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{{Short description|In mathematics, sum of products of variables, power of variables, and coefficients}}
[[File:Polynomialdeg3.svg|The [[graph of a function|graph]] of a polynomial function of degree 3|thumb]]

In [[mathematics]], a '''polynomial''' is an [[expression (mathematics)|expression]] consisting of [[variable (mathematics)|variables]] (also called [[indeterminate (variable)|indeterminates]]) and [[coefficient]]s, that involves only the operations of [[addition]], [[subtraction]], [[multiplication]], and non-negative [[integer]] [[exponentiation|exponents]] of variables. An example of a polynomial of a single indeterminate, {{math|''x''}}, is {{math|''x''<sup>2</sup> − 4''x'' + 7}}. An example in three variables is {{math|''x''<sup>3</sup> + 2''xyz''<sup>2</sup> − ''yz'' + 1}}.

Polynomials appear in many areas of mathematics and science. For example, they are used to form [[polynomial equation]]s, which encode a wide range of problems, from elementary [[word problem (mathematics education)|word problems]] to complicated scientific problems; 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]].

==Etymology==
The word ''polynomial'' [[hybrid word|joins two diverse roots]]: the Greek ''poly'', meaning "many," and the Latin ''nomen'', or name. It was derived from the term ''[[binomial (polynomial)|binomial]]'' by replacing the Latin root ''bi-'' with the Greek ''poly-''. The word ''polynomial'' was first used in the 17th century.<ref>See "polynomial" and "binomial", ''Compact Oxford English Dictionary''</ref>
{{anchor|Polynomial notation}}

==Notation and terminology==
The ''x'' occurring in a polynomial is commonly called either a ''variable'' or an ''indeterminate''. When the polynomial is considered as an expression, ''x'' is a fixed symbol which does not have any value (its value is "indeterminate"). However, when one considers 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 interchangeably.

It is common to use uppercase letters for indeterminates and corresponding lowercase letters for the variables (or arguments) of the associated function.{{citation needed|date=February 2019}}

A polynomial ''P'' in the indeterminate ''x'' is commonly denoted either as ''P'' or as ''P''(''x''). Formally, the name of the polynomial is ''P'', not ''P''(''x''), but the use of the [[functional notation]] ''P''(''x'') date from the time where the distinction between a polynomial and the associated function was unclear. Moreover, the functional notation is often useful for specifying, in a single phrase, a polynomial and its indeterminate. For example, "let ''P''(''x'') be a polynomial" is 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.

The ambiguity of having two notations for a single mathematical object may be formally resolved by considering the general meaning of the functional notation for polynomials.
If ''a'' denotes a number, a variable, another polynomial, or, more generally any expression, then ''P''(''a'') denotes, by convention, the result of substituting ''a'' for ''x'' in ''P''. Thus, the polynomial ''P'' defines the function
:<math>a\mapsto P(a),</math>
which is the ''polynomial function'' associated to ''P''.
Frequently, when using this notation, one supposes that ''a'' is a number. However one may use it over any domain where addition and multiplication are defined (that is, any [[ring (mathematics)|ring]]). In particular, if ''a'' is a polynomial then ''P''(''a'') is also a polynomial.
 
More specifically, when ''a'' is the indeterminate ''x'', then the [[Image (mathematics)|image]] of ''x'' by this function is the polynomial ''P'' itself (substituting ''x'' to ''x'' does not change anything). In other words,
:<math>P(x)=P,</math>
which justifies formally the existence of two notations for the same polynomial.

==Definition==
A polynomial is an [[expression (mathematics)|expression]] that can be built from [[constant (mathematics)|constants]] and symbols called indeterminates or variables by means of [[addition]], [[multiplication]] and [[exponentiation]] to a [[non-negative integer]] power. Two such expressions that may be transformed, one to the other, by applying the usual properties of [[commutative property|commutativity]], [[associative property|associativity]] and [[distributive property|distributivity]] of addition and multiplication are considered as defining the same polynomial.

