Polynomial

In mathematics, a polynomial is an expression constructed from one or more variables and constants, using the operations of addition, subtraction, multiplication, and constant positive whole number exponents. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In mathematics the word expression is a term for any well-formed combination of mathematical symbols A variable (ˈvɛərɪəbl is an Attribute of a physical or an abstract System which may change its Value while it is under Observation. For example, $x^2 - 4x + 7\,$ is a polynomial, but $x^2 - 4/x + 7x^{3/2}\,$ is not because it involves division by a variable and has an exponent that is not a positive whole number.

Polynomials are one of the most important concepts in algebra and throughout mathematics and science. They are used to form polynomial equations, which encode a wide range of problems, from elementary story problems to complicated problems in the sciences; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics, and are used in calculus and numerical analysis to approximate other functions. Abstract algebra has an unrelated term Word problem for groups. Chemistry (from Egyptian kēme (chem meaning "earth") is the Science concerned with the composition structure and properties Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Economics is the social science that studies the production distribution, and consumption of goods and services. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives Numerical analysis is the study of Algorithms for the problems of continuous mathematics (as distinguished from Discrete mathematics) Polynomials are used to construct polynomial rings, one of the most powerful concepts in algebra and algebraic geometry. In Mathematics, especially in the field of Abstract algebra, a polynomial ring is a ring formed from the set of Polynomials in one or more variables Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with

1 Overview
2 Elementary properties of polynomials
3 More advanced examples of polynomials
4 History
- 4.1 Notation
5 Solving polynomial equations
6 Graphs
7 Polynomials and calculus
8 Abstract algebra
9 Divisibility
10 Classifications
11 Extensions of the concept of a polynomial
12 See also
13 References
14 External links

Overview

A polynomial is either zero, or can be written as the sum of one or more non-zero terms. The word term is from the Latin terminus "boundary line limit" from the Indo-European root ter- "peg post boundary" The number of terms is finite. These terms consist of a constant (called the coefficient of the term) multiplied by zero or more variables (which are usually represented by letters). In Mathematics, a coefficient is a Constant multiplicative factor of a certain object A variable (ˈvɛərɪəbl is an Attribute of a physical or an abstract System which may change its Value while it is under Observation. Each variable may have an exponent which is a non-negative integer. The exponent on a variable in a term is equal to the degree of that variable in that term. When a Polynomial is expressed as a sum or difference of terms (e Since $x = x 1$ , the degree of a variable without a written exponent is one. A term with no variables is called a constant term, or just a constant. In Mathematics, the constant term of a Polynomial is the term of degree 0 The degree of a constant term is 0. The coefficient of a term may be any number, including fractions, irrational numbers, negative numbers, and complex numbers.

For example,

$-5x^2y\,$

is a term. The coefficient is –5, the variables are x and y, the degree of x is two, and the degree of y is one. In Mathematics, a coefficient is a Constant multiplicative factor of a certain object

The degree of the entire term is the sum of the degrees of each variable in it. In the example above, the degree is 2 + 1 = 3.

A polynomial is a sum of terms. For example, the following is a polynomial:

$3x^2 - 5x + 4\,.$

It consists of three terms: the first is degree two, the second is degree one, and the third is degree zero. Here "− 5x" stands for "+ (−5)x", so the coefficient of the middle term is −5.

When a polynomial in one variable is arranged in the traditional order, the terms of higher degree come before the terms of lower degree. In the first term above, the coefficient is 3, the variable is x, and the exponent is 2. In the second term, the coefficient is –5. The third term is a constant. The degree of a non-zero polynomial is the largest degree of any one term. In the example, the polynomial has degree two.

Alternative forms

An expression that can be converted to polynomial form through a sequence of applications of the commutative, associative, and distributive laws is usually considered to be a polynomial. In Mathematics, commutativity is the ability to change the order of something without changing the end result In Mathematics, associativity is a property that a Binary operation can have In Mathematics, and in particular in Abstract algebra, distributivity is a property of Binary operations that generalises the distributive law For instance

(x + 1) 3

is a polynomial because it can be worked out to $x 3 + 3 x 2 + 3 x + 1$ . Similarly

$\frac{x^3}{12}$

is considered a valid term in a polynomial, even though it involves a division, because it is equivalent to $\tfrac{1}{12}x^3$ and $\tfrac{1}{12}$ is just a constant. The coefficient of this term is therefore $\tfrac{1}{12}$ . For similar reasons, if complex coefficients are allowed, one may have a single term like $(2 + 3 i) x 3$ ; even though it looks like it should be worked out to two terms, the complex number 2+3i is in fact just a single coefficient in this case that happens to require a "+" to be written down.

