Polynomial

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For less elementary aspects of the subject, see Polynomial ring. Polynomial_sentence_0

In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. Polynomial_sentence_1

An example of a polynomial of a single indeterminate x is x − 4x + 7. Polynomial_sentence_2

An example in three variables is x + 2xyz − yz + 1. Polynomial_sentence_3

Polynomials appear in many areas of mathematics and science. Polynomial_sentence_4

For example, they are used to form polynomial equations, 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 and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. Polynomial_sentence_5

In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry. Polynomial_sentence_6

Etymology Polynomial_section_0

The word polynomial joins two diverse roots: the Greek poly, meaning "many", and the Latin nomen, or name. Polynomial_sentence_7

It was derived from the term binomial by replacing the Latin root bi- with the Greek poly-. Polynomial_sentence_8

The word polynomial was first used in the 17th century. Polynomial_sentence_9

Notation and terminology Polynomial_section_1

The x occurring in a polynomial is commonly called a variable or an indeterminate. Polynomial_sentence_10

When the polynomial is considered as an expression, x is a fixed symbol which does not have any value (its value is "indeterminate"). Polynomial_sentence_11

However, when one considers the function defined by the polynomial, then x represents the argument of the function, and is therefore called a "variable". Polynomial_sentence_12

Many authors use these two words interchangeably. Polynomial_sentence_13

It is common to use uppercase letters for indeterminates and corresponding lowercase letters for the variables (or arguments) of the associated function. Polynomial_sentence_14

A polynomial P in the indeterminate x is commonly denoted either as P or as P(x). Polynomial_sentence_15

Formally, the name of the polynomial is P, not P(x), but the use of the functional notation P(x) dates from a time when the distinction between a polynomial and the associated function was unclear. Polynomial_sentence_16

Moreover, the functional notation is often useful for specifying, in a single phrase, a polynomial and its indeterminate. Polynomial_sentence_17

For example, "let P(x) be a polynomial" is a shorthand for "let P be a polynomial in the indeterminate x". Polynomial_sentence_18

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. Polynomial_sentence_19

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. Polynomial_sentence_20

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 Polynomial_sentence_21

which is the polynomial function associated to P. Frequently, when using this notation, one supposes that a is a number. Polynomial_sentence_22

However, one may use it over any domain where addition and multiplication are defined (that is, any ring). Polynomial_sentence_23

In particular, if a is a polynomial then P(a) is also a polynomial. Polynomial_sentence_24

More specifically, when a is the indeterminate x, then the image of x by this function is the polynomial P itself (substituting x for x does not change anything). Polynomial_sentence_25

In other words, Polynomial_sentence_26

which justifies formally the existence of two notations for the same polynomial. Polynomial_sentence_27

Definition Polynomial_section_2

A polynomial is an expression that can be built from constants and symbols called variables or indeterminates by means of addition, multiplication and exponentiation to a non-negative integer power. Polynomial_sentence_28

Two such expressions that may be transformed, one to the other, by applying the usual properties of commutativity, associativity and distributivity of addition and multiplication, are considered as defining the same polynomial. Polynomial_sentence_29

A polynomial in a single indeterminate x can always be written (or rewritten) in the form Polynomial_sentence_30

This can be expressed more concisely by using summation notation: Polynomial_sentence_31

That is, a polynomial can either be zero or can be written as the sum of a finite number of non-zero terms. Polynomial_sentence_32

Each term consists of the product of a number – called the coefficient of the term – and a finite number of indeterminates, raised to nonnegative integer powers. Polynomial_sentence_33

Classification Polynomial_section_3

Further information: Degree of a polynomial Polynomial_sentence_34

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. Polynomial_sentence_35

Because x = x, the degree of an indeterminate without a written exponent is one. Polynomial_sentence_36

A term with no indeterminates and a polynomial with no indeterminates are called, respectively, a constant term and a constant polynomial. Polynomial_sentence_37

The degree of a constant term and of a nonzero constant polynomial is 0. Polynomial_sentence_38

The degree of the zero polynomial 0 (which has no terms at all) is generally treated as not defined (but see below). Polynomial_sentence_39

For example: Polynomial_sentence_40

is a term. Polynomial_sentence_41

The coefficient is −5, the indeterminates are x and y, the degree of x is two, while the degree of y is one. Polynomial_sentence_42

