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 midline dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...
, and nonnegative
integer
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the IndoEuropean 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=aljabr) 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
:
$a\backslash mapsto\; P(a),$
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,
:
$P(x)=P,$
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 midline 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
nonnegative 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
$(x1)(x2)$ and
$x^23x+2$ are two polynomial expressions that represent the same polynomial; so, one writes
$(x1)(x2)=x^23x+2.$
A polynomial in a single indeterminate can always be written (or rewritten) in the form
:
$a\_n\; x^n\; +\; a\_x^\; +\; \backslash dotsb\; +\; a\_2\; x^2\; +\; a\_1\; x\; +\; a\_0,$
where
$a\_0,\; \backslash ldots,\; a\_n$ are constants that are called the ''coefficients'' of the polynomial, and
$x$ is the indeterminate.
The word "indeterminate" means that
$x$ 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:
:
$\backslash sum\_^n\; a\_k\; x^k$
That is, a polynomial can either be zero or can be written as the sum of a finite number of nonzero
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:
:
$5x^2y$
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:
:
$\backslash underbrace\_\; \backslash underbrace\_\; \backslash underbrace\_.$
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=aljabr) 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 nonzero 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 nonzero 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 oneterm 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 twoterm polynomial is called a binomial, and a threeterm polynomial is called a ''trinomial''. The term "quadrinomial" is occasionally used for a fourterm 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 IndoEuropean 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 nonconstant 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:
:$(((((a\_n\; x\; +\; a\_)x\; +\; a\_)x\; +\; \backslash dotsb\; +\; a\_3)x\; +\; a\_2)x\; +\; a\_1)x\; +\; a\_0.$
]
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
:$P\; =\; 3x^2\; \; 2x\; +\; 5xy\; \; 2$ and $Q\; =\; 3x^2\; +\; 3x\; +\; 4y^2\; +\; 8$
then the sum
:$P\; +\; Q\; =\; 3x^2\; \; 2x\; +\; 5xy\; \; 2\; \; 3x^2\; +\; 3x\; +\; 4y^2\; +\; 8$
can be reordered and regrouped as
:$P\; +\; Q\; =\; (3x^2\; \; 3x^2)\; +\; (\; 2x\; +\; 3x)\; +\; 5xy\; +\; 4y^2\; +\; (8\; \; 2)$
and then simplified to
:$P\; +\; Q\; =\; x\; +\; 5xy\; +\; 4y^2\; +\; 6.$
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
:$\backslash begin\; \backslash color\; P\; \&\backslash color\; \backslash \backslash \; \backslash color\; Q\; \&\backslash color\; \backslash end$
then
:$\backslash begin\; \&\; \&\&(\backslash cdot)\; \&+\&(\backslash cdot)\&+\&(\backslash cdot\; )\&+\&(\backslash cdot)\; \backslash \backslash \&\&+\&(\backslash cdot)\&+\&(\backslash cdot)\&+\&(\backslash cdot\; )\&+\&\; (\backslash cdot)\; \backslash \backslash \&\&+\&(\backslash cdot)\&+\&(\backslash cdot)\&+\&\; (\backslash cdot\; )\&+\&(\backslash cdot)\; \backslash end$
Carrying out the multiplication in each term produces
:$\backslash begin\; PQ\; \&\; =\; \&\&\; 4x^2\; \&+\&\; 10xy\; \&+\&\; 2x^2y\; \&+\&\; 2x\; \backslash \backslash \; \&\&+\&\; 6xy\; \&+\&\; 15y^2\; \&+\&\; 3xy^2\; \&+\&\; 3y\; \backslash \backslash \; \&\&+\&\; 10x\; \&+\&\; 25y\; \&+\&\; 5xy\; \&+\&\; 5.\; \backslash end$
Combining similar terms yields
:$\backslash begin\; PQ\; \&\; =\; \&\&\; 4x^2\; \&+\&(\; 10xy\; +\; 6xy\; +\; 5xy\; )\; \&+\&\; 2x^2y\; \&+\&\; (\; 2x\; +\; 10x\; )\; \backslash \backslash \; \&\&\; +\; \&\; 15y^2\; \&+\&\; 3xy^2\; \&+\&(\; 3y\; +\; 25y\; )\&+\&5\; \backslash end$
which can be simplified to
:$PQ\; =\; 4x^2\; +\; 21xy\; +\; 2x^2y\; +\; 12x\; +\; 15y^2\; +\; 3xy^2\; +\; 28y\; +\; 5.$
As in the example, the product of polynomials is always a polynomial.][
]
Composition
Given a polynomial $f$ 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 ...
