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[[File:Polynomialdeg3.svg|The [[graph of a function|graph]] of a polynomial function of degree 3|thumb|upright]]
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In [[mathematics]], a '''polynomial''' is an [[mathematical expression|expression]] consisting of [[variable (mathematics)|variables]], called [[indeterminate (variable)|indeterminates]], and [[coefficient]]s that involves only the operations of [[addition]], [[subtraction]], [[multiplication]], and non-negative [[integer]] [[exponent]]s. An example of a polynomial of a single indeterminate (or variable), {{math|''x''}}, is {{math|''x''<sup>2</sup> &minus; 4''x'' + 7}}, which is a [[Quadratic function|quadratic polynomial]].


Polynomials appear in a wide variety of areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary [[word problem (mathematics education)|word problems]] to complicated problems in the sciences; they are used to define polynomial functions, which appear in settings ranging from basic [[chemistry]] and [[physics]] to [[economics]] and [[social science]]; they are used in [[calculus]] and [[numerical analysis]] to approximate other functions. In advanced mathematics, polynomials are used to construct [[polynomial ring]]s and [[algebraic variety|algebraic varieties]], central concepts in [[algebra]] and [[algebraic geometry]].
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==Etymology==
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According to the [[Oxford English Dictionary]], ''polynomial'' succeeded the term ''[[binomial]]'', and was made simply by replacing the Latin root ''bi-'' with the Greek ''poly-'', which comes from the Greek word for ''many''. The word ''polynomial'' was first used in the 17th century.<ref>Etymology of "polynomial". ''Compact Oxford English Dictionary''</ref>


=={{anchor|Polynomial notation}} Notation and terminology==
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The ''x'' occurring in a polynomial is commonly called either a ''variable'' or an ''indeterminate''. When the polynomial is considered for itself, ''x'' is a fixed symbol which does not have any value (its value is "indeterminate"). It is thus more correct to call it an "indeterminate". However, when one consider the [[function (mathematics)|function]] defined by the polynomial, then ''x'' represents the argument of the function, and is therefore called a "variable". Many authors use these two words indifferently, but this may be sometimes confusing and is not done in this article.
 
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It is a common convention to use upper case letters for the indeterminates and the corresponding lower case letters for the variables (arguments) of the associated function.
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It may be confusing that a polynomial ''P'' in the indeterminate ''X'' may appear in the formulas either as ''P'' or as ''P''(''X'').
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Normally, the name of the polynomial is ''P'', not ''P''(''X''). However, if ''a'' denotes a number, a variable, another polynomial, or, more generally any expression, then ''P''(''a'') denotes, by convention, the result of substituting ''X'' by ''a'' in ''P''. For example, the polynomial ''P'' defines the function
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:<math>x\mapsto P(x)</math>


In particular, if ''a'' = ''X'', then the definition of ''P''(''a'') implies
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:<math>P=P(X).</math>
 
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This equality allows writing "let ''P''(''X'') be a polynomial" as a shorthand for "let ''P'' be a polynomial in the indeterminate ''X''". On the other hand, when it is not necessary to emphasize the name of the indeterminate, many formulas are much simpler and easier to read if the name(s) of the indeterminate(s) do not appear at each occurrence of the polynomial.
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A polynomial in a single indeterminate can be written in the form
 
:<math>a_n x^n + a_{n-1}x^{n-1} + \dotsb + a_2 x^2 + a_1 x + a_0,</math>
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where <math>a_0, \ldots, a_n</math> are numbers, or more generally elements of a [[ring (mathematics)|ring]], and <math>x</math> is a symbol which is called an [[indeterminate (variable)|indeterminate]] or, for historical reasons, a [[variable (mathematics)|variable]]. The symbol <math>x</math> does not represent any value, although the usual (commutative, distributive) laws valid for [[arithmetic]] operations also apply to it.
 
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This can be expressed more concisely by using [[Capital-sigma notation#Capital-sigma notation|summation notation]]:
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:<math>\sum_{i=0}^n a_i x^i</math>
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That is, a polynomial can either be zero or can be written as the sum of a finite number of non-zero [[term (mathematics)|terms]]. Each term consists of the product of a number—called the [[coefficient]] of the term<ref>The coefficient of a term may be any number from a specified set.  If that set is the set of real numbers, we speak of "polynomials over the reals".  Other common kinds of polynomials are polynomials with integer coefficients, polynomials with complex coefficients, and polynomials with coefficients that are integers [[modular arithmetic|modulo]] of some [[prime number]] {{math|''p''}}.</ref>—and a finite number of indeterminates, raised to integer powers. The exponent on an indeterminate in a term is called the [[Degree of a polynomial|degree]] of that indeterminate in that term; the degree of the term is the sum of the degrees of the indeterminates in that term, and the degree of a polynomial is the largest degree of any one term with nonzero coefficient.  Since {{math|''x'' {{=}} ''x''<sup>1</sup>}}, the degree of an indeterminate without a written exponent is one. A term and a polynomial with no indeterminates are called respectively a [[constant term]] and a constant polynomial;<ref>This terminology date from the time where the distinction was not clear between a polynomial and the function that it defines: a constant term and a constant polynomial define [[constant function]]s.</ref> the degree of a constant term and of a nonzero constant polynomial is 0. The degree of the zero polynomial (which has no term) is not defined.<ref name=Barbeau-2003-pp1-2>{{cite book|author=Barbeau, E.J.|title=Polynomials|publisher=Springer|year=2003|isbn=9780387406275|pages=1–2|url=http://books.google.com/books?id=CynRMm5qTmQC&pg=PA1}}</ref>
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: <math> -5x^2y\,</math>
  <li>[http://www.zskqsjzp.com/forum.php?mod=viewthread&tid=2314644&fromuid=285047 http://www.zskqsjzp.com/forum.php?mod=viewthread&tid=2314644&fromuid=285047]</li>
 
 
is a term. The coefficient is {{math|&minus;5}}, the indeterminates are {{math|''x''}} and {{math|''y''}}, the degree of {{math|''x''}} is two, while the degree of {{math|''y''}} is one. The degree of the entire term is the sum of the degrees of each indeterminate in it, so in this example the degree is {{math|2 + 1 {{=}} 3}}.
  <li>[http://www.deipei.com/forum.php?mod=viewthread&tid=82214&fromuid=4169 http://www.deipei.com/forum.php?mod=viewthread&tid=82214&fromuid=4169]</li>
 
 
Forming a sum of several terms produces a polynomial. For example, the following is a polynomial:
  <li>[http://www.fabzine.it/forum/longchamp-bags-give-our-freedoms-t88393.html http://www.fabzine.it/forum/longchamp-bags-give-our-freedoms-t88393.html]</li>
 
 
:<math>\underbrace{_\,3x^2}_{\begin{smallmatrix}\mathrm{term}\\\mathrm{1}\end{smallmatrix}} \underbrace{-_\,5x}_{\begin{smallmatrix}\mathrm{term}\\\mathrm{2}\end{smallmatrix}} \underbrace{+_\,4}_{\begin{smallmatrix}\mathrm{term}\\\mathrm{3}\end{smallmatrix}}. </math>
  <li>[http://rhymerockrecords.net/bulletin//read.php?1,650014 http://rhymerockrecords.net/bulletin//read.php?1,650014]</li>
 