A polynomial in a single indeterminate ''x'' can always be written (or rewritten) 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>
where <math>a_0, \ldots, a_n</math> are constants and <math>x</math> is the indeterminate. The word "indeterminate" means that <math>x</math> represents no particular value, although any value may be substituted for it. The mapping that associates the result of this substitution to the substituted value is a [[function (mathematics)|function]], called a ''polynomial function''.

This can be expressed more concisely by using [[summation#Capital-sigma notation|summation notation]]:
:<math>\sum_{k=0}^n a_k x^k</math>

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{{snd}} 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>{{snd}} and a finite number of indeterminates, raised to nonnegative integer powers.

==Classification==
{{Further|Degree of a polynomial}}
The exponent on an indeterminate in a term is called the 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. Because {{math|''x'' {{=}} ''x''<sup>1</sup>}}, the degree of an indeterminate without a written exponent is one.

A term with no indeterminates and a polynomial with no indeterminates are called, respectively, a [[constant term]] and a constant polynomial.<ref>This terminology dates from the time when 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, 0, (which has no terms at all) is generally treated as not defined (but see below).<ref name=Barbeau-2003-pp1-2>{{cite book |last=Barbeau |first=E.J. |title=Polynomials |publisher=Springer |year=2003 |isbn=978-0-387-40627-5 |pages=1–2 |url=https://books.google.com/?id=CynRMm5qTmQC&pg=PA1}}</ref>

For example:
:<math> -5x^2y </math>

is a term. The coefficient is {{math|−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}}.

Forming a sum of several terms produces a polynomial. For example, the following is a polynomial:
:<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>

It consists of three terms: the first is degree two, the second is degree one, and the third is degree zero.

{{anchor|linear polynomial}}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.

{{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 in one indeterminate having an infinite number of [[root of a function|roots]]. The graph of the zero polynomial, {{math|''f''(''x'') {{=}} 0}}, is the {{math|''X''}}-axis.

In the case of polynomials in more than one indeterminate, a polynomial is called ''homogeneous'' of {{nowrap|degree {{math|''n''}}}} if ''all'' its non-zero terms have {{nowrap|degree {{math|''n''}}}}. The zero polynomial is homogeneous, and, as homogeneous polynomial, its degree is undefined.<ref>In fact, as [[homogeneous function]], it is homogeneous of ''every'' degree</ref> 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|−5}}}}. The third term is a constant. Because the ''degree'' of a non-zero polynomial is the largest degree of any one term, this polynomial has degree two.<ref>{{cite book |last=Edwards |first=Harold M. |title=Linear Algebra |publisher=Springer |year=1995 |isbn=978-0-8176-3731-6 |page=78 |url=https://books.google.com/?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 |first=Anthony W. |last=Knapp |title=Advanced Algebra: Along with a Companion Volume Basic Algebra |page=457 |year=2007 |publisher=Springer |isbn=978-0-8176-4522-9}}</ref> a two-term polynomial is called a [[binomial (polynomial)|binomial]], and a three-term polynomial is called a ''trinomial''. The term "quadrinomial" is occasionally used for a four-term polynomial.

{{anchor|real polynomial|complex polynomial|integer polynomial}}A '''real polynomial''' is a polynomial with [[real number|real]] coefficients. When it is used to define a [[function (mathematics)|function]], the [[domain (function)|domain]] is not  so restricted. However, a '''real polynomial function''' is a function from the reals to the reals that is defined by a real polynomial. Similarly, an '''integer polynomial''' is a polynomial with [[integer]] coefficients, and a '''complex polynomial''' is a polynomial with [[complex number|complex]] coefficients.

{{anchor|univariate|bivariate|Number of variables|Multivariate polynomial}}A polynomial in one indeterminate is called a ''[[univariate]] polynomial'', a polynomial in more than one indeterminate is called a '''multivariate polynomial'''. A polynomial with two indeterminates is called a '''bivariate 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.