Division by an expression containing a variable is not generally allowed in polynomials. ^[1] For example,

${1 \over x^2 + 1} \,$

is not a polynomial because it includes division by a variable. Similarly,

$( 5 + y ) ^ x ,\,$

is not a polynomial, because it has a variable exponent.

Since subtraction can be treated as addition of the additive opposite, and since exponentiation to a constant positive whole number power can be treated as repeated multiplication, polynomials can be constructed from constants and variables with just the two operations addition and multiplication.

Polynomial functions

A polynomial function is a function defined by evaluating a polynomial. In mathematics the word expression is a term for any well-formed combination of mathematical symbols A function ƒ of one argument is called a polynomial function if it satisfies

ƒ(x) = a_nxⁿ + a_n₋₁xⁿ⁻¹ + . . . + a₂x² + a₁x + a₀

for all arguments x, where n is a nonnegative integer and a₀, a₁,a₂, . . . , a_n are constant coefficients.

For example, the function f, taking real numbers to real numbers, defined by

f (x) = x 3 - x

is a polynomial function of one argument. Polynomial functions of multiple arguments can also be defined, using polynomials in multiple variables, as in

f (x, y) = 2 x 3 + 4 x 2 y + x y 5 + y 2 - 7

Polynomial functions are an important class of smooth functions. In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability

Polynomial equations

A polynomial equation is an equation in which a polynomial is set equal to another polynomial. An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent

$3x^2 + 4x -5 = 0 \,$

is a polynomial equation. In case of a polynomial equation the variable is considered an unknown, and one seeks to find the possible values for which both members of the equation evaluate to the same value (in general more than one solution may exist). A polynomial equation is to be contrasted with a polynomial identity like (x + y)(x – y) = x²–y², where both members represent the same polynomial in different forms, and as a consequence any evaluation of both members will give a valid equality.

Elementary properties of polynomials

A sum of polynomials is a polynomial
A product of polynomials is a polynomial
The derivative of a polynomial function is a polynomial function
Any primitive or antiderivative of a polynomial function is a polynomial function

Polynomials serve to approximate other functions, such as sine, cosine, and exponential. In Mathematics, a product is the Result of multiplying, or an expression that identifies factors to be multiplied In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose Derivative The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x)

All polynomials have an expanded form, in which the distributive law has been used to remove all parentheses. In Mathematics, and in particular in Abstract algebra, distributivity is a property of Binary operations that generalises the distributive law All polynomials with real or complex coefficients also have a factored form in which the polynomial is written as a product of linear polynomials. In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted For example, the polynomial

$x^2 - 2x - 3 \,$

is the expanded form of the polynomial

$(x - 3)(x + 1)\,$ ,

which is written in factored form. Note that the constants in the linear polynomials (like -3 and +1 in the above example) may be complex numbers in certain cases, even if all coefficients of the expanded form are real numbers. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted This is because the field of real numbers is not algebraically closed; however, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed. In Mathematics, a field F is said to be algebraically closed if every Polynomial in one Variable of degree at least 1 with Coefficients In Mathematics, the Fundamental theorem of algebra states that every non-constant single-variable Polynomial with complex coefficients has at Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted

In school algebra, students learn to move easily from one form to the other (see: factoring). In Mathematics, factorization ( also factorisation in British English) or factoring is the decomposition of an object (for

Every polynomial in one variable is equivalent to a polynomial with the form

$a_n x^n + a_{n-1}x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0$ .

This form is sometimes taken as the definition of a polynomial in one variable.