 
It consists of three terms: the first is degree two, the second is degree one, and the third is degree zero.
  <li>[http://yegnaforum.com/viewtopic.php?f=12&t=17954 http://yegnaforum.com/viewtopic.php?f=12&t=17954]</li>
 
 
Polynomials of small degree have been given specific names. A polynomial of degree zero is a ''constant polynomial'' or simply a ''constant''. Polynomials of degree one, two or three are respectively ''linear polynomials,'' ''quadratic polynomials'' and ''cubic polynomials''. For higher degrees the specific names are not commonly used, although ''quartic polynomial'' (for degree four) and ''quintic polynomial'' (for degree five) are sometimes used. The names for the degrees may be applied to the polynomial or to its terms. For example, in {{math|''x''<sup>2</sup> + 2''x'' + 1}} the term {{math|2''x''}} is a linear term in a quadratic polynomial.
</ul>
 
{{anchor|zero polynomial}}The polynomial 0, which may be considered to have no terms at all, is called the ''zero polynomial''. Unlike other constant polynomials, its degree is not zero. Rather the degree of the zero polynomial is either left explicitly undefined, or defined as negative (either −1 or −∞).<ref>{{MathWorld|urlname=ZeroPolynomial|title=Zero Polynomial}}</ref> These conventions are useful when defining [[Euclidean division of polynomials]]. The zero polynomial is also unique in that it is the only polynomial having an infinite number of [[Root of a function|roots]]. In the case of polynomials in more than one indeterminate, a polynomial is called ''homogeneous'' of {{nowrap|degree {{math|''n''}}}} if ''all'' its terms have {{nowrap|degree {{math|''n''}}}}. For example, {{math|''x''<sup>3</sup>''y''<sup>2</sup> + 7''x''<sup>2</sup>''y''<sup>3</sup> - 3''x''<sup>5</sup>}} is homogeneous of degree 5. For more details, see [[homogeneous polynomial]].
 
The [[commutative law]] of addition can be used to rearrange terms into any preferred order. In polynomials with one indeterminate, the terms are usually ordered according to degree, either in "descending powers of {{math|''x''}}", with the term of largest degree first, or in "ascending powers of {{math|''x''}}". The polynomial in the example above is written in descending powers of {{math|''x''}}. The first term has coefficient {{math|3}}, indeterminate {{math|''x''}}, and exponent {{math|2}}. In the second term, the coefficient {{nowrap|is {{math|&minus;5}}}}. The third term is a constant.  Since the ''degree'' of a non-zero polynomial is the largest degree of any one term, this polynomial has degree two.<ref>{{cite book|author=Edwards, Harold M.|title=Linear Algebra|publisher=Springer|year=1995|isbn=9780817637316|page=78|url=http://books.google.com/books?id=ylFR4h5BIDEC&pg=PA78}}</ref>
 
Two terms with the same indeterminates raised to the same powers are called "similar terms" or "like terms", and they can be combined, using the [[distributive law]], into a single term  whose coefficient is the sum of the coefficients of the terms that were combined. It may happen that this makes the coefficient 0.<ref name="Edwards-1995-p47" /> Polynomials can be classified by the number of terms with nonzero coefficients, so that a one-term polynomial is called a [[monomial]],<ref>Some authors use "monomial" to mean "[[monic polynomial|monic]] monomial". See {{cite book |author=Anthony W. Knapp |title=Advanced Algebra: Along with a Companion Volume Basic Algebra |page=457 |year=2007 |publisher=Springer |isbn=0-8176-4522-5}}</ref> a two-term polynomial is called a [[binomial]], and so on.
{{anchor|univariate}}A polynomial in one indeterminate is called a ''[[univariate]] polynomial'', a polynomial in more than one indeterminate is called a ''multivariate polynomial''. These notions refer more to the kind of polynomials one is generally working with than to individual polynomials; for instance when working with univariate polynomials one does not exclude constant polynomials (which may result, for instance, from the subtraction of non-constant polynomials), although strictly speaking constant polynomials do not contain any indeterminates at all. It is possible to further classify multivariate polynomials as ''bivariate'', ''trivariate'', and so on, according to the maximum number of indeterminates allowed.  Again, so that the set of objects under consideration be closed under subtraction, a study of trivariate polynomials usually allows bivariate polynomials, and so on.  It is common, also, to say simply "polynomials in {{math|''x'', ''y''}}, and {{math|''z''}}", listing the indeterminates allowed.
 
The ''evaluation of a polynomial'' consists of substituting a numerical value to each indeterminate and carrying out the indicated multiplications and additions. For polynomials in one indeterminate, the evaluation is usually more efficient (lower number of arithmetic operations to perform) using the [[Horner scheme]]:
:<math>(((\dotsb((a_n x + a_{n-1})x + a_{n-2})x + \dotsb + a_3)x + a_2)x + a_1)x + a_0.</math>
 
==Arithmetic of polynomials==
Polynomials can be added using the [[associative law|associative]] law of addition (grouping all their terms together into a single sum), possibly followed by reordering, and combining of like terms.<ref name="Edwards-1995-p47">{{cite book|author=Edwards, Harold M.|title=Linear Algebra|publisher=Springer|year=1995|isbn=9780817637316|page=47|url=http://books.google.com/books?id=ylFR4h5BIDEC&pg=PA47}}</ref><ref>{{cite book|author=Salomon, David|title=Coding for Data and Computer Communications|publisher=Springer|year=2006|isbn=9780387238043|page=459|url=http://books.google.com/books?id=Zr9bjEpXKnIC&pg=PA459}}</ref> For example, if
:<math>\begin{align}
  P &=  3x^2 - 2x + 5xy - 2 \\
  Q &= -3x^2 + 3x + 4y^2 + 8
\end{align}</math>
 
then
:<math>P + Q = 3x^2 - 2x + 5xy - 2 - 3x^2 + 3x + 4y^2 + 8 </math>
 
which can be simplified to
:<math>P + Q = x + 5xy + 4y^2 + 6 </math>
 
To work out the 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>\begin{align}
  \color{BrickRed} P &\color{BrickRed}{= 2x + 3y + 5} \\
  \color{RoyalBlue} Q &\color{RoyalBlue}{= 2x + 5y + xy + 1}
\end{align}</math>
 
then
:<math>\begin{array}{rccrcrcrcr}
{\color{BrickRed}P}{\color{RoyalBlue}Q}&{{=}}&&({\color{BrickRed}2x}\cdot{\color{RoyalBlue}2x})
&+&({\color{BrickRed}2x}\cdot{\color{RoyalBlue}5y})&+&({\color{BrickRed}2x}\cdot {\color{RoyalBlue}xy})&+&({\color{BrickRed}2x}\cdot{\color{RoyalBlue}1})
\\&&+&({\color{BrickRed}3y}\cdot{\color{RoyalBlue}2x})&+&({\color{BrickRed}3y}\cdot{\color{RoyalBlue}5y})&+&({\color{BrickRed}3y}\cdot {\color{RoyalBlue}xy})&+&
({\color{BrickRed}3y}\cdot{\color{RoyalBlue}1})
\\&&+&({\color{BrickRed}5}\cdot{\color{RoyalBlue}2x})&+&({\color{BrickRed}5}\cdot{\color{RoyalBlue}5y})&+&
({\color{BrickRed}5}\cdot {\color{RoyalBlue}xy})&+&({\color{BrickRed}5}\cdot{\color{RoyalBlue}1})
\end{array}</math>
 
which can be simplified to
:<math>PQ = 4x^2 + 21xy + 2x^2y + 12x + 15y^2 + 3xy^2 + 28y + 5</math>
 