The degree of the entire term is the sum of the degrees of each indeterminate in it, so in this example the degree is 2 + 1 = 3. Polynomial_sentence_43

Forming a sum of several terms produces a polynomial. Polynomial_sentence_44

For example, the following is a polynomial: Polynomial_sentence_45

It consists of three terms: the first is degree two, the second is degree one, and the third is degree zero. Polynomial_sentence_46

Polynomials of small degree have been given specific names. Polynomial_sentence_47

A polynomial of degree zero is a constant polynomial, or simply a constant. Polynomial_sentence_48

Polynomials of degree one, two or three are respectively linear polynomials, quadratic polynomials and cubic polynomials. Polynomial_sentence_49

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. Polynomial_sentence_50

The names for the degrees may be applied to the polynomial or to its terms. Polynomial_sentence_51

For example, the term 2x in x + 2x + 1 is a linear term in a quadratic polynomial. Polynomial_sentence_52

The polynomial 0, which may be considered to have no terms at all, is called the zero polynomial. Polynomial_sentence_53

Unlike other constant polynomials, its degree is not zero. Polynomial_sentence_54

Rather, the degree of the zero polynomial is either left explicitly undefined, or defined as negative (either −1 or −∞). Polynomial_sentence_55

The zero polynomial is also unique in that it is the only polynomial in one indeterminate that has an infinite number of roots. Polynomial_sentence_56

The graph of the zero polynomial, f(x) = 0, is the x-axis. Polynomial_sentence_57

In the case of polynomials in more than one indeterminate, a polynomial is called homogeneous of degree n if all of its non-zero terms have degree n. The zero polynomial is homogeneous, and, as a homogeneous polynomial, its degree is undefined. Polynomial_sentence_58

For example, xy + 7xy − 3x is homogeneous of degree 5. Polynomial_sentence_59

For more details, see Homogeneous polynomial. Polynomial_sentence_60

The commutative law of addition can be used to rearrange terms into any preferred order. Polynomial_sentence_61

In polynomials with one indeterminate, the terms are usually ordered according to degree, either in "descending powers of x", with the term of largest degree first, or in "ascending powers of x". Polynomial_sentence_62

The polynomial in the example above is written in descending powers of x. Polynomial_sentence_63

The first term has coefficient 3, indeterminate x, and exponent 2. Polynomial_sentence_64

In the second term, the coefficient is −5. Polynomial_sentence_65

The third term is a constant. Polynomial_sentence_66

Because the degree of a non-zero polynomial is the largest degree of any one term, this polynomial has degree two. Polynomial_sentence_67

Two terms with the same indeterminates raised to the same powers are called "similar terms" or "like terms", and they can be combined, using the distributive law, into a single term whose coefficient is the sum of the coefficients of the terms that were combined. Polynomial_sentence_68

It may happen that this makes the coefficient 0. Polynomial_sentence_69

Polynomials can be classified by the number of terms with nonzero coefficients, so that a one-term polynomial is called a monomial, a two-term polynomial is called a binomial, and a three-term polynomial is called a trinomial. Polynomial_sentence_70

The term "quadrinomial" is occasionally used for a four-term polynomial. Polynomial_sentence_71

A real polynomial is a polynomial with real coefficients. Polynomial_sentence_72

When it is used to define a function, the domain is not so restricted. Polynomial_sentence_73

However, a real polynomial function is a function from the reals to the reals that is defined by a real polynomial. Polynomial_sentence_74

Similarly, an integer polynomial is a polynomial with integer coefficients, and a complex polynomial is a polynomial with complex coefficients. Polynomial_sentence_75

A polynomial in one indeterminate is called a univariate polynomial, a polynomial in more than one indeterminate is called a multivariate polynomial. Polynomial_sentence_76

A polynomial with two indeterminates is called a bivariate polynomial. Polynomial_sentence_77

These notions refer more to the kind of polynomials one is generally working with than to individual polynomials; for instance, when working with univariate polynomials, one does not exclude constant polynomials (which may result from the subtraction of non-constant polynomials), although strictly speaking, constant polynomials do not contain any indeterminates at all. Polynomial_sentence_78

It is possible to further classify multivariate polynomials as bivariate, trivariate, and so on, according to the maximum number of indeterminates allowed. Polynomial_sentence_79