Evaluation of a polynomial consists of assigning a number to each variable and carrying out the indicated multiplications and additions. Evaluation is sometimes performed more efficiently using the Horner scheme

$((\ldots(a_n x + a_{n-1})x + ... + a_2)x + a_1)x + a_0\,$ . In Numerical analysis, the Horner scheme or Horner algorithm, named after William George Horner, is an Algorithm for the efficient evaluation

In elementary algebra, methods are given for solving all first degree and second degree polynomial equations in one variable. Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. In the case of polynomial equations, the variable is often called an unknown. The number of solutions may not exceed the degree, and will equal the degree when multiplicity of solutions and complex number solutions are counted. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted This fact is called the fundamental theorem of algebra. In Mathematics, the Fundamental theorem of algebra states that every non-constant single-variable Polynomial with complex coefficients has at

A system of polynomial equations is a set of equations in which a given variable must take on the same value everywhere it appears in any of the equations. Systems of equations are usually grouped with a single open brace on the left. In elementary algebra, methods are given for solving a system of linear equations in several unknowns. Elementary algebra is a fundamental and relatively basic form of Algebra taught to students who are presumed to have little or no formal knowledge of Mathematics beyond In Mathematics, a system of linear equations (or linear system) is a collection of Linear equations involving the same set of Variables For example To get a unique solution, the number of equations should equal the number of unknowns. If there are more unknowns than equations, the system is called underdetermined. In Mathematics, a system of linear equations (or linear system) is a collection of Linear equations involving the same set of Variables For example If there are more equations than unknowns, the system is called overdetermined. In Mathematics, a System of linear equations is considered overdetermined if there are more equations than unknowns This important subject is studied extensively in the area of mathematics known as linear algebra. Linear algebra is the branch of Mathematics concerned with Overdetermined systems are common in practical applications. For example, one U. S. mapping survey used computers to solve 2. 5 million equations in 400,000 unknowns. ^[2]

More advanced examples of polynomials

In linear algebra, the characteristic polynomial of a square matrix encodes several important properties of the matrix. Linear algebra is the branch of Mathematics concerned with In Linear algebra, one associates a Polynomial to every Square matrix, its characteristic polynomial. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally

In graph theory the chromatic polynomial of a graph encodes the different ways to vertex color the graph using x colors. In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects In Graph theory, graph coloring is a special case of Graph labeling; it is an assignment of labels traditionally called "colors" to elements of a In Mathematics and Computer science, a graph is the basic object of study in Graph theory. In Graph theory, graph coloring is a special case of Graph labeling; it is an assignment of labels traditionally called "colors" to elements of a

In abstract algebra, one may define polynomials with coefficients in any ring. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real

In knot theory the Alexander polynomial, the Jones polynomial, and the HOMFLY polynomial are important knot invariants. In Mathematics, knot theory is the area of Topology that studies mathematical knots While inspired by knots which appear in daily life in shoelaces In Mathematics, the Alexander polynomial is a Knot invariant which assigns a Polynomial with integer coefficients to each knot type In the mathematical field of Knot theory, the Jones polynomial is a Knot polynomial discovered by Vaughan Jones in 1983 In the mathematical field of Knot theory, the HOMFLY polynomial, sometimes called the HOMFLY-PT polynomial or the generalized Jones polynomial In the mathematical field of Knot theory, a knot invariant is a quantity (in a broad sense defined for each knot which is the same for equivalent knots

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 are 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. The Nine Chapters on the Mathematical Art ( is a Chinese Mathematics book composed by several generations of scholars from the 10th&ndash2nd century BC and " We would write $3 x + 2 y + z = 29$ .

Notation

Main article: History of mathematical notation

The earliest known use of the equal sign is in Robert Recorde's The Whetstone of Witte, 1557. Mathematical notation comprises the Symbols used to write mathematical Equations and Formulas It includes Arabic numerals, letters from the Robert Recorde (c 1510 &ndash 1558 was a Welsh Physician and Mathematician. The signs + for addition, − for subtraction, and the use of a letter for an unknown appear in Michael Stifel's Arithemetica integra, 1544. Michael Stifel or Styfel ( Esslingen 1486 or 1487 – April 19, 1567, Jena) was an Augustinian monk who became an early René Descartes, in La geometrie, 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 a 's denote constants and x denotes a variable. Descartes introduced the use of superscripts to denote exponents as well. ^[3]

Solving polynomial equations

Every polynomial corresponds to a polynomial function, where f(x) is set equal to the polynomial, and to a polynomial equation, where the polynomial is set equal to zero. The solutions to the equation are called the roots of the polynomial and they are the zeroes of the function and the x-intercepts of its graph. If x = a is a root of a polynomial, then (x - a) is a factor of that polynomial.

