
In
mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a polynomial is an
expression consisting of
indeterminates (also called
variables) and
coefficient
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s, that involves only the operations of
addition
Addition (usually signified by the plus symbol
The plus and minus signs, and , are mathematical symbol
A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object
A mathematical object is an ...

,
subtraction
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...

,
multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

, and non-negative
integer
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...
exponentiation
Exponentiation is a mathematical
Mathematics (from Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europ ...
of variables. An example of a polynomial of a single indeterminate is . An example in three variables is .
Polynomials appear in many areas of mathematics and science. For example, they are used to form
polynomial equation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s, which encode a wide range of problems, from elementary
word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic
chemistry
Chemistry is the study of the properties and behavior of . It is a that covers the that make up matter to the composed of s, s and s: their composition, structure, properties, behavior and the changes they undergo during a with other .
...

and
physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of eve ...

to
economics
Economics () is a social science
Social science is the branch
A branch ( or , ) or tree branch (sometimes referred to in botany
Botany, also called , plant biology or phytology, is the science of plant life and a bran ...

and
social science
Social science is the branch
A branch ( or , ) or tree branch (sometimes referred to in botany
Botany, also called , plant biology or phytology, is the science of plant life and a branch of biology. A botanist, plant scientist o ...

; they are used in
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

and
numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). Numerical analysis ...
to approximate other functions. In advanced mathematics, polynomials are used to construct
polynomial ring
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s and
algebraic varieties
Algebraic varieties are the central objects of study in algebraic geometry
Algebraic geometry is a branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures ...
, which are central concepts in
algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

and
algebraic geometry
Algebraic geometry is a branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

.
Etymology
The word ''polynomial''
joins two diverse roots: the Greek ''poly'', meaning "many", and the Latin ''nomen'', or name. It was derived from the term ''
binomial'' by replacing the Latin root ''bi-'' with the Greek ''poly-''. That is, it means a sum of many terms (many
monomial
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
s). The word ''polynomial'' was first used in the 17th century.
Notation and terminology
The ''x'' occurring in a polynomial is commonly called 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
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
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.
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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
''P''(''x'') dates from a time when 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
:
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). 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
An image (from la, imago) is an artifact that depicts visual perception
Visual perception is the ability to interpret the surrounding environment
Environment most often refers to:
__NOTOC__
* Natural environment, all living and non- ...
of ''x'' by this function is the polynomial ''P'' itself (substituting ''x'' for ''x'' does not change anything). In other words,
:
which justifies formally the existence of two notations for the same polynomial.
Definition
A ''polynomial expression'' is an
expression that can be built from
constants and symbols called ''variables'' or ''indeterminates'' by means of
addition
Addition (usually signified by the plus symbol
The plus and minus signs, and , are mathematical symbol
A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object
A mathematical object is an ...

,
multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

and
exponentiation
Exponentiation is a mathematical
Mathematics (from Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europ ...
to a
non-negative integer
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
power. The constants are generally
number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ...

s, but may be any expression that do not involve the indeterminates, and represent
mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical proofs ...
s that can be added and multiplied. Two polynomial expressions are considered as defining the same ''polynomial'' if they may be transformed, one to the other, by applying the usual properties of
commutativity
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
,
associativity
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule ...
and
distributivity
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of addition and multiplication. For example
and
are two polynomial expressions that represent the same polynomial; so, one writes
A polynomial in a single indeterminate can always be written (or rewritten) in the form
:
where
are constants that are called the ''coefficients'' of the polynomial, and
is the indeterminate.
The word "indeterminate" means that
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
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
, called a ''polynomial function''.
This can be expressed more concisely by using
summation notation:
:
That is, a polynomial can either be zero or can be written as the sum of a finite number of non-zero
terms. Each term consists of the product of a number called the
coefficient
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of the term and a finite number of indeterminates, raised to nonnegative integer powers.
Classification
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 term with nonzero coefficient.
Because , 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
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
and a constant polynomial. 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).
For example:
:
is a term. The coefficient is , the indeterminates are and , the degree of is two, while the degree of 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 .
Forming a sum of several terms produces a polynomial. For example, the following is a polynomial:
:
It consists of three terms: the first is degree two, the second is degree one, and the third is degree zero.
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 polynomial
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
s'' 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, the term in is a linear term in a quadratic 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 −∞). The zero polynomial is also unique in that it is the only polynomial in one indeterminate that has an infinite number of
roots
A root
In vascular plants, the roots are the plant organ, organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They most often ...
. The graph of the zero polynomial, , is the ''x''-axis.
In the case of polynomials in more than one indeterminate, a polynomial is called ''homogeneous'' of if ''all'' of its non-zero terms have . The zero polynomial is homogeneous, and, as a homogeneous polynomial, its degree is undefined. For example, is homogeneous of degree 5. For more details, see
Homogeneous polynomial
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
.
The
commutative law
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
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 ", with the term of largest degree first, or in "ascending powers of ". The polynomial is written in descending powers of . The first term has coefficient , indeterminate , and exponent . In the second term, the coefficient . 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.
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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
, 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.
Polynomials can be classified by the number of terms with nonzero coefficients, so that a one-term polynomial is called a
monomial
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
, a two-term polynomial is called a
binomial, and a three-term polynomial is called a ''trinomial''. The term "quadrinomial" is occasionally used for a four-term polynomial.
A real polynomial is a polynomial with
real
Real may refer to:
* Reality
Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
coefficients. When it is used to define a
function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
, the
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
*Doma ...
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
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...
coefficients, and a complex polynomial is a polynomial with
complex
The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London
, mottoeng = Let all come who by merit deserve the most reward
, established =
, type = Public university, Public rese ...