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. Polynomial_sentence_80

It is also common to say simply "polynomials in x, y, and z", listing the indeterminates allowed. Polynomial_sentence_81

The evaluation of a polynomial consists of substituting a numerical value to each indeterminate and carrying out the indicated multiplications and additions. Polynomial_sentence_82

For polynomials in one indeterminate, the evaluation is usually more efficient (lower number of arithmetic operations to perform) using Horner's method: Polynomial_sentence_83

Arithmetic Polynomial_section_4

Addition and subtraction Polynomial_section_5

Polynomials can be added using the associative law of addition (grouping all their terms together into a single sum), possibly followed by reordering (using the commutative law) and combining of like terms. Polynomial_sentence_84

For example, if Polynomial_sentence_85

then the sum Polynomial_sentence_86

can be reordered and regrouped as Polynomial_sentence_87

and then simplified to Polynomial_sentence_88

When polynomials are added together, the result is another polynomial. Polynomial_sentence_89

Subtraction of polynomials is similar. Polynomial_sentence_90

Multiplication Polynomial_section_6

Polynomials can also be multiplied. Polynomial_sentence_91

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. Polynomial_sentence_92

For example, if Polynomial_sentence_93

then Polynomial_sentence_94

Carrying out the multiplication in each term produces Polynomial_sentence_95

Combining similar terms yields Polynomial_sentence_96

which can be simplified to Polynomial_sentence_97

As in the example, the product of polynomials is always a polynomial. Polynomial_sentence_98

Composition Polynomial_section_7

Division Polynomial_section_8

The division of one polynomial by another is not typically a polynomial. Polynomial_sentence_99

Instead, such ratios are a more general family of objects, called rational fractions, rational expressions, or rational functions, depending on context. Polynomial_sentence_100

This is analogous to the fact that the ratio of two integers is a rational number, not necessarily an integer. Polynomial_sentence_101

For example, the fraction 1/(x + 1) is not a polynomial, and it cannot be written as a finite sum of powers of the variable x. Polynomial_sentence_102

For polynomials in one variable, there is a notion of Euclidean division of polynomials, generalizing the Euclidean division of integers. Polynomial_sentence_103

This notion of the division a(x)/b(x) results in two polynomials, a quotient q(x) and a remainder r(x), such that a = b q + r and degree(r) < degree(b). Polynomial_sentence_104

The quotient and remainder may be computed by any of several algorithms, including polynomial long division and synthetic division. Polynomial_sentence_105

When the denominator b(x) is monic and linear, that is, b(x) = x − c for some constant c, then the polynomial remainder theorem asserts that the remainder of the division of a(x) by b(x) is the evaluation f(c). Polynomial_sentence_106

In this case, the quotient may be computed by Ruffini's rule, a special case of synthetic division. Polynomial_sentence_107

Factoring Polynomial_section_9

All polynomials with coefficients in a unique factorization domain (for example, the integers or a field) also have a factored form in which the polynomial is written as a product of irreducible polynomials and a constant. Polynomial_sentence_108

This factored form is unique up to the order of the factors and their multiplication by an invertible constant. Polynomial_sentence_109

In the case of the field of complex numbers, the irreducible factors are linear. Polynomial_sentence_110

Over the real numbers, they have the degree either one or two. Polynomial_sentence_111

Over the integers and the rational numbers the irreducible factors may have any degree. Polynomial_sentence_112

For example, the factored form of Polynomial_sentence_113

is Polynomial_sentence_114

over the integers and the reals and Polynomial_sentence_115

over the complex numbers. Polynomial_sentence_116

The computation of the factored form, called factorization is, in general, too difficult to be done by hand-written computation. Polynomial_sentence_117

However, efficient polynomial factorization algorithms are available in most computer algebra systems. Polynomial_sentence_118

Calculus Polynomial_section_10

Main article: Calculus with polynomials Polynomial_sentence_119

For polynomials whose coefficients come from more abstract settings (for example, if the coefficients are integers modulo some prime number p, or elements of an arbitrary ring), the formula for the derivative can still be interpreted formally, with the coefficient kak understood to mean the sum of k copies of ak. Polynomial_sentence_120

For example, over the integers modulo p, the derivative of the polynomial x + x is the polynomial 1. Polynomial_sentence_121