Polynomial evaluation can be used to compute the remainder of [[Euclidean division of polynomials|polynomial division]] by a polynomial of degree one, since the remainder of the division of {{math|''f''(''x'')}} by {{math|(''x'' &minus; ''a'')}} is {{math|''f''(''a'')}}; see the [[polynomial remainder theorem]]. This is more efficient than the usual algorithm of division when the quotient is not needed.
 
* A [[sum]] of polynomials is a polynomial.<ref name=Barbeau-2003-pp1-2 />
* A [[product (mathematics)|product]] of polynomials is a polynomial.<ref name=Barbeau-2003-pp1-2 />
* A [[function composition|composition]] of two polynomials is a polynomial, which is obtained by substituting a variable of the first polynomial by the second polynomial.<ref name=Barbeau-2003-pp1-2 />
* The [[derivative]] of the polynomial {{math|''a''<sub>n</sub>''x''<sup>n</sup> + ''a''<sub>n&minus;1</sub>''x''<sup>n&minus;1</sup> + ... +  ''a''<sub>2</sub>''x''<sup>2</sup> + ''a''<sub>1</sub>''x'' + ''a''<sub>0</sub>}} is the polynomial {{math|n''a''<sub>n</sub>''x''<sup>n&minus;1</sup> + (n&minus;1)''a''<sub>n&minus;1</sub>''x''<sup>n&minus;2</sup>  + ... +  2''a''<sub>2</sub>''x'' + ''a''<sub>1</sub>}}. If the set of the coefficients does not contain the integers (for example if the coefficients are integers [[modular arithmetic|modulo]] some [[prime number]] {{math|''p''}}), then {{math|k''a''<sub>k</sub>}} should be interpreted as the sum of {{math|''a''<sub>k</sub>}} with itself, {{math|k}} times. For example, over the integers modulo {{math|''p''}}, the derivative of the polynomial {{math|''x''<sup>''p''</sup> + 1}} is the polynomial {{math|0}}.<ref name=Barbeau-2003-pp64-65>{{cite book|author=Barbeau, E.J.|title=Polynomials|publisher=Springer|year=2003|isbn=9780387406275|pages=64–65|url=http://books.google.com/books?id=CynRMm5qTmQC&pg=PA64}}</ref>
* A primitive or [[antiderivative]] of the polynomial {{math|''a''<sub>n</sub>''x''<sup>n</sup>  + ''a''<sub>n&minus;1</sub>''x''<sup>n&minus;1</sup>  + ... +  ''a''<sub>2</sub>''x''<sup>2</sup>  + ''a''<sub>1</sub>''x'' + ''a''<sub>0</sub>}}  is the polynomial  {{math|''a''<sub>n</sub>''x''<sup>n+1</sup>/(n+1) + ''a''<sub>n&minus;1</sub>''x''<sup>n</sup>/n  + ... +  ''a''<sub>2</sub>''x''<sup>3</sup>/3  + ''a''<sub>1</sub>''x''<sup>2</sup>/2 + ''a''<sub>0</sub>''x'' + ''c''}}, where {{math|''c''}} is an arbitrary constant. For instance, the antiderivatives of {{math|''x''<sup>2</sup> + 1}} have the form {{math|{{sfrac|3}}''x''<sup>3</sup> + ''x'' + ''c''}}.
 
As for the integers, two kinds of divisions are considered for the polynomials. The ''[[Euclidean division of polynomials]]'' that generalizes the [[Euclidean division]] of the integers. It results in two polynomials, a ''quotient'' and a ''remainder'' that are characterized by the following property of the polynomials: given two polynomials ''a'' and ''b'' such that ''b'' ≠ 0, there exists a unique pair of polynomials, ''q'', the quotient, and ''r'', the remainder, such that {{math|''a'' {{=}} ''b'' ''q'' + ''r''}} and {{math|degree(''r'') < degree(''b'')}} (here the polynomial zero is supposed to have a negative degree). By hand as well as with a computer, this division can be computed by the [[polynomial long division]] algorithm.<ref>Peter H. Selby, Steve Slavin, ''Practical Algebra: A Self-Teaching Guide, 2nd Edition'', Wiley, ISBN 0-471-53012-3 ISBN 978-0471530121</ref>
 
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>{{cite book|author=Barbeau, E.J.|title=Polynomials|publisher=Springer|year=2003|isbn=9780387406275|pages=80–82|url=http://books.google.com/books?id=CynRMm5qTmQC&pg=PA80}}</ref> For example, the factored form of
:<math> 5x^3-5</math>
is
:<math>5(x - 1)\left(x^2 + x + 1\right)</math>
 
over the integers and the reals and
:<math> 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, there are efficient [[algorithms]] (see [[Polynomial factorization]]) that are available in most [[computer algebra system]]s.
 
A formal quotient of polynomials, that is, an [[algebraic fraction]] where the numerator and denominator are polynomials, is called a "[[Rational function|rational expression]]" or "rational fraction" and is not, in general, a polynomial.  Division of a polynomial by a number, however, does yield another polynomial.  For example, {{math|''x''<sup>3</sup>/12}} is considered a valid term in a polynomial (and a polynomial by itself) because it is equivalent to {{math|(1/12)''x''<sup>3</sup>}} and 1/12 is just a constant. When this expression is used as a term, its coefficient is therefore 1/12. For similar reasons, if complex coefficients are allowed, one may have a single term like {{math|(2 + 3''i'') ''x''<sup>3</sup>}}; even though it looks like it should be expanded to two terms, the complex number 2&nbsp;+&nbsp;3''i'' is one complex number, and is the coefficient of that term. The expression {{math|1 {{=}} 1/(x<sup>2</sup> + 1)}} is not a polynomial because it includes division by a non-constant polynomial. The expression {{math|(5 + ''y'')<sup>''x''</sup>}} is not a polynomial, because it contains an indeterminate used as exponent.
 
Since subtraction can be replaced by addition of the opposite quantity, and since positive integer exponents can be replaced by repeated multiplication, all polynomials can be constructed from constants and indeterminates using only addition and multiplication.
 
==Polynomial functions==<!-- "Polynomial function" redirects here -->
{{see also|Ring of polynomial functions}}
A ''polynomial function'' is a function that can be defined by [[expression (mathematics)|evaluating]] a polynomial. A function {{math|''f''}} of one [[Argument of a function|argument]] is called a polynomial function if it satisfies{{citation needed|date=August 2013}}
 
: <math> f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0 \, </math>
 
for all arguments {{math|''x''}}, where {{math|''n''}} is a non-negative integer and {{math|''a''<sub>0</sub>,  ''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a<sub>n</sub>''}} are constant coefficients.
 