$f\; \backslash circ\; g$ is obtained by substituting each copy of the variable of the first polynomial by the second polynomial.[ For example, if $f(x)\; =\; x^2\; +\; 2x$ and $g(x)\; =\; 3x\; +\; 2$ then
$$(f\backslash circ\; g)(x)\; =\; f(g(x))\; =\; (3x\; +\; 2)^2\; +\; 2(3x\; +\; 2).$$
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=aljabr) 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 IndoEuropean 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=aljabr) 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=aljabr) 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 twoway 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
:$5x^35$
is
:$5(x\; \; 1)\backslash left(x^2\; +\; x\; +\; 1\backslash right)$
over the integers and the reals, and
:$5(x\; \; 1)\backslash left(x\; +\; \backslash frac\backslash right)\backslash left(x\; +\; \backslash frac\backslash right)$
over the complex numbers.
The computation of the factored form, called ''factorization'' is, in general, too difficult to be done by handwritten 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 , computerimplementable 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 $$P\; =\; a\_n\; x^n\; +\; a\_\; x^\; +\; \backslash dots\; +\; a\_2\; x^2\; +\; a\_1\; x\; +\; a\_0\; =\; \backslash sum\_^n\; a\_i\; x^i$$ with respect to is the polynomial
$$n\; a\_n\; x^\; +\; (n\; \; 1)a\_\; x^\; +\; \backslash dots\; +\; 2\; a\_2\; x\; +\; a\_1\; =\; \backslash sum\_^n\; i\; a\_i\; x^.$$
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 $P$ is
$$\backslash frac\; +\; \backslash frac\; +\; \backslash dots\; +\; \backslash frac\; +\; \backslash frac\; +\; a\_0\; x\; +\; c\; =\; c\; +\; \backslash sum\_^n\; \backslash frac$$
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
:$a\_n\; x^n\; +\; a\_\; x^\; +\; \backslash cdots\; +\; a\_2\; x^2\; +\; a\_1\; x\; +\; a\_0$
that evaluates to $f(x)$ 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 nonnegative 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
:$f(x)\; =\; x^3\; \; x,$
is a polynomial function of one variable. Polynomial functions of several variables are similarly defined, using polynomials in more than one indeterminate, as in
:$f(x,y)=\; 2x^3+4x^2y+xy^5+y^27.$
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 $\backslash left(\backslash sqrt\backslash right)^2,$ which takes the same values as the polynomial $1x^2$ on the interval $;\; href="/html/ALL/s/1,1.html"\; ;"title="1,1">1,1$continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ga ...
, smooth
Smooth may refer to:
Mathematics
* Smooth function
is a smooth function with compact support.
In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuo ...
, and entire
*In philately, see Cover (philately), Cover
*In mathematics, see Entire function
*In animal fancy and animal husbandry, entire (animal), entire indicates that an animal has not been desexed, that is, spayed or neutered
*In botany, an edge (such as o ...
.
Graphs
File:Algebra1 fnz fig037 pc.svg, Polynomial of degree 0:
File:Fonction de Sophie Germain.png, Polynomial of degree 1:
File:Polynomialdeg2.svg, Polynomial of degree 2:
File:Polynomialdeg3.svg, Polynomial of degree 3:
File:Polynomialdeg4.svg, Polynomial of degree 4:
File:Quintic polynomial.svg, Polynomial of degree 5:
File:Sextic Graph.svg, Polynomial of degree 6:
File:Septic graph.svg, Polynomial of degree 7:
A polynomial function in one real variable can be represented by a graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discret ...