Some polynomials, such as f(x) = x² + 1, do not have any roots among the real numbers. In Mathematics, the real numbers may be described informally in several different ways If, however, the set of allowed candidates is expanded to the complex numbers, every (non-constant) polynomial has at least one distinct root; this follows from the fundamental theorem of algebra. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, the Fundamental theorem of algebra states that every non-constant single-variable Polynomial with complex coefficients has at

There is a difference between approximating roots and finding exact roots. Formulas for the roots of polynomials up to a degree of 2 have been known since ancient times (see quadratic equation) and up to a degree of 4 since the 16th century (see Gerolamo Cardano, Niccolo Fontana Tartaglia). In Mathematics, there are several meanings of degree depending on the subject In Mathematics, a quadratic equation is a Polynomial Equation of the second degree. Niccolò Fontana Tartaglia (1499/1500 Brescia, Italy &ndash December 13, 1557, Venice, Italy was a Mathematician But formulas for degree 5 eluded researchers. In 1824, Niels Henrik Abel proved the striking result that there can be no general formula (involving only the arithmetical operations and radicals) for the roots of a polynomial of degree 5 or greater in terms of its coefficients (see Abel-Ruffini theorem). Niels Henrik Abel (August 5 1802 &ndash April 6 1829 was a noted Norwegian Mathematician who proved the impossibility of solving the Quintic equation The Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no general solution in radicals to Polynomial equations of This result marked the start of Galois theory which engages in a detailed study of relationships among roots of polynomials. In Mathematics, more specifically in Abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory

Numerically solving a polynomial equation in one unknown is easily done on computer by the Durand-Kerner method or by some other root-finding algorithm. In Numerical analysis, the Durand&ndashKerner method (established 1960&ndash66 ????? Durand and ????? Kerner, --> or method of A root-finding algorithm is a numerical method or Algorithm, for finding a value x such that f ( x) = 0 for a given function The reduction of equations in several unknowns to equations each in one unknown is discussed in the article on the Buchberger's algorithm. In computational Algebraic geometry and computational Commutative algebra, Buchberger's algorithm is a method of transforming a given set of generators for a polynomial The special case where all the polynomials are of degree one is called a system of linear equations, for which a range of different solution methods exist, including the classical gaussian elimination. In Mathematics, a system of linear equations (or linear system) is a collection of Linear equations involving the same set of Variables For example In Mathematics, a system of linear equations (or linear system) is a collection of Linear equations involving the same set of Variables For example In Linear algebra, Gaussian elimination is an efficient Algorithm for solving systems of linear equations, to find the rank of a matrix

It has been shown by Richard Birkeland and Karl Meyr that the roots of any polynomial may be expressed in terms of multivariate hypergeometric functions. Ferdinand von Lindemann and Hiroshi Umemura showed that the roots may also be expressed in terms of Siegel modular functions, generalizations of the theta functions that appear in the theory of elliptic functions. Carl Louis Ferdinand von Lindemann ( April 12, 1852 &ndash March 6 1939) was a German Mathematician, noted for his proof In Mathematics, theta functions are Special functions of Several complex variables. In Complex analysis, an elliptic function is a function defined on the Complex plane which is periodic in two directions (a Doubly-periodic These characterizations of the roots of arbitrary polynomials are generalizations of the methods previously discovered to solve the quintic equation. In Mathematics, a quintic equation is a Polynomial Equation of degree five

Graphs

A polynomial function in one real variable can be represented by a graph. In mathematics the graph of a function f is the collection of all Ordered pairs ( x, f ( x)

The graph of the zero polynomial

f(x) = 0

is the x-axis.

The graph of a degree 0 polynomial

f(x) = a₀ , where a₀ ≠ 0,

is a horizontal line with y-intercept a₀

The graph of a degree 1 polynomial (or linear function)

f(x) = a₀ + a₁x , where a₁ ≠ 0,

is an oblique line with y-intercept a₀ and slope a₁. Slope is used to describe the steepness incline gradient or grade of a straight line.