For example, the function {{math|''f''}}, taking real numbers to real numbers, defined by
 
:<math> f(x) = x^3 - x\,</math>
 
is a polynomial function of one variable. Polynomial functions of multiple variables can also be defined, using polynomials in multiple indeterminates, as in
 
: <math>f(x,y)= 2x^3+4x^2y+xy^5+y^2-7.\,</math>
An example is also the function <math>f(x)=\cos(2\arccos(x))</math> which, although it doesn't look like a polynomial, is a polynomial function on <math>[-1,1]</math> since for every <math>x</math> from <math>[-1,1]</math> it is true that <math>f(x)=2x^2-1</math> (see [[Chebyshev polynomials]]).
 
Polynomial functions are a class of functions having many important properties. They are all [[Continuous function|continuous]], [[smooth function|smooth]], [[entire function|entire]], [[computable function|computable]], etc.{{citation needed|date=August 2013}}
 
===Graphs of polynomial functions===
<div class="floatright">
<gallery perrow="2" widths="200px" heights="200px">
File:Polynomialdeg2.svg|Polynomial of degree 2:<br><small>{{math|''f''(''x'') {{=}} ''x''<sup>2</sup> &minus; ''x'' &minus; 2}}<br>{{math|{{=}} (''x'' + 1)(''x'' &minus; 2)}}</small>
File:Polynomialdeg3.svg|Polynomial of degree 3:<br><small>{{math|''f''(''x'') {{=}} ''x''<sup>3</sup>/4 + 3''x''<sup>2</sup>/4 &minus; 3''x''/2 &minus; 2}}<br>{{math|{{=}}  1/4 (''x'' + 4)(''x'' + 1)(''x'' &minus; 2)}}</small>
File:Polynomialdeg4.svg|Polynomial of degree 4:<br><small>{{math|''f''(''x'') {{=}} 1/14 (''x'' + 4)(''x'' + 1)(''x'' &minus; 1)(''x'' &minus; 3) + 0.5}}</small>
File:Polynomialdeg5.svg|Polynomial of degree 5:<br><small>{{math|''f''(''x'') {{=}} 1/20 (''x'' + 4)(''x'' + 2)(''x'' + 1 )(''x'' &minus; 1)(''x'' &minus; 3)}}<br>{{math|+ 2}}</small>
File:Sextic Graph.svg|Polynomial of degree 6:<br><small>{{math|''f''(''x'') {{=}} 1/30 (''x''+3.5)(''x''+2)(''x'' + 1)(''x'' &minus; 1)(''x'' &minus; 3)}}<br>{{math|(''x'' &minus; 4) + 2}}</small>
File:Septic graph.svg|Polynomial of degree 7:<br><small>{{math|''f''(''x'') {{=}} (''x'' &minus; 3)(''x'' &minus; 2)(''x'' &minus; 1)(''x'')(''x'' + 1)(''x'' + 2)}}<br>{{math|(''x'' + 3)}}</small>
</gallery>
</div>
A polynomial function in one real variable can be represented by a [[graph of a function|graph]].
* The graph of the zero polynomial
::{{math|''f''(''x'') {{=}} 0}}
:is the {{math|''x''}}-axis.
 
* The graph of a degree 0 polynomial
::{{math|''f''(''x'') {{=}} ''a''<sub>0</sub>}}, where {{math|''a''<sub>0</sub> ≠ 0}},
:is a horizontal line with {{math|''y''-intercept ''a''<sub>0</sub>}}
 
* The graph of a degree 1 polynomial (or linear function)
::{{math|''f''(''x'') {{=}} ''a''<sub>0</sub> + ''a''<sub>1</sub>''x''}} , where {{math|''a''<sub>1</sub> ≠ 0}},
:is an oblique line with {{math|''y''-intercept ''a''<sub>0</sub>}} and [[slope]] {{math|''a''<sub>1</sub>}}.
 
* The graph of a degree 2 polynomial
::{{math|''f''(''x'') {{=}} ''a''<sub>0</sub> + ''a''<sub>1</sub>''x'' + ''a''<sub>2</sub>''x''<sup>2</sup>}}, where {{math|''a''<sub>2</sub> ≠ 0}}
:is a [[parabola]].
 
* The graph of a degree 3 polynomial
::{{math|''f''(''x'') {{=}} ''a''<sub>0</sub> + ''a''<sub>1</sub>''x'' + ''a''<sub>2</sub>''x''<sup>2</sup>, + ''a''<sub>3</sub>''x''<sup>3</sup>}}, where {{math|''a''<sub>3</sub> ≠ 0}}
:is a cubic curve.
 
* The graph of any polynomial with degree 2 or greater
::{{math|''f''(''x'') {{=}} ''a''<sub>0</sub> + ''a''<sub>1</sub>''x'' + ''a''<sub>2</sub>''x''<sup>2</sup> + ... + ''a''<sub>''n''</sub>''x''<sup>''n''</sup>}} , where {{math|''a''<sub>''n''</sub> ≠ 0 and ''n'' ≥ 2}}
:is a continuous non-linear curve.
 
The graph of a non-constant (univariate) polynomial always [[Infinity#Calculus|tends to infinity]] when the variable increases indefinitely (in [[absolute value]]).{{citation needed|date=August 2013}}
 
Polynomial graphs are analyzed in calculus using intercepts, slopes, concavity, and end behavior.
 
==Polynomial equations==
{{main|Algebraic equation}}
A ''polynomial equation'', also called ''[[algebraic equation]]'', is an [[equation]] of the form<ref>{{Cite book|author=Proskuryakov, I.V.|chapter=Algebraic equation|editor=Hazewinkel, Michiel| editor-link=Michiel Hazewinkel |title=Encyclopaedia of Mathematics|volume=vol. 1|publisher=Springer|year=1994|isbn=9781556080104|url=http://books.google.com/books?id=PE1a-EIG22kC&pg=PA88}}</ref>
 
:<math>a_n x^n + a_{n-1}x^{n-1} + \dotsb + a_2 x^2 + a_1 x + a_0 = 0</math>
 
For example,
 
: <math> 3x^2 + 4x -5 = 0 \,</math>
 
is a polynomial equation.
 
In case of a univariate polynomial equation, the variable is considered an [[Variable (mathematics)|unknown]], and one seeks to find the possible values for which both members of the equation evaluate to the same value (in general more than one solution may exist). A polynomial equation stands in contrast to a ''polynomial identity'' like {{math|(''x'' + ''y'')(''x'' &minus; ''y'') {{=}} ''x''<sup>2</sup> &minus; ''y''<sup>2</sup>}}, where both members represent the same polynomial in different forms, and as a consequence any evaluation of both members gives a valid equality. This means that a polynomial identity is a polynomial equation for which all possible values of the unknowns are solutions.{{citation needed|date=August 2013}}
 
In elementary [[algebra]], methods such as the [[quadratic formula]] are given for solving all first degree and second degree polynomial equations in one variable. There are also formulas for the [[cubic function|cubic]] and [[quartic equations]]. For higher degrees, [[Abel-Ruffini theorem]] asserts that there can not exist a general formula. Therefore, only [[numerical approximation]]s of the roots may be computed (see [[Root-finding algorithm]]). The number of solutions may not exceed the degree, and equals the degree when the [[complex number|complex]] solutions are counted with their [[Multiplicity (mathematics)|multiplicity]]. This fact is called the [[fundamental theorem of algebra]].
 