.

The graph of the zero polynomial
is the axis.

The graph of a degree 0 polynomial
is a horizontal line with

The graph of a degree 1 polynomial (or linear function)
is an oblique line with and
slope
In mathematics, the slope or gradient of a line
Line, lines, The Line, or LINE may refer to:
Arts, entertainment, and media Films
* ''Lines'' (film), a 2016 Greek film
* ''The Line'' (2017 film)
* ''The Line'' (2009 film)
* ''The Line'', ...
.

The graph of a degree 2 polynomial
is a
parabola
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 ...
.

The graph of a degree 3 polynomial
is a
cubic curve
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 ...
.

The graph of any polynomial with degree 2 or greater
is a continuous nonlinear curve.
A nonconstant polynomial function when the variable increases indefinitely (in absolute value
In , the absolute value or modulus of a , denoted , is the value of without regard to its . Namely, if is , and if is (in which case is positive), and . For example, the absolute value of 3 is 3, and the absolute value of − ...
). If the degree is higher than one, the graph does not have any asymptote
In analytic geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo'' "earth", ''wikt:μέ ...
. It has two parabolic branches 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
A ''polynomial equation'', also called an ''algebraic 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 ...
'', is an equation
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 the form
:$a\_n\; x^n\; +\; a\_x^\; +\; \backslash dotsb\; +\; a\_2\; x^2\; +\; a\_1\; x\; +\; a\_0\; =\; 0.$
For example,
:$3x^2\; +\; 4x\; 5\; =\; 0$
is a polynomial equation.
When considering equations, the indeterminates (variables) of polynomials are also called unknown
Unknown or The Unknown may refer to:
Film
* The Unknown (1915 comedy film), ''The Unknown'' (1915 comedy film), a silent boxing film
* The Unknown (1915 drama film), ''The Unknown'' (1915 drama film)
* The Unknown (1927 film), ''The Unknown'' (19 ...
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
Identity may refer to:
Social sciences
* Identity (social science), personhood or group affiliation in psychology and sociology
Group expression and affiliation
* Cultural identity, a person's selfaffiliation (or categorization by others ...
'' like , 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
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=aljabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...
, methods such as the quadratic formula
In elementary algebra
Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas a ...
are taught for solving all first degree and second degree polynomial equations in one variable. There are also formulas for the cubic
Cubic may refer to:
Science and mathematics
* Cube (algebra), "cubic" measurement
* Cube, a threedimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex
** Cubic crystal system
200px, A network ...
and quartic 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. For higher degrees, the Abel–Ruffini theorem
In mathematics, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no solution in radicals to general algebraic equation, polynomial equations of quintic equation, degree five or higher with arbitrary coef ...
asserts that there can not exist a general formula in radicals. However, rootfinding algorithm
In mathematics and computing, a rootfinding algorithm is an algorithm for finding Zero of a function, zeroes, also called "roots", of continuous functions. A zero of a function , from the real numbers to real numbers or from the complex numbers to ...
s may be used to find numerical approximation
(c. 1800–1600 BC) with annotations. The approximation of the square root of 2 is four sexagesimal figures, which is about six decimal figures. 1 + 24/60 + 51/602 + 10/603 = 1.41421296...
Numerical analysis is the study of algorithms that use num ...
s of the roots of a polynomial expression of any degree.
The number of solutions of a polynomial equation with real coefficients may not exceed the degree, and equals the degree when the 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 ...
solutions are counted with their multiplicity. This fact is called the fundamental theorem of algebra
The fundamental theorem of algebra states that every non constant singlevariable polynomial
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, s ...
.
Solving equations
A ''root'' of a nonzero univariate polynomial is a value of such that . In other words, a root of is a solutions of the 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 ...
or a zero
0 (zero) is a 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 ...
of the polynomial function defined by . In the case of the zero polynomial, every number is ia zero of the corresponding function, and the concept of root is rarely cosidered.