The graph of a degree 2 polynomial

f(x) = a₀ + a₁x + a₂x², where a₂ ≠ 0

is a parabola. In Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular

The graph of any polynomial with degree 2 or greater

f(x) = a₀ + a₁x + a₂x² + . . . + a_nxⁿ , where a_n ≠ 0 and n ≥ 2

is a continuous non-linear curve.

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

The illustrations below show graphs of polynomials.

Polynomial of degree 2: f(x) = x² - x - 2 = (x+1)(x-2)	Polynomial of degree 3: f(x) = x³/5 + 4x²/5 - 7x/5 - 2 = 1/5 (x+5)(x+1)(x-2)
Polynomial of degree 4: f(x) = 1/14 (x+4)(x+1)(x-1)(x-3) + 0. 5	Polynomial of degree 5: f(x) = 1/20 (x+4)(x+2)(x+1)(x-1)(x-3) + 2

Polynomials and calculus

Main article: Calculus with polynomials

One important aspect of calculus is the project of analyzing complicated functions by means of approximating them with polynomials. In Mathematics, Polynomials are perhaps the simplest functions with which to do Calculus. The culmination of these efforts is Taylor's theorem, which roughly states that every differentiable function locally looks like a polynomial, and the Stone-Weierstrass theorem, which states that every continuous function defined on a compact interval of the real axis can be approximated on the whole interval as closely as desired by a polynomial. In Calculus, Taylor's theorem gives a sequence of approximations of a Differentiable function around a given point by Polynomials (the Taylor In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set Polynomials are also frequently used to interpolate functions. In the mathematical subfield of Numerical analysis, polynomial interpolation is the Interpolation of a given Data set by a Polynomial

Quotients of polynomials are called rational expressions, and functions that evaluate rational expressions are called rational functions. In Mathematics, a quotient is the result of a division. For example when dividing 6 by 3 the quotient is 2 while 6 is called the dividend, and 3 the In Mathematics, a rational function is any function which can be written as the Ratio of two Polynomial functions Definitions In In Mathematics, a rational function is any function which can be written as the Ratio of two Polynomial functions Definitions In Rational functions are the only functions that can be evaluated on a computer by a fixed sequence of instructions involving operations of addition, multiplication, division, which operations on floating point numbers are usually implemented in hardware. A computer is a Machine that manipulates data according to a list of instructions. Hardware is a general term that refers to the physical artifacts of a Technology. All the other functions that computers need to evaluate, such as trigonometric functions, logarithms and exponential functions, must then be computed in software that may use approximations to those functions on certain intervals by rational functions, and possibly iteration. In Mathematics, the logarithm of a number to a given base is the power or Exponent to which the base must be raised in order to produce The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x)

Calculating derivatives and integrals of polynomials is particularly simple. For the polynomial

$\sum_{i=0}^n a_i x^i$

the derivative with respect to x is

$\sum_{i=1}^n a_i i x^{i-1}$

and the indefinite integral is

$\sum_{i=0}^n {a_i\over i+1} x^{i+1}+c$ .

Abstract algebra

Main article: Polynomial ring

In abstract algebra, one must take care to distinguish between polynomials and polynomial functions. In Mathematics, especially in the field of Abstract algebra, a polynomial ring is a ring formed from the set of Polynomials in one or more variables Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules A polynomial f in one indeterminate $X$ over a ring $R$ is defined to be a formal expression of the form

$f = a_n X^n + a_{n - 1} X^{n - 1} + \cdots + a_1 X^1 + a_0X^0$

where $n$ is a natural number, the coefficients $a_0,\ldots,a_n$ are elements of $R$ , Here X is a formal symbol, whose powers Xⁱ are at this point just placeholders for the corresponding coefficients a_i, so that the given formal expression is just a way to encode the sequence (a_n,…,a₁,a₀). In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real Two polynomials sharing the same value of $n$ are considered to be equal if and only if the sequences of their coefficients are equal; furthermore any polynomial is equal to any polynomial with greater value of $n$ obtained from it by adding terms in front whose coefficient is zero. These polynomials can be added by simply adding corresponding coefficients (the rule for extending by terms with zero coefficients can be used to make sure such coefficients exist). Thus each polynomial is actually equal to the sum of the terms used in its formal expression, if such a term a_iXⁱ is interpreted as a polynomial that has zero coefficients at all powers of X other than Xⁱ. Then to define multiplication, it suffices by the distributive law to describe the product of any two such terms, which is given by the rule