{{anchor|evaluation}}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 [[Horner's method]]:
:<math>(((((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==
Polynomials can be added using the [[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 |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/?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/?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{Red} P &\color{Red}{= 2x + 3y + 5} \\
 \color{Blue} Q &\color{Blue}{= 2x + 5y + xy + 1}
\end{align}</math>

then
:<math>\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>

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, because the remainder of the division of {{math|''f''(''x'')}} by {{math|(''x'' − ''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 [[summation|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''−1</sub>''x''<sup>''n''−1</sup> + ... + ''a''<sub>2</sub>''x''<sup>2</sup> + ''a''<sub>1</sub>''x'' + ''a''<sub>0</sub>}} is the polynomial {{math|''na<sub>n</sub>x''<sup>''n''−1</sup> + (''n'' − 1)''a''<sub>''n''−1</sub>''x''<sup>''n''−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|''ka''<sub>''k''</sub>}} should be interpreted as the sum of {{math|''a''<sub>''k''</sub>}} with itself, {{mvar|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 |last=Barbeau |first=E.J. |title=Polynomials |publisher=Springer |year=2003 |isbn=978-0-387-40627-5 |pages=64–5 |url=https://books.google.com/?id=CynRMm5qTmQC&pg=PA64}}</ref>
*A primitive integral or [[antiderivative]] of the polynomial {{math|''a''<sub>''n''</sub>''x''<sup>''n''</sup> + ''a''<sub>''n''−1</sub>''x''<sup>''n''−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''−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 {{math|''a''}} and {{math|''b''}} such that {{math|''b'' ≠ 0}}, there exists a unique pair of polynomials, {{math|''q''}}, the quotient, and {{math|''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|978-0-471-53012-1}}</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 |last=Barbeau |first=E.J. |title=Polynomials |publisher=Springer |year=2003 |isbn=978-0-387-40627-5 |pages=80–2 |url=https://books.google.com/?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, efficient [[factorization of polynomials|polynomial factorization]] [[algorithm]]s are available in most [[computer algebra system]]s.

A formal quotient of polynomials, that is, an [[algebraic fraction]] wherein 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, yields 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 {{math|1/12}} is just a constant. When this expression is used as a term, its coefficient is therefore {{math|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 {{math|2 + 3''i''}} is one complex number, and is the coefficient of that term. The expression {{math|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.

Because subtraction can be replaced by addition of the opposite quantity, and because 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 [[#evaluation|evaluating]] a polynomial. More precisely, a function {{math|''f''}} of one [[argument of a function|argument]] from a given domain is a polynomial function if there exists a polynomial
:<math>a_n x^n + a_{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0 </math>
that evaluates to <math>f(x)</math> for all {{mvar|x}} in the [[domain of a function|domain]] of {{mvar|f}} (here, {{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).

Generally, unless otherwise specified, polynomial functions have [[complex number|complex]] coefficients, arguments, and values. In particular, a polynomial, restricted to have real coefficients, defines a function from the complex numbers to the complex numbers. If the domain of this function is also [[restriction of a function|restricted]] to the reals, the resulting function maps reals to reals.

For example, the function {{math|''f''}}, defined by
:<math> f(x) = x^3 - x,</math>
is a polynomial function of one variable. Polynomial functions of several variables are similarly defined, using polynomials in more than one indeterminate, as in
:<math>f(x,y)= 2x^3+4x^2y+xy^5+y^2-7.</math>

According to the definition of polynomial functions, there may be expressions that obviously are not polynomials but nevertheless define polynomial functions. An example is the expression <math>\left(\sqrt{1-x^2}\right)^2,</math> which takes the same values as the polynomial <math>1-x^2</math> on the interval <math>[-1,1]</math>, and thus both expressions define the same polynomial function on this interval.

Every polynomial function is [[continuous function|continuous]], [[smooth function|smooth]], and [[entire function|entire]].