Polynomial functions Polynomial_section_11

See also: Ring of polynomial functions Polynomial_sentence_122

A polynomial function is a function that can be defined by evaluating a polynomial. Polynomial_sentence_123

More precisely, a function f of one argument from a given domain is a polynomial function if there exists a polynomial Polynomial_sentence_124

For example, the function f, defined by Polynomial_sentence_125

is a polynomial function of one variable. Polynomial_sentence_126

Polynomial functions of several variables are similarly defined, using polynomials in more than one indeterminate, as in Polynomial_sentence_127

Every polynomial function is continuous, smooth, and entire. Polynomial_sentence_128

Graphs Polynomial_section_12

A polynomial function in one real variable can be represented by a graph. Polynomial_sentence_129

Polynomial_unordered_list_0

  • The graph of the zero polynomialPolynomial_item_0_0

Polynomial_description_list_1

  • Polynomial_item_1_1
    • f(x) = 0Polynomial_item_1_2
  • is the x-axis.Polynomial_item_1_3

Polynomial_unordered_list_2

  • The graph of a degree 0 polynomialPolynomial_item_2_4

Polynomial_description_list_3

  • Polynomial_item_3_5
    • f(x) = a0, where a0 ≠ 0,Polynomial_item_3_6
  • is a horizontal line with y-intercept a0Polynomial_item_3_7

Polynomial_unordered_list_4

  • The graph of a degree 1 polynomial (or linear function)Polynomial_item_4_8

Polynomial_description_list_5

  • Polynomial_item_5_9
    • f(x) = a0 + a1x , where a1 ≠ 0,Polynomial_item_5_10
  • is an oblique line with y-intercept a0 and slope a1.Polynomial_item_5_11

Polynomial_unordered_list_6

  • The graph of a degree 2 polynomialPolynomial_item_6_12

Polynomial_description_list_7

  • Polynomial_item_7_13
    • f(x) = a0 + a1x + a2x, where a2 ≠ 0Polynomial_item_7_14
  • is a parabola.Polynomial_item_7_15

Polynomial_unordered_list_8

  • The graph of a degree 3 polynomialPolynomial_item_8_16

Polynomial_description_list_9

  • Polynomial_item_9_17
    • f(x) = a0 + a1x + a2x + a3x, where a3 ≠ 0Polynomial_item_9_18
  • is a cubic curve.Polynomial_item_9_19

Polynomial_unordered_list_10

  • The graph of any polynomial with degree 2 or greaterPolynomial_item_10_20

Polynomial_description_list_11

  • Polynomial_item_11_21
    • f(x) = a0 + a1x + a2x + ... + anx , where an ≠ 0 and n ≥ 2Polynomial_item_11_22
  • is a continuous non-linear curve.Polynomial_item_11_23

A non-constant polynomial function tends to infinity when the variable increases indefinitely (in absolute value). Polynomial_sentence_130

If the degree is higher than one, the graph does not have any asymptote. Polynomial_sentence_131

It has two parabolic branches with vertical direction (one branch for positive x and one for negative x). Polynomial_sentence_132

Polynomial graphs are analyzed in calculus using intercepts, slopes, concavity, and end behavior. Polynomial_sentence_133

Equations Polynomial_section_13

Main article: Algebraic equation Polynomial_sentence_134

A polynomial equation, also called an algebraic equation, is an equation of the form Polynomial_sentence_135

For example, Polynomial_sentence_136

is a polynomial equation. Polynomial_sentence_137

When considering equations, the indeterminates (variables) of polynomials are also called unknowns, and the solutions are the possible values of the unknowns for which the equality is true (in general more than one solution may exist). Polynomial_sentence_138

A polynomial equation stands in contrast to a polynomial identity like (x + y)(x − y) = x − y, where both expressions represent the same polynomial in different forms, and as a consequence any evaluation of both members gives a valid equality. Polynomial_sentence_139

In elementary algebra, methods such as the quadratic formula are taught for solving all first degree and second degree polynomial equations in one variable. Polynomial_sentence_140

There are also formulas for the cubic and quartic equations. Polynomial_sentence_141

For higher degrees, the Abel–Ruffini theorem asserts that there can not exist a general formula in radicals. Polynomial_sentence_142

However, root-finding algorithms may be used to find numerical approximations of the roots of a polynomial expression of any degree. Polynomial_sentence_143