A number is a root of a polynomial if and only if the linear 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 ...
divides , that is if there is another polynomial such that . It may happen that a power (greater than ) of divides ; in this case, is a ''multiple root'' of , and otherwise is simple root of . If is a nonzero polynomial, there is a highest power such that divides , which is called the ''multiplicity'' of as a root of . The number of roots of a nonzero polynomial , counted with their respective multiplicities, cannot exceed the degree of , and equals this degree if all 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 ...
roots are considered (this is a consequence of the fundamental theorem of algebra
The fundamental theorem of algebra states that every non constant singlevariable polynomial
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, s ...
.
The the coefficients of a polynomial and its roots are related by Vieta's formulas
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 ...
.
Some polynomials, such as , do not have any roots among 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. If, however, the set of accepted solutions is expanded to the 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, every nonconstant polynomial has at least one root; this is the fundamental theorem of algebra
The fundamental theorem of algebra states that every non constant singlevariable polynomial
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, s ...
. By successively dividing out factors , 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 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 is . 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 expressionIn 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 ha ...
; for example the golden ratio
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 ...
$(1+\backslash sqrt\; 5)/2$ is the unique positive solution of $x^2x1=0.$ In the ancient times, they succeeded only for degrees one and two. For quadratic equation
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=aljabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. ...
s, the quadratic formula
In elementary algebra
Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas a ...
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
roots
A root is the part of a plant that most often lies below the surface of the soil but can also be aerial or aerating, that is, growing up above the ground or especially above water.
Root or roots may also refer to:
Art, entertainment, a ...
and quartic 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 ...
). But formulas for degree 5 and higher eluded researchers for several centuries. In 1824, Niels Henrik Abel
Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics ...
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 mathematics, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no solution in radicals to general algebraic equation, polynomial equations of quintic equation, degree five or higher with arbitrary coef ...
). In 1830, Évariste Galois
Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by Nth root, radical ...
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
In , Galois theory, originally introduced by , provides a connection between and . This connection, the , allows reducing certain problems in field theory to group theory, which makes them simpler and easier to understand.
Galois introduced the ...
and group theory
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 ...
, two important branches of modern algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=aljabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...
. 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
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=aljabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...
and sextic equation
233px, Graph of a sextic function, with 6 real number, real root of a function, roots (crossings of the axis) and 5 critical point (mathematics), critical points. Depending on the number and vertical locations of minimum, minima and maxima, the se ...
).
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
(c. 1800–1600 BC) with annotations. The approximation of the square root of 2 is four sexagesimal figures, which is about six decimal figures. 1 + 24/60 + 51/602 + 10/603 = 1.41421296...
Numerical analysis is the study of algorithms that use num ...
s of the solutions. There are many methods for that; some are restricted to polynomials and others may apply to any continuous function
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 ...
. The most efficient algorithm
In and , an algorithm () is a finite sequence of , computerimplementable 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 allow solving easily (on a computer
A computer is a machine that can be programmed to Execution (computing), carry out sequences of arithmetic or logical operations automatically. Modern computers can perform generic sets of operations known as Computer program, programs. These ...
) polynomial equations of degree higher than 1,000 (see Rootfinding algorithm
In mathematics and computing, a rootfinding algorithm is an algorithm for finding Zero of a function, zeroes, also called "roots", of continuous functions. A zero of a function , from the real numbers to real numbers or from the complex numbers to ...
).
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
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 ...
. For a set of polynomial equations in several unknowns, there are algorithm
In and , an algorithm () is a finite sequence of , computerimplementable 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 to decide whether they have a finite number of 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 ...
solutions, and, if this number is finite, for computing the solutions. See System of polynomial equations
A system of polynomial equations (sometimes simply a polynomial system) is a set of simultaneous equations
In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for ...
.
The special case where all the polynomials are of degree one is called a system of linear equations
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 which another range of different solution methods exist, including the classical Gaussian elimination.