===Graphs===
<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> − ''x'' − 2}}<br/>{{math|{{=}} (''x'' + 1)(''x'' − 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 − 3''x''/2 − 2}}<br/>{{math|{{=}} 1/4 (''x'' + 4)(''x'' + 1)(''x'' − 2)}}</small>
File:Polynomialdeg4.svg|Polynomial of degree 4:<br/><small>{{math|''f''(''x'') {{=}} 1/14 (''x'' + 4)(''x'' + 1)(''x'' − 1)(''x'' − 3) <br/>+ 0.5}}</small>
File:quintic polynomial.svg|Polynomial of degree 5:<br/><small>{{math|''f''(''x'') {{=}} 1/20 (''x'' + 4)(''x'' + 2)(''x'' + 1 )(''x'' − 1)<br/>(''x'' − 3)}}{{math|+ 2}}</small>
File:Sextic Graph.svg|Polynomial of degree 6:<br/><small>{{math|''f''(''x'') {{=}} 1/100 (''x''<sup>6</sup> − 2''x'' <sup>5</sup> − 26''x''<sup>4</sup> + 28''x''<sup>3</sup>}}<br/>{{math|+ 145''x''<sup>2</sup> - 26''x'' - 80)}}</small>
File:Septic graph.svg|Polynomial of degree 7:<br/><small>{{math|''f''(''x'') {{=}} (''x'' − 3)(''x'' − 2)(''x'' − 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 equation|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.

A non-constant polynomial function [[infinity#Calculus|tends to infinity]] when the variable increases indefinitely (in [[absolute value]]). If the degree is higher than one, the graph does not have any [[asymptote]]. It has two [[parabolic branch]]es with vertical direction (one branch for positive ''x'' and one for negative ''x'').

Polynomial graphs are analyzed in calculus using intercepts, slopes, concavity, and end behavior.

==Equations==
{{Main|Algebraic equation}}
A ''polynomial equation'', also called ''[[algebraic equation]]'', is an [[equation]] of the form<ref>{{Cite book |last=Proskuryakov |first=I.V. |chapter=Algebraic equation |editor=Hazewinkel, Michiel |editor-link=Michiel Hazewinkel |title=Encyclopaedia of Mathematics |volume=vol. 1 |publisher=Springer |year=1994 |isbn=978-1-55608-010-4 |chapter-url=https://books.google.com/?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.

When considering equations, the indeterminates (variables) of polynomials are also called [[variable (mathematics)|unknown]]s, and the ''solutions'' are the possible values of the unknowns for which the equality is true (in general more than one solution may exist). A polynomial equation stands in contrast to a ''polynomial [[identity (mathematics)|identity]]'' like {{math|(''x'' + ''y'')(''x'' − ''y'') {{=}} ''x''<sup>2</sup> − ''y''<sup>2</sup>}}, where both expressions represent the same polynomial in different forms, and as a consequence any evaluation of both members gives a valid equality.

In elementary [[algebra]], methods such as the [[quadratic formula]] are taught for solving all first degree and second degree polynomial equations in one variable. There are also formulas for the [[cubic equation|cubic]] and [[quartic equation]]s. For higher degrees, the [[Abel–Ruffini theorem]] asserts that there can not exist a general formula in radicals. However, [[root-finding algorithm]]s may be used to find [[numerical approximation]]s of the roots of a polynomial expression of any degree.

The number of real solutions of a polynomial equation with real coefficients 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 equations{{anchor|Solving_polynomial_equations}}===
<!-- "Simple root (polynomial)" redirects here -->
{{See also|Root-finding algorithm|Properties of polynomial roots}}

Every polynomial {{math|''P''}} in {{math|''x''}} defines a [[function (mathematics)|function]] <math>x\mapsto P(x),</math> called the ''polynomial function'' associated to {{math|''P''}}; the [[equation]] {{math|''P''(''x'') {{=}} 0}} is the ''polynomial equation'' associated to {{math|''P''}}. The solutions of this equation are called the ''roots'' of the polynomial, or the zeros of the associated function (they correspond to the points where the graph of the function meets the {{math|''x''}}-axis).