===Solving polynomial equations===<!-- "Simple root (polynomial)" redirects here -->
{{Further|Properties of polynomial roots}}
 
Every polynomial {{math|''P''}} in {{math|''x''}} corresponds to a function, {{math|''f''(''x'') {{=}} ''P''}} (where the occurrences of {{math|''x''}} in {{math|''P''}} are interpreted as the argument of {{math|''f''}}), called the ''polynomial function'' of {{math|''P''}}; the equation in {{math|''x''}} setting {{math|''f''(''x'') {{=}} 0}} is the ''polynomial equation'' corresponding to {{math|''P''}}. The solutions of this equation are called the ''roots'' of the polynomial; they are the ''zeroes'' of the function {{math|''f''}} (corresponding to the points where the graph of {{math|''f''}} meets the {{math|''x''}}-axis). A number {{math|''a''}} is a root of {{math|''P''}} if and only if the polynomial {{math|''x'' &minus; ''a''}} (of degree one in {{math|''x''}}) divides {{math|''P''}}. It may happen that {{math|''x'' &minus; ''a''}} divides {{math|''P''}} more than once: if {{math|(''x'' &minus; ''a'')<sup>2</sup>}} divides {{math|''P''}} then {{math|''a''}} is called a ''multiple root'' of {{math|''P''}}, and otherwise {{math|''a''}} is called a ''simple root'' of {{math|''P''}}. If {{math|''P''}} is a nonzero polynomial, there is a highest power {{math|''m''}} such that {{math|(''x''&nbsp;−&nbsp;''a'')<sup>''m''</sup>}} divides {{math|''P''}}, which is called the ''multiplicity'' of the root {{math|''a''}} in {{math|''P''}}. When {{math|''P''}} is the zero polynomial, the corresponding polynomial equation is trivial, and this case is usually excluded when considering roots: with the above definitions every number would be a root of the zero polynomial, with undefined (or infinite) multiplicity. With this exception made, the number of roots of {{math|''P''}}, even counted with their respective multiplicities, cannot exceed the degree of {{math|''P''}}.<ref>{{cite book|author=Leung, Kam-tim et al|title=Polynomials and Equations|publisher=Hong Kong University Press|year=1992|isbn=9789622092716|page=134|url=http://books.google.com/books?id=v5uXkwIUbC8C&pg=PA134}}</ref> The relation between the roots of a polynomial and its coefficients is described by [[Viète's formulas]].
 
Some polynomials, such as {{math|''x''<sup>2</sup> + 1}}, do not have any roots among the [[real number]]s. If, however, the set of allowed candidates is expanded to the [[complex number]]s, every non-constant polynomial has at least one root; this is the [[fundamental theorem of algebra]]. By successively dividing out factors {{math|''x'' &minus; ''a''}}, one sees that any polynomial with complex coefficients can be written as a constant (its leading coefficient) times a product of such polynomial factors of degree&nbsp;1; as a consequence, the number of (complex) roots counted with their multiplicities is exactly equal to the degree of the polynomial.
 
There is a difference between approximating roots and finding exact expressions for roots. Formulas for expressing the roots of polynomials of [[Degree of a polynomial|degree]] 2 in terms of square roots have been known since ancient times (see [[quadratic equation]]), and for polynomials of degree 3 or 4 similar formulas (using cube roots in addition to square roots) were found in the 16th century (see [[cubic function]] and [[quartic function]] for the formulas and [[Niccolo Fontana Tartaglia]], [[Lodovico Ferrari]], [[Gerolamo Cardano]], and [[Franciscus Vieta|Vieta]] for historical details). But formulas for degree 5 eluded researchers. In 1824, [[Niels Henrik Abel]] proved the striking result that there can be no general (finite) formula, involving only arithmetic operations and radicals, that expresses the roots of a polynomial of degree 5 or greater in terms of its coefficients (see [[Abel-Ruffini theorem]]). In 1830, [[Évariste Galois]], studying the permutations of the roots of a polynomial, extended the [[Abel-Ruffini theorem]] by showing that, given a polynomial equation, one may decide if it is solvable by radicals, and, if it is, solve it. This result marked the start of [[Galois theory]] and [[Group theory]], two important branches of modern mathematics. Galois himself noted that the computations implied by his method were impracticable. Nevertheless, formulas for solvable equations of degrees 5 and 6 have been published (see [[quintic function]] and [[sextic equation]]).
 
Numerical approximations of roots of polynomial equations in one unknown is easily done on a computer by the [[Jenkins-Traub method]], [[Laguerre's method]], [[Durand–Kerner method]] or by some other [[root-finding algorithm]].<ref>{{cite book|author=McNamee, J.M.|title=Numerical Methods for Roots of Polynomials, Part 1|publisher=Elsevier|year=2007|isbn=9780080489476|url=http://books.google.com/books?id=4PMqxwG-eqQC&printsec=frontcover}}</ref>
 
For polynomials in more than one indeterminate the notion of root does not exist, and there are usually infinitely many combinations of values for the variables for which the polynomial function takes the value zero. However for certain ''sets'' of such polynomials it may happen that for only finitely many combinations all polynomial functions take the value zero.
 
For a set of polynomial equations in several unknowns, there are [[algorithm]]s to decide if they have a finite number of complex solutions. If the number of solutions is finite, there are algorithms to compute the solutions. The methods underlying these algorithms are described in the article [[systems of polynomial equations]].
 
The special case where all the polynomials are of degree one is called a [[system of linear equations]], for which another range of different [[System of linear equations#Solving a linear system|solution methods]] exist, including the classical [[Gaussian elimination]].
 
==Polynomials associated to other objects==
 
===Calculus===
{{Main|Calculus with polynomials}}
{{See also|Polynomial interpolation}}
The simple structure of polynomial functions makes them quite useful in analyzing general functions using polynomial approximations. An important example in [[calculus]] is [[Taylor's theorem]], which roughly states that every [[differentiable]] function locally looks like a polynomial function, and the [[Stone-Weierstrass theorem]], which states that every [[continuous function]] defined on a [[compact space|compact]] [[interval (mathematics)|interval]] of the real axis can be approximated on the whole interval as closely as desired by a polynomial function.
 
