Editing Polynomial (section)

=== Divisibility ===
{{Main|Polynomial greatest common divisor|Factorization of polynomials}}
If {{math|''R''}} is an [[integral domain]] and {{math|''f''}} and {{math|''g''}} are polynomials in {{math|''R''[''x'']}}, it is said that {{math|''f''}} ''divides'' {{math|''g''}} or {{math|''f''}} is a divisor of {{math|''g''}} if there exists a polynomial {{math|''q''}} in {{math|''R''[''x'']}} such that {{math|''f'' ''q'' {{=}} ''g''}}. If <math>a\in R,</math> then {{mvar|a}} is a root of {{mvar|f}} if and only <math>x-a</math> divides {{mvar|f}}. In this case, the quotient can be computed using the [[polynomial long division]].<ref>{{Cite book |last=Irving |first=Ronald S. |title=Integers, Polynomials, and Rings: A Course in Algebra |publisher=Springer |year=2004 |isbn=978-0-387-20172-6 |page=129 |url=https://books.google.com/books?id=B4k6ltaxm5YC&pg=PA129}}</ref><ref>{{cite book |last=Jackson |first=Terrence H. |title=From Polynomials to Sums of Squares |publisher=CRC Press |year=1995 |isbn=978-0-7503-0329-3 |page=143 |url=https://books.google.com/books?id=LCEOri2-doMC&pg=PA143}}</ref>

If {{math|''F''}} is a [[field (mathematics)|field]] and {{math|''f''}} and {{math|''g''}} are polynomials in {{math|''F''[''x'']}} with {{math|''g'' ≠ 0}}, then there exist unique polynomials {{math|''q''}} and {{math|''r''}} in {{math|''F''[''x'']}} with
<math display="block"> f = q \, g + r </math>
and such that the degree of {{math|''r''}} is smaller than the degree of {{math|''g''}} (using the convention that the polynomial 0 has a negative degree). The polynomials {{math|''q''}} and {{math|''r''}} are uniquely determined by {{math|''f''}} and {{math|''g''}}. This is called ''[[Euclidean division of polynomials|Euclidean division]], division with remainder'' or ''polynomial long division'' and shows that the ring {{math|''F''[''x'']}} is a [[Euclidean domain]].

Analogously, ''prime polynomials'' (more correctly, ''[[irreducible polynomial]]s'') can be defined as ''non-zero polynomials which cannot be factorized into the product of two non-constant polynomials''. In the case of coefficients in a ring, ''"non-constant"'' must be replaced by ''"non-constant or non-[[unit (ring theory)|unit]]"'' (both definitions agree in the case of coefficients in a field). Any polynomial may be decomposed into the product of an invertible constant by a product of irreducible polynomials. If the coefficients belong to a field or a [[unique factorization domain]] this decomposition is unique up to the order of the factors and the multiplication of any non-unit factor by a unit (and division of the unit factor by the same unit). When the coefficients belong to integers, rational numbers or a finite field, there are algorithms to test irreducibility and to compute the factorization into irreducible polynomials (see ''[[Factorization of polynomials]]''). These algorithms are not practicable for hand-written computation, but are available in any [[computer algebra system]]. [[Eisenstein's criterion]] can also be used in some cases to determine irreducibility.