiley |isbn=978-0-471-53012-1 |edition=2nd}}</ref>

When the denominator {{math|''b''(''x'')}} is [[monic polynomial|monic]] and linear, that is, {{math|1=''b''(''x'') = ''x'' − ''c''}} for some constant {{mvar|c}}, then the [[polynomial remainder theorem]] asserts that the remainder of the division of {{math|''a''(''x'')}} by {{math|''b''(''x'')}} is the [[#evaluation|evaluation]] {{math|''a''(''c'')}}.<ref name = openstax/>  In this case, the quotient may be computed by [[Ruffini's rule]], a special case of synthetic division.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Ruffini's Rule|url=https://mathworld.wolfram.com/RuffinisRule.html|access-date=2020-07-25|website=mathworld.wolfram.com|language=en}}</ref>