Calculating derivatives and integrals of polynomial functions is particularly simple. For the polynomial function
:<math>\sum_{i=0}^n a_i x^i</math>
the derivative with respect to ''x'' is
:<math>\sum_{i=1}^n a_i i x^{i-1}</math>
and the indefinite integral is
:<math>\sum_{i=0}^n {a_i\over i+1} x^{i+1}+c.</math>
 
=== Abstract algebra ===
{{Main|Polynomial ring}}
 
In [[abstract algebra]], one distinguishes between ''polynomials'' and ''polynomial functions''. A ''polynomial'' {{math|''f''}} in one indeterminate {{math|''X''}} over a [[ring (mathematics)|ring]] {{math|''R''}} is defined as a formal expression of the form
: <math>f = a_n X^n + a_{n - 1} X^{n - 1} + \cdots + a_1 X^1 + a_0X^0</math>
where {{math|''n''}} is a natural number, the coefficients {{math|''a''<sub>0</sub>, . . ., ''a''<sub>n</sub>}} are elements of {{math|''R''}}, and {{math|''X''}} is a formal symbol, whose powers {{math|''X<sup>i</sup>''}} are just placeholders for the corresponding coefficients {{math|''a<sub>i</sub>''}}, so that the given formal expression is just a way to encode the sequence {{math|(''a''<sub>0</sub>, ''a''<sub>1</sub>, . . .)}}, where there is an {{math|''n''}} such that {{math|''a<sub>i</sub>'' {{=}} 0}}  for all {{math|''i'' > ''n''}}. Two polynomials sharing the same value of ''n'' are considered equal if and only if the sequences of their coefficients are equal; furthermore any polynomial is equal to any polynomial with greater value of {{math|''n''}} obtained from it by adding terms in front whose coefficient is zero. These polynomials can be added by simply adding corresponding coefficients (the rule for extending by terms with zero coefficients can be used to make sure such coefficients exist). Thus each polynomial is actually equal to the sum of the terms used in its formal expression, if such a term {{math|''a<sub>i</sub>X<sup>i</sup>''}} is interpreted as a polynomial that has zero coefficients at all powers of {{math|''X''}} other than {{math|''X<sup>i</sup>''}}. Then to define multiplication, it suffices by the [[distributive law]] to describe the product of any two such terms, which is given by the rule
 
:<div style="vertical-align:30%;display:inline"><math>
a X^k \; b X^l = ab X^{k+l}</math></div>{{nbsp|2}} for all elements ''a'', ''b'' of the ring ''R'' and all [[natural numbers]] ''k'' and ''l''.
 
Thus the set of all polynomials with coefficients in the ring {{math|''R''}} forms itself a ring, the ''ring of polynomials'' over {{math|''R''}}, which is denoted by {{math|''R''[''X'']}}. The map from  {{math|''R''}} to {{math|''R''[''X'']}} sending  {{math|''R''}} to {{math|''rX''<sup>0</sup>}} is an injective homomorphism of rings, by which  {{math|''R''}} is viewed as a subring of {{math|''R''[''X'']}}. If  {{math|''R''}} is [[commutative ring|commutative]], then {{math|''R''[''X'']}} is an [[Algebra (ring theory)|algebra]] over {{math|''R''}}.
 
One can think of the ring {{math|''R''[''X'']}} as arising from  {{math|''R''}} by adding one new element ''X'' to ''R'', and extending in a minimal way to a ring in which {{math|''X''}} satisfies no other relations than the obligatory ones, plus commutation with all elements of  {{math|''R''}} (that is {{math|''Xr'' {{=}} ''rX''}}). To do this, one must add all powers of {{math|''X''}} and their linear combinations as well.
 
Formation of the polynomial ring, together with forming factor rings by factoring out [[ideal (ring theory)|ideals]], are important tools for constructing new rings out of known ones. For instance, the ring (in fact field) of complex numbers, which can be constructed from the polynomial ring {{math|''R''[''X'']}} over the real numbers by factoring out the ideal of multiples of the polynomial {{math|''X''<sup>2</sup> + 1}}. Another example is the construction of [[finite field]]s, which proceeds similarly, starting out with the field of integers modulo some [[prime number]] as the coefficient ring  {{math|''R''}} (see [[modular arithmetic]]).
 
If  {{math|''R''}} is commutative, then one can associate to every polynomial {{math|''P''}} in {{math|''R''[''X'']}}, a ''polynomial function'' {{math|''f''}} with domain and range equal to  {{math|''R''}} (more generally one can take domain and range to be the same [[unital algebra|unital]] [[associative algebra]] over {{math|''R''}}). One obtains the value {{math|''f''(''r'')}} by [[substitution (algebra)|substitution]] of the value  {{math|''R''}} for the symbol {{math|''X''}} in {{math|''P''}}. One reason to distinguish between polynomials and polynomial functions is that over some rings different polynomials may give rise to the same polynomial function (see [[Fermat's little theorem]] for an example where  {{math|''R''}} is the integers modulo {{math|''p''}}). This is not the case when  {{math|''R''}} is the real or complex numbers, whence the two concepts are not always distinguished in [[analysis (mathematics)|analysis]]. An even more important reason to distinguish between polynomials and polynomial functions is that many operations on polynomials (like [[Euclidean division]]) require looking at what a polynomial is composed of as an expression rather than evaluating it at some constant value for {{math|''X''}}.
 
====Divisibility====
{{Main|Polynomial greatest common divisor|Factorization of polynomials}}
In [[commutative algebra]], one major focus of study is ''divisibility'' among 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''}}. One can show that every zero gives rise to a linear divisor, or more formally, if {{math|''f''}} is a polynomial in {{math|''R''[''X'']}} and {{math|''r''}} is an element of {{math|''R''}} such that {{math|''f''(''r'') {{=}} 0}}, then the polynomial ({{math|''X'' &minus; ''r''}}) divides {{math|''f''}}. The converse is also true. The quotient can be computed using the [[polynomial long division]].<ref>{{Cite book|author=Irving, Ronald S.|title=Integers, Polynomials, and Rings: A Course in Algebra|publisher=Springer|year=2004|isbn=9780387201726|page=129|url=http://books.google.com/books?id=B4k6ltaxm5YC&pg=PA129}}</ref><ref>{{cite book|author=Jackson, Terrence H.|title=From Polynomials to Sums of Squares|publisher=CRC Press|year=1995|isbn=9780750303293|page=143|url=http://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> 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 element|irreducible]] polynomials'') 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.
 
===Other applications===
{{see also|Orthogonal polynomial|B-spline|spline interpolation}}
Polynomials serve to approximate other [[function (mathematics)|functions]],<ref>{{cite book|author=de Villiers, Johann |title=Mathematics of Approximation|publisher=Springer|year=2012|isbn=9789491216503|url=http://books.google.com/books?id=l5mIro_6RlUC&printsec=frontcover}}</ref> such as the use of [[Spline (mathematics)|splines]].
 
Polynomials are frequently used to encode information about some other object. The [[characteristic polynomial]] of a matrix or linear operator contains information about the operator's [[eigenvalue]]s. The [[minimal polynomial (field theory)|minimal polynomial]] of an [[algebraic element]] records the simplest algebraic relation satisfied by that element. The [[chromatic polynomial]] of a [[Graph (mathematics)|graph]] counts the number of proper colourings of that graph.
 
The term "polynomial", as an adjective, can also be used for quantities or functions that can be written in polynomial form. For example, in [[computational complexity theory]] the phrase ''[[polynomial time]]'' means that the time it takes to complete an [[algorithm]] is bounded by a polynomial function of some variable, such as the size of the input.
 