A polynomial equation for which one is interested only in the solutions which are integer
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the IndoEuropean languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...
s is called a Diophantine 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). I ...
. Solving Diophantine equations is generally a very hard task. It has been proved that there cannot be any general algorithm
In and , an algorithm () is a finite sequence of , computerimplementable 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 ...
for solving them, and even for deciding whether the set of solutions is empty (see Hilbert's tenth problem
Hilbert's tenth problem is the tenth on the list of mathematical problems that the German mathematician David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and one of the most influential mathemat ...
). 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
In number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number ...
.
Generalizations
There are several generalizations of the concept of polynomials.
Trigonometric polynomials
A trigonometric polynomial is a finite linear combination
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 ...
of functions
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 ...
sin(''nx'') and cos(''nx'') with ''n'' taking on the values of one or more natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
s. The coefficients may be taken as real numbers, for realvalued 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#Multipleangle formulae). Conversely, every polynomial in sin(''x'') and cos(''x'') may be converted, with Producttosum identities, into a linear combination of functions sin(''nx'') and cos(''nx''). This equivalence explains why linear combinations are called polynomials.
For , there is no difference between such a function and a finite Fourier series
In mathematics, a Fourier series () is a periodic function composed of harmonically related Sine wave, sinusoids combined by a weighted summation. With appropriate weights, one cycle (or ''period'') of the summation can be made to approximate an ...
.
Trigonometric polynomials are widely used, for example in trigonometric interpolation applied to the interpolation
In the 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 Europe. Its populatio ...
of periodic function
A periodic function 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 logica ...
s. They are used also in the discrete Fourier transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equallyspaced Sampling (signal processing), samples of a function (mathematics), function into a samelength sequence of equallyspaced samples of the discret ...
.
Matrix polynomials
A matrix polynomial is a polynomial with square matrices
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 ...
as variables. Given an ordinary, scalarvalued polynomial
:$P(x)\; =\; \backslash sum\_^n\; =a\_0\; +\; a\_1\; x+\; a\_2\; x^2\; +\; \backslash cdots\; +\; a\_n\; x^n,$
this polynomial evaluated at a matrix ''A'' is
:$P(A)\; =\; \backslash sum\_^n\; =a\_0\; I\; +\; a\_1\; A\; +\; a\_2\; A^2\; +\; \backslash cdots\; +\; a\_n\; A^n,$
where ''I'' is the identity matrix
In linear algebra, the identity matrix of size ''n'' is the ''n'' × ''n'' square matrix
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...
.
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
In abstract algebra, a matrix ring is a set of matrices
Matrix or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics)
In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of ...
''M_{n}''(''R'').
Laurent polynomials
Laurent 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). It ...
s are like polynomials, but allow negative powers of the variable(s) to occur.
Rational functions
A rational fraction
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=aljabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
is the quotient
In arithmetic
Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', ...
(algebraic fraction
In algebra, an algebraic fraction is a fraction (mathematics), fraction whose numerator and denominator are algebraic expressions. Two examples of algebraic fractions are \frac and \frac. Algebraic fractions are subject to the same laws as arithmeti ...
) of two polynomials. Any algebraic expressionIn 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 ha ...
that can be rewritten as a rational fraction is a 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 ...
.
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
Formal power series
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 gener ...
are like polynomials, but allow infinitely many nonzero 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
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 cannot), but the rules for manipulating their terms are the same as for polynomials. Nonformal power series
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 ...
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 , may be called an exponential polynomial.
Polynomial ring
A ''polynomial'' over a commutative ring
In , a branch of , a commutative ring is a in which the multiplication operation is . The study of commutative rings is called . Complementarily, is the study of s where multiplication is not required to be commutative.
Definition and first e ...
is a polynomial whose all coefficients belong to . It is straightforward to verify that the polynomials in a given set of indeterminates over form a commutative ring, called the ''polynomial ring'' in these indeterminates, denoted $R;\; href="/html/ALL/s/.html"\; ;"title="">$
So, most of the theory of the multivariate case can be reduced to an iterated univariate case.
