Editing Polynomial (section)

== Operations ==
=== Addition and subtraction ===
Polynomials can be added using the [[associative law]] of addition (grouping all their terms together into a single sum), possibly followed by reordering (using the [[commutative law]]) and combining of like terms.<ref name="Edwards-1995-p47">{{cite book |last=Edwards |first=Harold M. |title=Linear Algebra |publisher=Springer |year=1995 |isbn=978-0-8176-3731-6 |page=47 |url=https://books.google.com/books?id=ylFR4h5BIDEC&pg=PA47}}</ref><ref>{{cite book |last=Salomon |first=David |title=Coding for Data and Computer Communications |publisher=Springer |year=2006 |isbn=978-0-387-23804-3 |page=459 |url=https://books.google.com/books?id=Zr9bjEpXKnIC&pg=PA459}}</ref> For example, if
<math display="block"> P = 3x^2 - 2x + 5xy - 2 </math> and <math display="block"> Q = -3x^2 + 3x + 4y^2 + 8</math>
then the sum
<math display="block">P + Q = 3x^2 - 2x + 5xy - 2 - 3x^2 + 3x + 4y^2 + 8 </math>
can be reordered and regrouped as
<math display="block">P + Q = (3x^2 - 3x^2) +  (- 2x + 3x) + 5xy + 4y^2 + (8 - 2) </math>
and then simplified to
<math display="block">P + Q = x + 5xy + 4y^2 + 6.</math>
When polynomials are added together, the result is another polynomial.<ref name=":0">{{Cite book|url=https://books.google.com/books?id=PagNAQAAIAAJ&q=the+addition+of+polynomials+is+an+operation+that+takes+any+two+polynomials+and+produce+always+another+polynomial,|title=Introduction to Algebra|date=1965|publisher=Yale University Press|pages=621|language=en|quote=Any two such polynomials can be added, subtracted, or multiplied. Furthermore, the result in each case is another polynomial}}</ref>

Subtraction of polynomials is similar.

=== Multiplication ===
{{Further|Polynomial expansion}}
Polynomials can also be multiplied.  To expand the [[product (mathematics)|product]] of two polynomials into a sum of terms, the distributive law is repeatedly applied, which results in each term of one polynomial being multiplied by every term of the other.<ref name="Edwards-1995-p47"/> For example, if
<math display="block">\begin{align}
 \color{Red} P &\color{Red}{= 2x + 3y + 5} \\
 \color{Blue} Q &\color{Blue}{= 2x + 5y + xy + 1}
\end{align}</math>
then
<math display="block">\begin{array}{rccrcrcrcr}
{\color{Red}{P}} {\color{Blue}{Q}} & {{=}}&&({\color{Red}{2x}}\cdot{\color{Blue}{2x}})
&+&({\color{Red}{2x}}\cdot{\color{Blue}{5y}})&+&({\color{Red}{2x}}\cdot {\color{Blue}{xy}})&+&({\color{Red}{2x}}\cdot{\color{Blue}{1}})
\\&&+&({\color{Red}{3y}}\cdot{\color{Blue}{2x}})&+&({\color{Red}{3y}}\cdot{\color{Blue}{5y}})&+&({\color{Red}{3y}}\cdot {\color{Blue}{xy}})&+&
({\color{Red}{3y}}\cdot{\color{Blue}{1}})
\\&&+&({\color{Red}{5}}\cdot{\color{Blue}{2x}})&+&({\color{Red}{5}}\cdot{\color{Blue}{5y}})&+&
({\color{Red}{5}}\cdot {\color{Blue}{xy}})&+&({\color{Red}{5}}\cdot{\color{Blue}{1}})
\end{array}</math>
Carrying out the multiplication in each term produces
<math display="block">\begin{array}{rccrcrcrcr}
PQ & = && 4x^2 &+& 10xy &+& 2x^2y &+& 2x \\
&&+& 6xy &+& 15y^2 &+& 3xy^2 &+& 3y \\
&&+& 10x &+& 25y &+& 5xy &+& 5.
\end{array}</math>
Combining similar terms yields
<math display="block">\begin{array}{rcccrcrcrcr}
PQ & = && 4x^2 &+&( 10xy + 6xy + 5xy ) &+& 2x^2y &+& ( 2x + 10x ) \\
&& + & 15y^2 &+& 3xy^2 &+&( 3y + 25y )&+&5
\end{array}</math>
which can be simplified to
<math display="block">PQ = 4x^2 + 21xy + 2x^2y + 12x + 15y^2 + 3xy^2 + 28y + 5.</math>
As in the example, the product of polynomials is always a polynomial.<ref name=":0" /><ref name=Barbeau-2003-pp1-2/>