==Extensions of the concept of a polynomial==
Rings of polynomials in a finite number of variables are of fundamental importance in [[algebraic geometry]] which studies the simultaneous zero sets of several such multivariate polynomials. These rings can alternatively be constructed by repeating the construction of univariate polynomials with as coefficient ring another ring of polynomials: thus the ring {{math|''R''[''X'', ''Y'']}} of polynomials in {{math|''X''}} and {{math|''Y''}} can be viewed as the ring {{math|(''R''[''X''])[''Y'']}} of polynomials in {{math|''Y''}} with as coefficients polynomials in {{math|''X''}}, or as the ring
{{math|(''R''[''Y''])[''X'']}} of polynomials in ''X'' with as coefficients polynomials in {{math|''Y''}}. These identifications are compatible with arithmetic operations (they are [[isomorphism]]s of rings), but some notions such as degree or whether a polynomial is considered monic can change between these points of view. One can construct rings of polynomials in infinitely many indeterminates, but since polynomials are (finite) expressions, any individual polynomial can only contain finitely many indeterminates.{{citation needed|date=August 2013}}
 
A bivariate polynomial where the second variable is substituted by an exponential function applied to the first variable, for example {{math|''P''(''X'', ''e''<sup>''X''</sup>)}}, may be called an [[exponential polynomial]].
 
[[Laurent polynomial]]s are like polynomials, but allow negative powers of the variable(s) to occur.
 
[[Quotient]]s of polynomials are called [[rational expression]]s (or rational fractions), and functions that evaluate rational expressions are called [[rational function]]s. Rational fractions are formal quotients of polynomials (they are formed from polynomials just as [[rational number]]s are formed from [[integer]]s, writing a [[Algebraic fraction|fraction]] of two of them; fractions related by the canceling of common factors are identified with each other). The rational function defined by a rational fraction is the quotient of the polynomial functions defined by the numerator and the denominator of the rational fraction. The rational fractions contain the Laurent polynomials, but do not limit denominators to powers of an indeterminate. While polynomial functions are defined for all values of the variables, a rational function is defined only for the values of the variables for which the denominator is not null. A rational function produces rational output for any rational input for which it is defined; this is not true of other functions such as [[trigonometric function]]s, [[logarithm]]s and [[exponential function]]s.{{citation needed|date=August 2013}}
 
[[Formal power series]] are like polynomials, but allow infinitely many non-zero terms to occur, so that they do not have finite degree. Unlike polynomials they cannot in general be explicitly and fully written down (just like [[real number]]s cannot), but the rules for manipulating their terms are the same as for polynomials.
 
==History==
Determining the roots of polynomials, or "solving algebraic equations", is among the oldest problems in mathematics. However, the elegant and practical notation we use today only developed beginning in the 15th century. Before that, equations were written out in words. For example, an algebra problem from the Chinese [[The Nine Chapters on the Mathematical Art|Arithmetic in Nine Sections]], circa 200 BCE, begins "Three sheafs of good crop, two sheafs of mediocre crop, and one sheaf of bad crop are sold for 29 dou." We would write {{math|3''x''&nbsp;+&nbsp;2''y''&nbsp;+&nbsp;''z'' {{=}}&nbsp;29}}.
 
===History of the notation===
{{Main|History of mathematical notation}}
The earliest known use of the equal sign is in [[Robert Recorde]]'s ''[[The Whetstone of Witte]]'', 1557. The signs + for addition, &minus; for subtraction, and the use of a letter for an unknown appear in [[Michael Stifel]]'s ''Arithemetica integra'', 1544. [[René Descartes]], in ''La géometrie'', 1637, introduced the concept of the graph of a polynomial equation. He popularized the use of letters from the beginning of the alphabet to denote constants and letters from the end of the alphabet to denote variables, as can be seen above, in the general formula for a polynomial in one variable, where the {{math|''a''}}'s denote constants and {{math|''x''}} denotes a variable. Descartes introduced the use of superscripts to denote exponents as well.<ref>Howard Eves, ''An Introduction to the History of Mathematics, Sixth Edition, Saunders, ISBN 0-03-029558-0</ref>
 
==See also==
*[[Lill's method]]
*[[List of polynomial topics]]
*[[Polynomials on vector spaces]]
*[[Power series]]
 
==Notes==
{{Reflist|colwidth=35em}}
 
==References==
<!-- * {{cite book|author=|title=|publisher=|year=|isbn=|url=}} -->
{{Refbegin}}
* {{cite book|author=Barbeau, E.J.|title=Polynomials|publisher=Springer|year=2003|isbn=9780387406275|url=http://books.google.com/books?id=CynRMm5qTmQC&printsec=frontcover}}
* {{cite book|editors=Bronstein, Manuel et al|title=Solving Polynomial Equations: Foundations, Algorithms, and Applications|publisher=Springer|year=2006|isbn=9783540273578|url=http://books.google.com/books?id=aIlSmBV3yf8C&printsec=frontcover}}
* {{cite book|authors=Cahen, Paul-Jean & Chabert, Jean-Luc|title=Integer-Valued Polynomials|publisher=American Mathematical Society|year=1997|isbn=9780821803882|url=http://books.google.com/books?id=AlAluH5is6AC&printsec=frontcover}}
* {{Lang Algebra}}. This classical book covers most of the content of this article.
* {{cite book|author=Leung, Kam-tim et al|title=Polynomials and Equations|publisher=Hong Kong University Press|year=1992|isbn=9789622092716|url=http://books.google.com/books?id=v5uXkwIUbC8C&printsec=frontcover}}
* Mayr, K. Über die Auflösung algebraischer Gleichungssysteme durch hypergeometrische Funktionen. ''Monatshefte für Mathematik und Physik'' vol. 45, (1937) pp.&nbsp;280–313.
* {{cite book|author=Prasolov, Victor V.|title=Polynomials|publisher=Springer|year=2005|isbn=9783642040122|url=http://books.google.com/books?id=qIJPxdwSqlcC&printsec=frontcover}}
* {{cite book|author=Sethuraman, B.A.|chapter=Polynomials|title=Rings, Fields, and Vector Spaces: An Introduction to Abstract Algebra Via Geometric Constructibility|publisher=Springer|year=1997|isbn=9780387948485|url=http://books.google.com/books?id=yWnTIqmUOFgC&pg=PA119}}
* Umemura, H. Solution of algebraic equations in terms of theta constants. In D. Mumford, ''Tata Lectures on Theta II'', Progress in Mathematics 43, Birkhäuser, Boston, 1984.
* von Lindemann, F. [http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?ht=VIEW&did=D55215 Über die Auflösung der algebraischen Gleichungen durch transcendente Functionen]. Nachrichten von der Königl. Gesellschaft der Wissenschaften, vol. 7, 1884. Polynomial solutions in terms of theta functions.
* von Lindemann, F. [http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?did=D55847 Über die Auflösung der algebraischen Gleichungen durch transcendente Functionen II]. Nachrichten von der Königl. Gesellschaft der Wissenschaften und der Georg-Augusts-Universität zu Göttingen, 1892 edition.
{{Refend}}
 