The number of solutions of a polynomial equation with real coefficients may not exceed the degree, and equals the degree when the complex solutions are counted with their multiplicity. Polynomial_sentence_144

This fact is called the fundamental theorem of algebra. Polynomial_sentence_145

Solving equations Polynomial_section_14

See also: Root-finding of polynomials and Properties of polynomial roots Polynomial_sentence_146

A number a is a root of a polynomial P if and only if the linear polynomial x − a divides P, that is if there is another polynomial Q such that P = (x – a) Q. Polynomial_sentence_147

It may happen that x − a divides P more than once: if (x − a) divides P then a is called a multiple root of P, and otherwise a is called a simple root of P. If P is a nonzero polynomial, there is a highest power m such that (x − a) divides P, which is called the multiplicity of the root a in P. When P is the zero polynomial, the corresponding polynomial equation is trivial, and this case is usually excluded when considering roots, as, with the above definitions, every number is a root of the zero polynomial, with an undefined multiplicity. Polynomial_sentence_148

With this exception made, the number of roots of P, even counted with their respective multiplicities, cannot exceed the degree of P. The relation between the coefficients of a polynomial and its roots is described by Vieta's formulas. Polynomial_sentence_149

Some polynomials, such as x + 1, do not have any roots among the real numbers. Polynomial_sentence_150

If, however, the set of accepted solutions is expanded to the complex numbers, every non-constant polynomial has at least one root; this is the fundamental theorem of algebra. Polynomial_sentence_151

By successively dividing out factors x − a, one sees that any polynomial with complex coefficients can be written as a constant (its leading coefficient) times a product of such polynomial factors of degree 1; as a consequence, the number of (complex) roots counted with their multiplicities is exactly equal to the degree of the polynomial. Polynomial_sentence_152

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 approximations of the solutions. Polynomial_sentence_153

There are many methods for that; some are restricted to polynomials and others may apply to any continuous function. Polynomial_sentence_154

The most efficient algorithms allow solving easily (on a computer) polynomial equations of degree higher than 1,000 (see Root-finding algorithm). Polynomial_sentence_155

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". Polynomial_sentence_156

The study of the sets of zeros of polynomials is the object of algebraic geometry. Polynomial_sentence_157

For a set of polynomial equations in several unknowns, there are algorithms to decide whether they have a finite number of complex solutions, and, if this number is finite, for computing the solutions. Polynomial_sentence_158

See System of polynomial equations. Polynomial_sentence_159

The special case where all the polynomials are of degree one is called a system of linear equations, for which another range of different solution methods exist, including the classical Gaussian elimination. Polynomial_sentence_160

A polynomial equation for which one is interested only in the solutions which are integers is called a Diophantine equation. Polynomial_sentence_161

Solving Diophantine equations is generally a very hard task. Polynomial_sentence_162

It has been proved that there cannot be any general algorithm for solving them, and even for deciding whether the set of solutions is empty (see Hilbert's tenth problem). Polynomial_sentence_163

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. Polynomial_sentence_164

Generalizations Polynomial_section_15

There are several generalizations of the concept of polynomials. Polynomial_sentence_165

Trigonometric polynomials Polynomial_section_16

Main article: Trigonometric polynomial Polynomial_sentence_166

A trigonometric polynomial is a finite linear combination of functions sin(nx) and cos(nx) with n taking on the values of one or more natural numbers. Polynomial_sentence_167

The coefficients may be taken as real numbers, for real-valued functions. Polynomial_sentence_168

If sin(nx) and cos(nx) are expanded in terms of sin(x) and cos(x), a trigonometric polynomial becomes a polynomial in the two variables sin(x) and cos(x) (using List of trigonometric identities#Multiple-angle formulae). Polynomial_sentence_169

Conversely, every polynomial in sin(x) and cos(x) may be converted, with Product-to-sum identities, into a linear combination of functions sin(nx) and cos(nx). Polynomial_sentence_170

This equivalence explains why linear combinations are called polynomials. Polynomial_sentence_171

For complex coefficients, there is no difference between such a function and a finite Fourier series. Polynomial_sentence_172

Trigonometric polynomials are widely used, for example in trigonometric interpolation applied to the interpolation of periodic functions. Polynomial_sentence_173