$a X^k \; b X^l = ab X^{k+l}$

for all elements a, b of the ring R and all natural numbers k and l. In Mathematics, and in particular in Abstract algebra, distributivity is a property of Binary operations that generalises the distributive law In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an

Thus the set of all polynomials with coefficients in the ring R forms itself a ring, the ring of polynomials over R, which is denoted by R[X]. The map from R to R[X] sending r to rX⁰ is an injective homomorphism of rings, by which R is viewed as a subring of R[X]. If R is commutative, then R[X] is an algebra over R. In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property In Mathematics, specifically in Ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the

One can think of the ring R[X] as arising from R by adding one new element X to R, and extending in a minimal way to a ring in which X satisfies no other relations than the obligatory ones, plus commutation with all elements of R (that is Xr = rX). To do this, one must add all powers of X and their linear combinations as well.

Formation of the polynomial ring, together with forming factor rings by factoring out ideals, are important tools for constructing new rings out of known ones. In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. For instance, the ring (in fact field) of complex numbers, which can be constructed from the polynomial ring R[X] over the real numbers by factoring out the ideal of multiples of the polynomial X² + 1. Another example is the construction of finite fields, which proceeds similarly, starting out with the field of integers modulo some prime number as the coefficient ring R (see modular arithmetic). In Abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers

If $R$ is commutative, then one can associate to every polynomial P in $R [X]$ , a polynomial function f with domain and range equal to $R$ (more generally one can take domain and range to be the same unital associative algebra over $R$ ). In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property In Mathematics, an algebra is unital (some authors say unitary) if it contains a multiplicative Identity element (or unit) i In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive One obtains the value f(r) by everywhere replacing the symbol X in P by r. One reason that algebraists distinguish between polynomials and polynomial functions is that over some rings different polynomials may give rise to the same polynomial function (see Fermat's little theorem for an example where R is the integers modulo p). Fermat's little theorem (not to be confused with Fermat's last theorem) states that if p is a Prime number, then for any Integer a This is not the case when $R$ is the real or complex numbers and therefore many analysts often don't separate the two concepts. An even more important reason to distinguish between polynomials and polynomial functions is that many operations on polynomials (like Euclidean division) require looking at what a polynomial is composed of as an expression rather than evaluating it at some constant value for $X$ . In Abstract algebra, a Euclidean domain (also called a Euclidean ring) is a type of ring in which the Euclidean algorithm applies And it should be noted that if $R$ is not commutative, there is no (well behaved) notion of polynomial function at all.

Divisibility

In commutative algebra, one major focus of study is divisibility among polynomials. Commutative algebra is the branch of Abstract algebra that studies Commutative rings their ideals, and modules over such rings If R is an integral domain and f and g are polynomials in R[X], it is said that f divides g if there exists a polynomial q in R[X] such that f q = g. In Abstract algebra, a branch of Mathematics, an integral domain is a Commutative ring with an additive identity 0 and a multiplicative identity 1 such One can then show that "every zero gives rise to a linear factor", or more formally: if f is a polynomial in R[X] and r is an element of R such that f(r) = 0, then the polynomial (X − r) divides f. The converse is also true. The quotient can be computed using the Horner scheme. In Numerical analysis, the Horner scheme or Horner algorithm, named after William George Horner, is an Algorithm for the efficient evaluation

If F is a field and f and g are polynomials in F[X] with g ≠ 0, then there exist unique polynomials q and r in F[X] with

$f = q \, g + r$

and such that the degree of r is smaller than the degree of g. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division The polynomials q and r are uniquely determined by f and g. This is called "division with remainder" or "polynomial long division" and shows that the ring F[X] is a Euclidean domain. In Algebra, polynomial long division is an Algorithm for dividing a Polynomial by another polynomial of the same or lower degree, a generalised In Abstract algebra, a Euclidean domain (also called a Euclidean ring) is a type of ring in which the Euclidean algorithm applies