==External links==
{{Commons category|Polynomials}}
{{Wiktionary|polynomial}}
*{{springer|title=Polynomial|id=p/p073690}}
*[http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=640&bodyId=1038 Euler's work on Imaginary Roots of Polynomials]
*{{MathWorld |title=Polynomial |id=Polynomial}}
 
{{Polynomials}}
 
[[Category:Polynomials| ]]
[[Category:Algebra]]
 
{{Link GA|yo}}

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As this is researching my practice, Laurene suggested that I understand my community of design practitioners as well. Most of my design knowledge comes from 10 years of design practic, 6 years of studying and 4 years training, and in that time, the conversations and observations of other design practitioners have given me an annecdotal impression of my practice. Laurene argues that I need to make this more rigorous, and document my conversation with other designers, which makes sense. I need to figure out whether my assumption Jeremy Scott Adidas of the graphic designers model is how it is by using it as a starting point and to develop many variations of this model. I need to understand how other design practitioners would connect with my DFP model. The audience of my scientific studies are the community of design practitioners afterall, and that i would like to provide ways for designers to effectively communicate the things they're doing that has less emphasis on artefact, but more about communication.

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Better late than never, In my opinion. I don't write as much as Let me for the site because I have a hard time trying to guess what the public wants, however i will try to submit more content. Recently, news of a Bic pen specifically designed "for her" spawned a hilarious cacophony of snarky, sarcastic comments around the product Amazon page. Now, Ellen DeGeneres has joined the party, adding her characteristic sarcasmdisguisedassweetness towards the fray on Friday Ellen.

Examples of demo could be:1) demo 1 shows rain or wave being trained to clear a dirty surface then green grass Polo Ralph Lauren Nz is proved to be growing back with an office background.2) demo 2 show water failing from the sky to clean away dirt then green grass starts to grow back in someones gargen.3) demo 3 shows an individual character using a pipe linked to a natural fountain or spring to clear away some dirt after which grass starts to grows back in Cheap Nike Air Max Australia a lawn,(garden).I think a bit of imagination is needed that is all.The website will have About Us, Service, Product, Contact Us in its tool line. The writeup or contents is going Nike Free Run to be short.

He looks full of joyful young life. But Henio continues to be dead since 1942, killed inside a gas chamber at the Majdanek concentration camp as he was 9. Along Ocean Terrace, the nearest road to the beach, employees of Vinny's Games have set up a temporary booth to try Oakley Frogskins to woo those passing by into playing a dart game involving balloons.Joe Franzi, a supervisor at Vinny's, explained they were attempting to raise money to rebuild their stand along the boardwalk, which had been heavily damaged by the eightfoot storm surge. Recent business, Franzi said, has been "pretty good some days and just OK on others.""There are a few people who walk by and prevent in during the day, and some people just give us donations," Franzi said.

To become successful in school, our children need fuel. School lunches can be tough. 2nd set: 21 Milos, but it looks like a matador versus a really slow, lethargic bull out there. After some of his groundstrokes, Milos seems to take forever to reposition himself on the baseline.

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Prior to 1989, births were recognized as either occurring in or out Nike Air Yeezy 2 of a hospital (eg, home, car, office). Since 1989, a particular category for home births was included on the Standard Certificate. Since 2003, some states have changed their Standard Certificate to identify planned versus unplanned home births and, by 2008, there were 27 states which reported planning status, comprising 68 percent of all home births.ACOG Committee on Obstetric Practice.

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With regards to gadgets be it camera, phones, laptops or even tablets and music players typically, anything "compact", "shiny" and in flashy "colours" is just for the ladies. And even the largest brands have fallen within this trap. Even a company like Apple took the bait because it serenaded us with the flashy pink range of iPods. We hope we aren alive to determine the launch of a pink iPhone. Visiting laptops, the brand designers have interpreted Nike Air Max 90 its larger area as a field to spew their imagination, from tacky floral prints to abstract art to even jewel finish glossiness and, of course, the colours they think we want. One brand that went all out was Sony, who launched its Vaio range in vivid colours, seemingly to ensure they are femalefriendly.

HOWARD, Harvey M. 80, of Medway, Ohio passed away peacefully Monday, November 14, 2011. He was created March 26, 1931 in Olive Hill, Kentucky the son from the late Autie Vesta (Adkins) Howard. Army Air Corp veteran and retired from Wright Patterson Air Force Base. Harvey was an avid fisherman, storyteller and was an amazing needle work craftsman. He's survived by his children; Rickie Howard, Keith Howard, Christina Howard; grandchildren, Justin, William, Megan, Rachel, Sarah, Leah, Taylor Wyatt; great grandson, Zen; sisters, Faye, Gladys Bonnie; nieces, nephews, other relatives and friends. In addition to his parents, he is preceded in death by his wife, Janet L. Howard; a son, Harvey Howard II; a daughter, Karen D. Howard; brothers, Donald David Howard; sisters, Jewell, Fern, Beatrice Glenna Jean. Visitation Louis Vuitton Auckland is going to be 68 PM Thursday, November 17, 2011 in the TROSTEL, CHAPMAN, DUNBAR FRALEY FUNERAL HOME, New Carlisle, Ohio. Burial is going to be 10:00 AM Friday November 18, 2011 at the Dayton National Cemetery.

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Drumline is a dramatic comedy that spotlights the world of university marching bands. Miles (Cannon) is a Harlem teenager who receives a full scholarship to attend Atlanta A University based on his excellent percussion talents. making the transition from hiphop street drumming to the drumline of the school's legendary marching band is much more challenging than he expected. Nike Free Run Australia The premiere of School Gyrls will Nike Free Run 5.0 Australia coincide with the release of a brandnew multimedia book series from Simon Schuster Children's Division as well as the release of a debut album along with a musical tour by the pop trio. from Nike High Heels Nz the School Gyrls book series is a movie novelization that will be published within this month by Simon Spotlight, an imprint of Simon Schuster Children's Publishing. Aladdin Books will publish six original novels based on the three girls. the School Gyrls, signed to Cannon's NCredible Entertainment, will even embark on a 10week crosscountry tour with Cannon in spring 2010 timed towards the March 23 release of their debut album on Island Records. first single from the album, "Something Like a Party" will be on sale digitally on Feb. 9. has also partnered with Archie Comics introducing the School Gyrls into their iconic comic book series, with the cartoon characterization from the girls being introduced into Archie issue 607 which will hit newsstands nationwide on March 24 and be available for download on iTunes and Archie Digital.

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Colorful worshippers always help to make for great photo opportunities at Angkor. One of my favorite wideangle views is of this statue of Vishnu thought to have been the original centerpiece of Angkor Wat prior to being moved by a Buddhist king in the Western gatehouse (Gopura). It can be a little tricky to time the image capture when ever no one is in the field of view as you can tell from the leg peeking into the scene on the right. Fortunately, a small job with the array of tools in Photoshop CS6. My Sigma 1224mm fullframe lens on my small Nikon D600 is perfect for capturing this shot:

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