The map from to sending to itself considered as a constant polynomial is an injective ring homomorphism
In ring theory, a branch of abstract algebra, a ring homomorphism is a structurepreserving function (mathematics), function between two ring (algebra), rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function s ...
, by which is viewed as a subring of . In particular, is an algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=aljabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...
over .
One can think of the ring as arising from by adding one new element ''x'' to ''R'', and extending in a minimal way to a ring in which satisfies no other relations than the obligatory ones, plus commutation with all elements of (that is ). To do this, one must add all powers of and their linear combinations as well.
Formation of the polynomial ring, together with forming factor rings by factoring out ideals
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...
, 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 over the real numbers by factoring out the ideal of multiples of the polynomial . Another example is the construction of finite field
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, which proceeds similarly, starting out with the field of 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 ...
as the coefficient ring (see modular arithmetic #REDIRECT Modular arithmetic#REDIRECT Modular arithmetic
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), a ...
).
If is commutative, then one can associate with every polynomial in a ''polynomial function'' with domain and range equal to . (More generally, one can take domain and range to be any same unital associative algebra
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 ...
over .) One obtains the value by substitution
Substitution may refer to:
Arts and media
*Chord substitution, in music, swapping one chord for a related one within a chord progression
*Substitution (poetry), a variation in poetic scansion
*Substitution (song), "Substitution" (song), a 2009 so ...
of the value for the symbol in . 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
Fermat's little theorem states that if is a prime number, then for any integer , the number is an integer multiple of . In the notation of modular arithmetic, this is expressed as
:a^p \equiv a \pmod p.
For example, if = 2 and = 7, then 27 = ...
for an example where is the integers modulo ). This is not the case when is the real or complex numbers, whence the two concepts are not always distinguished in analysis
Analysis is the process of breaking a complex topic or substance
Substance may refer to:
* Substance (Jainism), a term in Jain ontology to denote the base or owner of attributes
* Chemical substance, a material with a definite chemical composit ...
. An even more important reason to distinguish between polynomials and polynomial functions is that many operations on polynomials (like 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 ...
) require looking at what a polynomial is composed of as an expression rather than evaluating it at some constant value for .
Divisibility
If is an integral domain
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 ...
and and are polynomials in , it is said that ''divides'' or is a divisor of if there exists a polynomial in such that . If $a\backslash in\; R,$ then is a root of if and only $xa$ divides . In this case, the quotient can be computed using the polynomial long division
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=aljabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
.
If is 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 ...
and and are polynomials in with , then there exist unique polynomials and in with
:$f\; =\; q\; \backslash ,\; g\; +\; r$
and such that the degree of is smaller than the degree of (using the convention that the polynomial 0 has a negative degree). The polynomials and are uniquely determined by and . This is called ''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 ...
, division with remainder'' or ''polynomial long division'' and shows that the ring is a Euclidean domain
In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a #Definition, Euclidean function which allows a suitable generalization of the Euclidean division of ...
.
Analogously, ''prime polynomials'' (more correctly, ''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'') can be defined as ''nonzero polynomials which cannot be factorized into the product of two nonconstant polynomials''. In the case of coefficients in a ring, ''"nonconstant"'' must be replaced by ''"nonconstant or nonunit
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* Unit (album), ...
"'' (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
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 ...
this decomposition is unique up to the order of the factors and the multiplication of any nonunit 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
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 ...
). These algorithms are not practicable for handwritten computation, but are available in any 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 ...
. Eisenstein's criterionIn 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 ha ...
can also be used in some cases to determine irreducibility.
Applications
Positional notation
In modern positional numbers systems, such as the decimal systemDecimal system may refer to:
* Decimal (base ten) number system, used in mathematics for writing numbers and performing arithmetic
* Dewey Decimal Classification, Dewey Decimal System, a subject classification system used in libraries
* Decimal curr ...