Analogously, polynomial "primes" (more correctly, irreducible polynomials) can be defined which cannot be factorized into the product of two polynomials of lesser degree. In Mathematics, the adjective irreducible means that an object cannot be expressed as a product of at least two non-trivial factors in a given set In Mathematics, there are several meanings of degree depending on the subject It is not easy to determine if a given polynomial is irreducible. One can start by simply checking if the polynomial has linear factors. Then, one can check divisibility by some other irreducible polynomials. Eisenstein's criterion can also be used in some cases to determine irreducibility. In Mathematics, Eisenstein's criterion gives sufficient conditions for a Polynomial to be irreducible over the rational

See also: Greatest common divisor of two polynomials. Informally the greatest common divisor (GCD of two polynomials p ( x) and q ( x) is the "biggest" polynomial that divides

Classifications

The most important classification of polynomials is based of the number of distinct variables. A polynomial in one variable is called a univariate polynomial, a polynomial in more than one variable is called a multivariate polynomial. These notions refer more to the kind of polynomials one is generally working with than to individual polynomials; for instance when working with univariate polynomials one does not exclude constant polynomials (which may result for instance from the subtraction of non-constant polynomials), although strictly speaking constant polynomials do not contain any variables at all. It is possible to further classify multivariate polynomials as bivariate, trivariate etc. , according to the number of variables, but this is rarely done; it is more common for instance to say simply "polynomials in x, y, and z". A (usually mulitvariate) polynomial is called homogeneous of degree n if all its terms have degree n.

Univariate polynomials have many properties not shared by multivariate polynomials. For instance, the terms of a univariate polynomial are completely ordered by their degree, and it is conventional to always write them in order of decreasing degree. A univariate polynomial in x of degree n then takes the general form

$c_nx^n+c_{n-1}x^{n-1}+\cdots+c_2x^2+c_1x+c_0$

where c_n, c_n-1, …, c₂, c₁ and c₀ are constants, the coefficients of this polynomial. Here the term c_nxⁿ is called the leading term and its coefficient c_n the leading coefficient; if the leading coefficient is 1, the univariate polynomial is called monic. Note that apart from the leading coefficient c_n (which must be non-zero or else the polynomial would not be of degree n) this general form allows for coefficients to be zero; when this happens the corresponding term is zero and may be removed from the sum without changing the polynomial. It is nevertheless common to refer to c_i as the coeffient of xⁱ, even when c_i happens to be 0, so that xⁱ does not really occur in any term; for instance one can speak of the constant term the polynomial, meaning c₀ even if it should be zero. In Mathematics, the constant term of a Polynomial is the term of degree 0

Polynomials can similarly be classified by the kind of constant values allowed as coefficients. One can work with polynomials with integral, rational, real or complex coefficients, and in abstract algebra polynomials with many other types of coefficients can be defined. Like for the previous classification, this is about the coefficients one is generally working with; for instance when working with polynomials with complex coefficients one includes polynomials whose coefficients happen to all be real, even though such polynomials can also be considered to be a polynomials with real coefficients.

Polynomials can further be classified by their degree and/or the number of non-zero terms they contain.

Polynomials classified by degree
Degree	Name	Example
${^{-\infty}}$	zero	$0$
$0$	(non-zero) constant	$1$
$1$	linear	$x + 1$
$2$	quadratic	$x 2 + 1$
$3$	cubic	$x 3 + 1$
$4$	quartic or biquadratic	$x 4 + 1$
$5$	quintic	$x 5 + 1$
$6$	sextic or hexic	$x 6 + 1$
$7$	septic or heptic	$x 7 + 1$
$8$	octic	$x 8 + 1$
$9$	nonic	$x 9 + 1$
$10$	decic	$x 10 + 1$

The names for degrees higher than $3$ are less common. A quadratic function, in Mathematics, is a Polynomial function of the form f(x=ax^2+bx+c \\! where a \ne 0 \\! This article discusses cubic equations in one variable For a discussion of cubic equations in two variables see Elliptic curve. The names for the degrees may be applied to the polynomial or to its terms. For example, a constant may refer to a zero degree polynomial or to a zero degree term.