A number {{math|''a''}} is a root of a polynomial {{math|''P''}} if and only if the [[#linear polynomial|linear polynomial]] {{math|''x'' − ''a''}} divides {{math|''P''}}, that is if there is another polynomial {{math|''Q''}} such that {{math|1=''P'' = (''x'' – ''a'') Q}}. It may happen that {{math|''x'' − ''a''}} divides {{math|''P''}} more than once: if {{math|(''x'' − ''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'' − ''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, as, with the above definitions, every number is a root of the zero polynomial, with an undefined 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 |last=Leung |first=Kam-tim |title=Polynomials and Equations |publisher=Hong Kong University Press |year=1992 |isbn=9789622092716 |page=134 |url=https://books.google.com/?id=v5uXkwIUbC8C&pg=PA134|display-authors=etal}}</ref> 
The relation between the coefficients of a polynomial and its roots is described by [[Vieta'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 accepted solutions 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'' − ''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 may be several meanings of "solving an equation". One may want to express the solutions as explicit numbers; for example, the unique solution of {{math|1=2''x'' – 1 = 0}} is {{math|1/2}}. Unfortunately, this is, in general, impossible for equations of degree greater than one, and, since the ancient times, mathematicians have searched to express the solutions as [[algebraic expression]]; for example the [[golden ratio]] <math>(1+\sqrt 5)/2</math> is the unique positive solution of <math>x^2-x-1=0.</math> In the ancient times, they succeeded only for degrees one and two. For [[quadratic equation]]s, the [[quadratic formula]] provides such expressions of the solutions. Since the 16th century, similar formulas (using cube roots in addition to square roots), but much more complicated are known for equations of degree three and four (see [[cubic equation]] and [[quartic equation]]). But formulas for degree 5 and higher eluded researchers for several centuries. In 1824, [[Niels Henrik Abel]] proved the striking result that there are equations of degree 5 whose solutions cannot be expressed by a (finite) formula, involving only arithmetic operations and radicals (see [[Abel–Ruffini theorem]]). In 1830, [[Évariste Galois]] proved that most equations of degree higher than four cannot be solved by radicals, and showed that for each equation, one may decide whether 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 [[algebra]]. 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]]).

When there is no algebraic expression for the roots, and when such an algebraic expression exists but is too complicated to be useful, the unique way of solving is to compute [[numerical approximation]]s of the solutions.<ref>{{cite book |last=McNamee |first=J.M. |title=Numerical Methods for Roots of Polynomials, Part 1 |publisher=Elsevier |year=2007 |isbn=978-0-08-048947-6 |url=https://books.google.com/?id=4PMqxwG-eqQC&printsec=frontcover}}</ref> There are many methods for that; some are restricted to polynomials and others may apply to any [[continuous function]]. The most efficient [[algorithm]]s allow solving easily (on a [[computer]]) polynomial equations of degree higher than 1,000 (see [[Root-finding algorithm]]).

For polynomials in more than one indeterminate, the combinations of values for the variables for which the polynomial function takes the value zero are generally called ''zeros'' instead of "roots". The study of the sets of zeros of polynomials is the object of [[algebraic geometry]]. For a set of polynomial equations in several unknowns, there are [[algorithm]]s to decide whether they have a finite number of [[complex number|complex]] solutions, and, if this number is finite, for computing the solutions. See [[System 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]].

A polynomial equation for which one is interested only in the solutions which are [[integer]]s is called a [[Diophantine equation]]. Solving Diophantine equations is generally a very hard task. It has been proved that there cannot be any general [[algorithm]] for solving them, and even for deciding whether the set of solutions is empty (see [[Hilbert's tenth problem]]). Some of the most famous problems that have been solved during the fifty last years are related to Diophantine equations, such as [[Fermat's Last Theorem]].

==Generalizations==
{{anchor|Generalizations of polynomials}}<!-- [[Polynomial expression]] redirects to this section -->
There are several generalizations of the concept of polynomials.

===Trigonometric polynomials===
{{Main|Trigonometric polynomial}}
A '''trigonometric polynomial''' is a finite [[linear combination]] of [[function (mathematics)|functions]] sin(''nx'') and cos(''nx'') with ''n'' taking on the values of one or more [[natural number]]s.<ref>{{cite book |last1=Powell |first1=Michael J. D. |author1-link=Michael J. D. Powell |title=Approximation Theory and Methods |publisher=[[Cambridge University Press]] |isbn=978-0-521-29514-7 |year=1981}}</ref> The coefficients may be taken as real numbers, for real-valued functions.