coefficients.
A polynomial in one indeterminate is called a ''
univariate In mathematics, a univariate object is an expression, equation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geom ...
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 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 also common to say simply "polynomials in , and ", 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
Horner's method:
:
Arithmetic
Addition and subtraction
Polynomials can be added using the
associative law
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
of addition (grouping all their terms together into a single sum), possibly followed by reordering (using the
commutative law
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
) and combining of like terms.
For example, if
:
and
then the sum
:
can be reordered and regrouped as
:
and then simplified to
:
When polynomials are added together, the result is another polynomial.
Subtraction of polynomials is similar.
Multiplication
Polynomials can also be multiplied. To expand 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.
For example, if
:
then
:
Carrying out the multiplication in each term produces
:
Combining similar terms yields
:
which can be simplified to
:
As in the example, the product of polynomials is always a polynomial.
[
]
Composition
Given a polynomial of a single variable and another polynomial of any number of variables, the composition
Composition or Compositions may refer to:
Arts
* Composition (dance), practice and teaching of choreography
* Composition (music), an original piece of music and its creation
*Composition (visual arts)
The term composition means "putting togethe ...
is obtained by substituting each copy of the variable of the first polynomial by the second polynomial.[ For example, if and then
A composition may be expanded to a sum of terms using the rules for multiplication and division of polynomials. The composition of two polynomials is another polynomial.
]
Division
The division of one polynomial by another is not typically a polynomial. Instead, such ratios are a more general family of objects, called ''rational fraction
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
s'', ''rational expressions'', or ''rational function
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s'', depending on context. This is analogous to the fact that the ratio of two integer
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...
s is a rational number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...
, not necessarily an integer. For example, the fraction is not a polynomial, and it cannot be written as a finite sum of powers of the variable .
For polynomials in one variable, there is a notion of Euclidean division of polynomials
In algebra, the greatest common divisor (frequently abbreviated as GCD) of two polynomials is a polynomial, of the highest possible degree, that is a factor of both the two original polynomials. This concept is analogous to the greatest common ...
, generalizing the Euclidean division
In arithmetic, Euclidean division – or division with remainder – is the process of division (mathematics), dividing one integer (the dividend) by another (the divisor), in a way that produces a quotient and a remainder smaller than the divisor ...
of integers. This notion of the division results in two polynomials, a ''quotient'' and a ''remainder'' , such that and . The quotient and remainder may be computed by any of several algorithms, including polynomial long division
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
and synthetic division
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
.
When the denominator is monic and linear, that is, for some constant , then the polynomial remainder theorem
In algebra, the polynomial remainder theorem or little Bézout's theorem (named after Étienne Bézout) is an application of Euclidean division of polynomials. It states that the remainder of the division of a polynomial f(x) by a linear polynomia ...
asserts that the remainder of the division of by is the evaluation
Evaluation is a
system
A system is a group of interacting
Interaction is a kind of action that occurs as two or more objects have an effect upon one another. The idea of a two-way effect is essential in the concept of interaction, as oppos ...
.[ In this case, the quotient may be computed by ]Ruffini's rule
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, a special case of synthetic division.
Factoring
All polynomials with coefficients in a unique factorization domain
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
(for example, the integers or a field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grassl ...
) also have a factored form in which the polynomial is written as a product of irreducible polynomial
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
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
In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

s, the irreducible factors are linear. Over the real number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s, they have the degree either one or two. Over the integers and the rational number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...
s the irreducible factors may have any degree. For example, the factored form of
:
is
:
over the integers and the reals, and
:
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 polynomial factorization
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
algorithm
In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always and are used as specifications for performing s, , , and other ...

s are available in most computer algebra system
A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software
Mathematical software is software used to mathematical model, model, analyze or calculate numeric, symbolic or geometric data.
It is a type of applica ...

s.
Calculus
Calculating derivative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

s and integrals of polynomials is particularly simple, compared to other kinds of functions.
The derivative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

of the polynomial with respect to is the polynomial
Similarly, the general antiderivative
In calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zer ...
(or indefinite integral) of is
where is an arbitrary constant. For example, antiderivatives of have the form .
For polynomials whose coefficients come from more abstract settings (for example, if the coefficients are integers modulo some prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
, or elements of an arbitrary ring), the formula for the derivative can still be interpreted formally, with the coefficient understood to mean the sum of copies of . For example, over the integers modulo , the derivative of the polynomial is the polynomial .
Polynomial functions
A ''polynomial function'' is a function that can be defined by evaluating a polynomial. More precisely, a function of one argument
In logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, lab ...
from a given domain is a polynomial function if there exists a polynomial
:
that evaluates to for all in the domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
*Doma ...
of (here, is a non-negative integer and are constant coefficients).
Generally, unless otherwise specified, polynomial functions have complex
The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London
, mottoeng = Let all come who by merit deserve the most reward
, established =
, type = Public university, Public rese ...

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 restricted to the reals, the resulting function is a real function
In mathematical analysis, and applications in geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. ...
that maps reals to reals.
For example, the function , defined by
:
is a polynomial function of one variable. Polynomial functions of several variables are similarly defined, using polynomials in more than one indeterminate, as in
:
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 which takes the same values as the polynomial on the interval