, the digits and their positions in the representation of an integer, for example, 45, are a shorthand notation for a polynomial in the radix
In a positional numeral system
Positional notation (or placevalue notation, or positional numeral system) usually denotes the extension to any of the (or ). More generally, a positional system is a numeral system in which the contribution ...
or base, in this case, . As another example, in radix 5, a string of digits such as 132 denotes the (decimal) number = 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
:$a\; =\; r\_m\; b^m\; +\; r\_\; b^\; +\; \backslash dotsb\; +\; r\_1\; b\; +\; r\_0,$
where ''m'' is a nonnegative integer and the ''rs are integers such that
: and for .
Interpolation and approximation
The simple structure of polynomial functions makes them quite useful in analyzing general functions using polynomial approximations. An important example 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 ...
is Taylor's theorem
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 zero. ...
, which roughly states that every differentiable 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). It ...
locally looks like a polynomial function, and the Stone–Weierstrass theoremIn mathematical analysis, the Weierstrass approximation theorem states that every continuous function
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical struct ...
, which states that every continuous function
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 ...
defined on a compact
Compact as used in politics may refer broadly to a pact
A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations
International relations (IR), international affairs (IA) or internationa ...
interval of the real axis can be approximated on the whole interval as closely as desired by a polynomial function. Practical methods of approximation include polynomial interpolation
In numerical analysis
(c. 1800–1600 BC) with annotations. The approximation of the square root of 2 is four sexagesimal figures, which is about six decimal figures. 1 + 24/60 + 51/602 + 10/603 = 1.41421296...
Numerical analysis is the study of ...
and the use of splines.
Other applications
Polynomials are frequently used to encode information about some other object. The characteristic polynomial
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and ...
of a matrix or linear operator contains information about the operator's eigenvalue
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces an ...
s. The minimal polynomial
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (mathematics), ring (which is also a commutative algebra (structure), commutative algebra) formed from the Set (mathematics), set of polynomial ...
of an algebraic element
In mathematics, if is a field extension of , then an element of is called an algebraic element over , or just algebraic over , if there exists some nonzero polynomial with coefficients in such that . Elements of which are not algebraic over ...
records the simplest algebraic relation satisfied by that element. The chromatic polynomial
The chromatic polynomial is a graph polynomialIn mathematics, a graph polynomial is a Graph property, graph invariant whose values are polynomials. Invariants of this type are studied in algebraic graph theory.
Important graph polynomials include: ...
of a graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discret ...
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
Computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved by a computer. A computation problem is solvable by ...
the phrase ''polynomial time
In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by th ...
'' means that the time it takes to complete an algorithm
In and , an algorithm () is a finite sequence of , computerimplementable 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 ...
is bounded by a polynomial function of some variable, such as the size of the input.
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 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 .
History of the notation
The earliest known use of the equal sign is in Robert Recorde
Robert Recorde (c. 1512 – 1558) was a Welsh physician and mathematician. He invented the equals sign (=) and also introduced the preexisting plus sign
The plus and minus signs, and , are mathematical symbols used to represent the notions ...
's ''The Whetstone of Witte
400 px, The passage in ''The Whetstone of Witte'' introducing the equals sign
''The Whetstone of Witte'' is the shortened title of Robert Recorde's mathematics book published in 1557, the full title being ''The whetstone of witte, whiche is the se ...
'', 1557. The signs + for addition, − for subtraction, and the use of a letter for an unknown appear in Michael Stifel
Michael Stifel or Styfel (1487 – April 19, 1567) was a German monk, Protestant reformer and mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the ...
's ''Arithemetica integra'', 1544. René Descartes
René Descartes ( or ; ; Latinisation of names, Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a Frenchborn philosopher, Mathematics, mathematician, and scientist who spent a large portion of his working life in the Du ...
, 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 's denote constants and denotes a variable. Descartes introduced the use of superscripts to denote exponents as well.
See also
* List of polynomial topics
*
*Polynomial sequence
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 ...
*
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Notes
References
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External links
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Algebra