They are used also in the discrete Fourier transform. Polynomial_sentence_174

Matrix polynomials Polynomial_section_17

Main article: Matrix polynomial Polynomial_sentence_175

A matrix polynomial is a polynomial with square matrices as variables. Polynomial_sentence_176

Given an ordinary, scalar-valued polynomial Polynomial_sentence_177

this polynomial evaluated at a matrix A is Polynomial_sentence_178

where I is the identity matrix. Polynomial_sentence_179

A matrix polynomial equation is an equality between two matrix polynomials, which holds for the specific matrices in question. Polynomial_sentence_180

A matrix polynomial identity is a matrix polynomial equation which holds for all matrices A in a specified matrix ring Mn(R). Polynomial_sentence_181

Laurent polynomials Polynomial_section_18

Main article: Laurent polynomial Polynomial_sentence_182

Laurent polynomials are like polynomials, but allow negative powers of the variable(s) to occur. Polynomial_sentence_183

Rational functions Polynomial_section_19

Main article: Rational function Polynomial_sentence_184

A rational fraction is the quotient (algebraic fraction) of two polynomials. Polynomial_sentence_185

Any algebraic expression that can be rewritten as a rational fraction is a rational function. Polynomial_sentence_186

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. Polynomial_sentence_187

The rational fractions include the Laurent polynomials, but do not limit denominators to powers of an indeterminate. Polynomial_sentence_188

Power series Polynomial_section_20

Main article: Formal power series Polynomial_sentence_189

Formal power series are like polynomials, but allow infinitely many non-zero terms to occur, so that they do not have finite degree. Polynomial_sentence_190

Unlike polynomials they cannot in general be explicitly and fully written down (just like irrational numbers cannot), but the rules for manipulating their terms are the same as for polynomials. Polynomial_sentence_191

Non-formal power series also generalize polynomials, but the multiplication of two power series may not converge. Polynomial_sentence_192

Other examples Polynomial_section_21

A bivariate polynomial where the second variable is substituted by an exponential function applied to the first variable, for example P(x, e), may be called an exponential polynomial. Polynomial_sentence_193

Applications Polynomial_section_22

Abstract algebra Polynomial_section_23

Main article: Polynomial ring Polynomial_sentence_194

In abstract algebra, one distinguishes between polynomials and polynomial functions. Polynomial_sentence_195

A polynomial f in one indeterminate x over a ring R is defined as a formal expression of the form Polynomial_sentence_196

where n is a natural number, the coefficients a0, . Polynomial_sentence_197

. Polynomial_sentence_198

., an are elements of R, and x is a formal symbol, whose powers x are just placeholders for the corresponding coefficients ai, so that the given formal expression is just a way to encode the sequence (a0, a1, . Polynomial_sentence_199

. Polynomial_sentence_200

. Polynomial_sentence_201

), where there is an n such that ai = 0 for all i > n. Two polynomials sharing the same value of n are considered equal if and only if the sequences of their coefficients are equal; furthermore any polynomial is equal to any polynomial with greater value of n obtained from it by adding terms in front whose coefficient is zero. Polynomial_sentence_202

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). Polynomial_sentence_203

Thus each polynomial is actually equal to the sum of the terms used in its formal expression, if such a term aix is interpreted as a polynomial that has zero coefficients at all powers of x other than x. Polynomial_sentence_204

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 Polynomial_sentence_205

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]. Polynomial_sentence_206

The map from R to R[x] sending r to rx is an injective homomorphism of rings, by which R is viewed as a subring of R[x]. Polynomial_sentence_207

If R is commutative, then R[x] is an algebra over R. Polynomial_sentence_208

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). Polynomial_sentence_209

To do this, one must add all powers of x and their linear combinations as well. Polynomial_sentence_210

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. Polynomial_sentence_211

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 x + 1. Polynomial_sentence_212

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). Polynomial_sentence_213

If R is commutative, then one can associate with 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 any same unital associative algebra over R.) One obtains the value f(r) by substitution of the value r for the symbol x in P. One reason to distinguish between polynomials and polynomial functions is that, over some rings, different polynomials may give rise to the same polynomial function (see Fermat's little theorem for an example where R is the integers modulo p). Polynomial_sentence_214

This is not the case when R is the real or complex numbers, whence the two concepts are not always distinguished in analysis. Polynomial_sentence_215