If sin(''nx'') and cos(''nx'') are expanded in terms of sin(''x'') and cos(''x''), a trigonometric polynomial becomes a polynomial in the two variables sin(''x'') and cos(''x'') (using [[List of trigonometric identities#Multiple-angle formulae]]). Conversely, every polynomial in sin(''x'') and cos(''x'') may be converted, with [[List of trigonometric identities#Product-to-sum and sum-to-product identities|Product-to-sum identities]], into a linear combination of functions sin(''nx'') and cos(''nx''). This equivalence explains why linear combinations are called polynomials.

For [[complex number|complex coefficients]], there is no difference between such a function and a finite [[Fourier series]].

Trigonometric polynomials are widely used, for example in [[trigonometric interpolation]] applied to the [[interpolation]] of [[periodic function]]s. They are used also in the [[discrete Fourier transform]].

===Matrix polynomials===
{{Main|Matrix polynomial}}
A [[matrix polynomial]] is a polynomial with [[square matrix|square matrices]] as variables.<ref>{{cite book |title=Matrix Polynomials |volume=58 |series=Classics in Applied Mathematics |first1=Israel |last1=Gohberg |first2=Peter |last2=Lancaster |first3=Leiba |last3=Rodman |publisher=[[Society for Industrial and Applied Mathematics]] |location=Lancaster, PA |year=2009 |origyear=1982 |isbn=978-0-89871-681-8 |zbl=1170.15300}}</ref> Given an ordinary, scalar-valued polynomial
:<math>P(x) = \sum_{i=0}^n{ a_i x^i} =a_0 + a_1 x+ a_2 x^2 + \cdots + a_n x^n, </math>
this polynomial evaluated at a matrix ''A'' is
:<math>P(A) = \sum_{i=0}^n{ a_i A^i} =a_0 I + a_1 A + a_2 A^2 + \cdots + a_n A^n,</math>
where ''I'' is the [[identity matrix]].{{sfn|Horn|Johnson|1990|p=36}}

A '''matrix polynomial equation''' is an equality between two matrix polynomials, which holds for the specific matrices in question. A '''matrix polynomial identity''' is a matrix polynomial equation which holds for all matrices ''A'' in a specified [[matrix ring]] ''M<sub>n</sub>''(''R'').

===Laurent polynomials===
{{Main|Laurent polynomial}}
[[Laurent polynomial]]s are like polynomials, but allow negative powers of the variable(s) to occur.

===Rational functions===
{{Main|Rational function}}
A [[rational fraction]] is the [[quotient]] ([[algebraic fraction]]) of two polynomials. Any [[algebraic expression]] that can be rewritten as a rational fraction is a [[rational function]].

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 zero.

The rational fractions include the Laurent polynomials, but do not limit denominators to powers of an indeterminate.

===Power series===
{{Main|Formal power series}}
[[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 [[irrational number]]s cannot), but the rules for manipulating their terms are the same as for polynomials. Non-formal [[power series]] also generalize polynomials, but the multiplication of two power series may not converge.

===Other examples===
* 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]].

==Applications==

===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|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'' − ''r''}}) divides {{math|''f''}}. The converse is also true. 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/?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/?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 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.

===Positional notation===
{{Main|Positional notation}}
In modern positional numbers systems, such as the [[Decimal|decimal system]], the digits and their positions in the representation of an integer, for example, 45, are a shorthand notation for a polynomial in the [[radix]] or base, in this case, {{nowrap|4 × 10<sup>1</sup> + 5 × 10<sup>0</sup>}}. As another example, in radix 5, a string of digits such as 132 denotes the (decimal) number {{nowrap|1 × 5<sup>2</sup> + 3 × 5<sup>1</sup> + 2 × 5<sup>0</sup>}} = 42. This representation is unique.  Let ''b'' be a positive integer greater than 1.  Then every positive integer ''a'' can be expressed uniquely in the form

:<math>a = r_m b^m + r_{m-1} b^{m-1} + \dotsb + r_1 b + r_0,</math>

where ''m'' is a nonnegative integer and the ''r'''s are integers such that

:{{math|0 < ''r''<sub>''m''</sub> < ''b''}} and {{math|0 ≤ ''r''<sub>''i''</sub> < ''b''}} for {{math|1=''i'' = 0, 1, . . . , ''m'' − 1}}.<ref>{{harvtxt|McCoy|1968|p=75}}</ref>

===Other applications===
{{See also|Orthogonal polynomials|B-spline|spline interpolation}}
Polynomials serve to approximate other [[function (mathematics)|functions]],<ref>{{cite book |last=de Villiers |first=Johann |title=Mathematics of Approximation |publisher=Springer |year=2012 |isbn=9789491216503 |url=https://books.google.com/?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 (discrete 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.