=== Composition ===
Given a polynomial <math>f</math> of a single variable and another polynomial {{mvar|g}} of any number of variables, the [[function composition|composition]] <math>f \circ g</math> is obtained by substituting each copy of the variable of the first polynomial by the second polynomial.<ref name=Barbeau-2003-pp1-2/>  For example, if <math>f(x) = x^2 + 2x</math> and <math>g(x) = 3x + 2</math> then
<math display = "block"> (f\circ g)(x) = f(g(x)) = (3x + 2)^2 + 2(3x + 2).</math>
A composition may be expanded to a sum of terms using the rules for multiplication and division of polynomials.  The composition of two polynomials is another polynomial.<ref>{{Cite book|last=Kriete|first=Hartje|url=https://books.google.com/books?id=HwqjxJOLLOoC&q=The+composition+of+two+polynomials+is+always+another+polynomial.&pg=PA159|title=Progress in Holomorphic Dynamics|date=1998-05-20|publisher=CRC Press|isbn=978-0-582-32388-9|pages=159|language=en|quote=This class of endomorphisms is closed under composition,}}</ref>
<!--something about the composition where ''f'' has many variables? <ref name=Barbeau-2003-pp1-2/>-->

=== 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>

=== Factoring ===
All polynomials with coefficients in a [[unique factorization domain]] (for example, the integers or a [[field (mathematics)|field]]) also have a factored form in which the polynomial is written as a product of [[irreducible polynomial]]s and a constant. This factored form is unique up to the order of the factors and their multiplication by an invertible constant. In the case of the field of [[complex number]]s, the irreducible factors are linear. Over the [[real number]]s, they have the degree either one or two. Over the integers and the [[rational number]]s the irreducible factors may have any degree.<ref name=Barbeau-2003-pp80-82>{{harvnb|Barbeau|2003|pp=[https://books.google.com/books?id=CynRMm5qTmQC&pg=PA80 80]–2}}</ref> For example, the factored form of
<math display="block"> 5x^3-5</math>
is
<math display="block">5(x - 1)\left(x^2 + x + 1\right)</math>
over the integers and the reals, and
<math display="block"> 5(x - 1)\left(x + \frac{1 + i\sqrt{3}}{2}\right)\left(x + \frac{1 - i\sqrt{3}}{2}\right)</math>
over the complex numbers.

The computation of the factored form, called ''factorization'' is, in general, too difficult to be done by hand-written computation. However, efficient [[factorization of polynomials|polynomial factorization]] [[algorithm]]s are available in most [[computer algebra system]]s.

=== Calculus ===
{{Main|Calculus with polynomials}}
Calculating [[derivative]]s and integrals of polynomials is particularly simple, compared to other kinds of functions.
The [[derivative]] of the polynomial <math display="block">P = a_n x^n + a_{n - 1} x^{n - 1} + \dots + a_2 x^2 + a_1 x + a_0 = \sum_{i=0}^n a_i x^i</math> with respect to {{mvar|x}} is the polynomial 
<math display="block"> n a_n x^{n - 1} + (n - 1)a_{n - 1} x^{n - 2} + \dots + 2 a_2 x + a_1 = \sum_{i=1}^n i a_i x^{i-1}.</math>
Similarly, the general [[antiderivative]] (or indefinite integral) of <math>P</math> is
<math display="block"> \frac{a_n x^{n + 1}}{n + 1} + \frac{a_{n - 1} x^{n}}{n} + \dots + \frac{a_2 x^3}{3} + \frac{a_1 x^2}{2} + a_0 x + c = c + \sum_{i = 0}^n \frac{a_i x^{i + 1}}{i + 1}</math>
where {{mvar|c}} is an arbitrary constant.  For example, antiderivatives of {{math|''x''<sup>2</sup> + 1}} have the form {{math|{{sfrac|3}}''x''<sup>3</sup> + ''x'' + ''c''}}.

For polynomials whose coefficients come from more abstract settings (for example, if the coefficients are integers [[modular arithmetic|modulo]] some [[prime number]] {{math|''p''}}, or elements of an arbitrary ring), the formula for the derivative can still be interpreted formally, with the coefficient {{math|''ka''<sub>''k''</sub>}} understood to mean the sum of {{mvar|k}} copies of {{math|''a''<sub>''k''</sub>}}. For example, over the integers modulo {{math|''p''}}, the derivative of the polynomial {{math|''x''<sup>''p''</sup> + ''x''}} is the polynomial {{math|1}}.<ref name=Barbeau-2003-pp64-65>{{harvnb|Barbeau|2003|pp=[https://books.google.com/books?id=CynRMm5qTmQC&pg=PA64 64]–5}}</ref>