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 to be negative (either –1 or –∞)[1]. The latter convention is important when defining Euclidean division of polynomials. In Abstract algebra, a Euclidean domain (also called a Euclidean ring) is a type of ring in which the Euclidean algorithm applies

Polynomials classified by number of non-zero terms
Number of non-zero terms	Name	Example
$0$	zero polynomial	$0$
$1$	monomial	$x 2$
$2$	binomial	$x 2 + 1$
$3$	trinomial	$x 2 + x + 1$

The word monomial can be ambiguous, as it is also often used to denote just a power of the variable, or in the multivariate case product of such powers, without any coefficient. Two or more terms which involve the same monomial in the latter sense, in other words which differ only in the value of their coefficients, are called similar terms; they can be combined into a single term by adding their coefficients; if the resulting term has coefficient zero, it may be removed altogether. The above classification according to the number of terms assumes that similar terms have been combined first.

Extensions of the concept of a polynomial

One also speaks of polynomials in several variables, obtained by taking the ring of polynomials of a ring of polynomials: R[X,Y] = (R[X])[Y] = (R[Y])[X]. These are of fundamental importance in algebraic geometry which studies the simultaneous zero sets of several such multivariate polynomials. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with

Polynomials are frequently used to encode information about some other object. The characteristic polynomial of a matrix or linear operator contains information about the operator's eigenvalues. In Linear algebra, one associates a Polynomial to every Square matrix, its characteristic polynomial. In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes The minimal polynomial of an algebraic element records the simplest algebraic relation satisfied by that element. In Linear algebra, the minimal polynomial of an n -by- n matrix A over a field F is the Monic polynomial In Mathematics, the roots of Polynomials are in Abstract algebra called algebraic elements.

Other related objects studied in abstract algebra are formal power series, which are like polynomials but may have infinite degree, and the rational functions, which are ratios of polynomials. In Mathematics, formal power series are devices that make it possible to employ much of the analytical machinery of Power series in settings that do not In Mathematics, a rational function is any function which can be written as the Ratio of two Polynomial functions Definitions In

References

^ Peter H. This is a list of Polynomial topics, by Wikipedia page See also Trigonometric polynomial, List of algebraic geometry topics. Selby, Steve Slavin, Practical Algebra: A Self-Teaching Guide, 2nd Edition, Wiley, ISBN-10 0471530123 ISBN-13 978-0471530121
^ Gilbert Strang, Linear Algebra and its Applications, Fourth Edition, Thompson Brooks/Cole, ISBN 0030105676.
^ Howard Eves, An Introduction to the History of Mathematics, Sixth Edition, Saunders, ISBN 0030295580

R. Birkeland. Über die Auflösung algebraischer Gleichungen durch hypergeometrische Funktionen. Mathematische Zeitschrift vol. 26, (1927) pp. 565-578. Shows that the roots of any polynomial may be written in terms of multivariate hypergeometric functions. Paper is available here.
F. von Lindemann. Über die Auflösung der algebraischen Gleichungen durch transcendente Functionen. Nachrichten von der Königl. Gesellschaft der Wissenschaften, vol. 7, 1884. Polynomial solutions in terms of theta functions. Paper available here.
F. von Lindemann. Über die Auflösung der algebraischen Gleichungen durch transcendente Functionen II. Nachrichten von der Königl. Gesellschaft der Wissenschaften und der Georg-Augusts-Universität zu Göttingen, 1892 edition. Paper available here.
K. Mayr. Über die Auflösung algebraischer Gleichungssysteme durch hypergeometrische Funktionen. Monatshefte für Mathematik und Physik vol. 45, (1937) pp. 280-313.
H. Umemura. Solution of algebraic equations in terms of theta constants. In D. Mumford, Tata Lectures on Theta II, Progress in Mathematics 43, Birkhäuser, Boston, 1984.

External links

Dictionary

-adjective

(algebra) that can be described or limited by a polynomial.
(taxonomy) of a polynomial name or entity

-noun

(algebra) An expression consisting of a sum of a finite number of terms, each term being the product of a constant coefficient and one or more variables raised to a non-negative integer power, such as <math>a_n x^n + a_{n-1}x^{n-1} + ... + a_0 x^0</math>.
(taxonomy) A taxonomic designation (such as of a subspecies) consisting of more than two terms.

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