In [[computer graphics]] they are used to interpolate between values to evaluate animation of dynamic graphical objects.{{citation needed|date=February 2019}}

==History==
{{Main|Cubic function#History|Quartic function#History|Abel–Ruffini theorem#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, − 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]]
*[[Template:Mathematical expressions|Table of mathematical expressions]]
*[[Polynomial transformations]]
*[[Polynomial mapping]]
*[[Polynomial functor]]

==Notes==
{{Reflist|30em}}

==References==
<!-- *{{cite book |title= |publisher= |year= |isbn= |url=}} -->
{{Refbegin}}
*{{cite book |last=Barbeau |first=E.J. |title=Polynomials |publisher=Springer |year=2003 |isbn=978-0-387-40627-5 |url=https://books.google.com/?id=CynRMm5qTmQC&printsec=frontcover}}
*{{cite book |editor-last=Bronstein |editor-first=Manuel |title=Solving Polynomial Equations: Foundations, Algorithms, and Applications |publisher=Springer |year=2006 |isbn=978-3-540-27357-8 |url=https://books.google.com/?id=aIlSmBV3yf8C&printsec=frontcover|display-editors=etal}}
*{{cite book |last1=Cahen |first1=Paul-Jean |last2=Chabert |first2=Jean-Luc |title=Integer-Valued Polynomials |publisher=American Mathematical Society |year=1997 |isbn=978-0-8218-0388-2 |url=https://books.google.com/?id=AlAluH5is6AC&printsec=frontcover}}
*{{Lang Algebra}}. This classical book covers most of the content of this article.
*{{cite book |last=Leung |first=Kam-tim |title=Polynomials and Equations |publisher=Hong Kong University Press |year=1992 |isbn=9789622092716 |url=https://books.google.com/?id=v5uXkwIUbC8C&printsec=frontcover|display-authors=etal}}
*Mayr, K. Über die Auflösung algebraischer Gleichungssysteme durch hypergeometrische Funktionen. ''Monatshefte für Mathematik und Physik'' vol. 45, (1937) pp.&nbsp;280–313.
*{{citation | last1 = McCoy | first1 = Neal H. | title = Introduction To Modern Algebra, Revised Edition | location = Boston | publisher = [[Allyn and Bacon]] | year = 1968 | lccn = 68015225 }}
*{{cite book |last=Prasolov |first=Victor V. |title=Polynomials |publisher=Springer |year=2005 |isbn=978-3-642-04012-2 |url=https://books.google.com/?id=qIJPxdwSqlcC&printsec=frontcover}}
*{{cite book |last=Sethuraman |first=B.A. |chapter=Polynomials |title=Rings, Fields, and Vector Spaces: An Introduction to Abstract Algebra Via Geometric Constructibility |publisher=Springer |year=1997 |isbn=978-0-387-94848-5 |chapter-url=https://books.google.com/?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://gdz.sub.uni-goettingen.de/index.php?id=11&PID=GDZPPN00252161X&L=1 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://gdz.sub.uni-goettingen.de/index.php?id=11&PID=GDZPPN002526069&L=1 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.
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==External links==
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*{{cite web |url=http://mathdl.maa.org/mathDL/46/?pa=content&sa=viewDocument&nodeId=640&pf=1 |title=Euler's Investigations on the Roots of Equations |archiveurl=https://web.archive.org/web/20120924140505/http://mathdl.maa.org/mathDL/46/?pa=content&sa=viewDocument&nodeId=640&pf=1 |archivedate=September 24, 2012 |url-status=dead}}
*{{MathWorld |title=Polynomial |id=Polynomial}}

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