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. Polynomial_sentence_216

Divisibility Polynomial_section_24

Main articles: Polynomial greatest common divisor and Factorization of polynomials Polynomial_sentence_217

In commutative algebra, one major focus of study is divisibility among polynomials. Polynomial_sentence_218

If R is an integral domain and f and g are polynomials in R[x], it is said that f divides g or f is a divisor of g if there exists a polynomial q in R[x] such that f q = g. One can show that every zero gives rise to a linear divisor, 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. Polynomial_sentence_219

The quotient can be computed using the polynomial long division. Polynomial_sentence_220

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 Polynomial_sentence_221

and such that the degree of r is smaller than the degree of g (using the convention that the polynomial 0 has a negative degree). Polynomial_sentence_222

The polynomials q and r are uniquely determined by f and g. This is called Euclidean division, division with remainder or polynomial long division and shows that the ring F[x] is a Euclidean domain. Polynomial_sentence_223

Analogously, prime polynomials (more correctly, irreducible polynomials) can be defined as non-zero polynomials which cannot be factorized into the product of two non-constant polynomials. Polynomial_sentence_224

In the case of coefficients in a ring, "non-constant" must be replaced by "non-constant or non-unit" (both definitions agree in the case of coefficients in a field). Polynomial_sentence_225

Any polynomial may be decomposed into the product of an invertible constant by a product of irreducible polynomials. Polynomial_sentence_226

If the coefficients belong to a field or a unique factorization domain this decomposition is unique up to the order of the factors and the multiplication of any non-unit factor by a unit (and division of the unit factor by the same unit). Polynomial_sentence_227

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). Polynomial_sentence_228

These algorithms are not practicable for hand-written computation, but are available in any computer algebra system. Polynomial_sentence_229

Eisenstein's criterion can also be used in some cases to determine irreducibility. Polynomial_sentence_230

Positional notation Polynomial_section_25

Main article: Positional notation Polynomial_sentence_231

In modern positional numbers systems, such as the decimal system, the digits and their positions in the representation of an integer, for example, 45, are a shorthand notation for a polynomial in the radix or base, in this case, 4 × 10 + 5 × 10. Polynomial_sentence_232

As another example, in radix 5, a string of digits such as 132 denotes the (decimal) number 1 × 5 + 3 × 5 + 2 × 5 = 42. Polynomial_sentence_233

This representation is unique. Polynomial_sentence_234

Let b be a positive integer greater than 1. Polynomial_sentence_235

Then every positive integer a can be expressed uniquely in the form Polynomial_sentence_236

where m is a nonnegative integer and the r's are integers such that Polynomial_sentence_237

Polynomial_description_list_12

  • 0 < rm < b and 0 ≤ ri < b for i = 0, 1, . . . , m − 1.Polynomial_item_12_24

Interpolation and approximation Polynomial_section_26

See also: Polynomial interpolation, Orthogonal polynomials, B-spline, and spline interpolation Polynomial_sentence_238

The simple structure of polynomial functions makes them quite useful in analyzing general functions using polynomial approximations. Polynomial_sentence_239

An important example in calculus is Taylor's theorem, which roughly states that every differentiable function locally looks like a polynomial function, and the Stone–Weierstrass theorem, which states that every continuous function defined on a compact interval of the real axis can be approximated on the whole interval as closely as desired by a polynomial function. Polynomial_sentence_240

Practical methods of approximation include polynomial interpolation and the use of splines. Polynomial_sentence_241

Other applications Polynomial_section_27

Polynomials are frequently used to encode information about some other object. Polynomial_sentence_242

The characteristic polynomial of a matrix or linear operator contains information about the operator's eigenvalues. Polynomial_sentence_243

The minimal polynomial of an algebraic element records the simplest algebraic relation satisfied by that element. Polynomial_sentence_244

The chromatic polynomial of a graph counts the number of proper colourings of that graph. Polynomial_sentence_245

The term "polynomial", as an adjective, can also be used for quantities or functions that can be written in polynomial form. Polynomial_sentence_246

For example, in computational complexity theory the phrase polynomial time means that the time it takes to complete an algorithm is bounded by a polynomial function of some variable, such as the size of the input. Polynomial_sentence_247

History Polynomial_section_28

Main articles: Cubic function § History, Quartic function § History, and Abel–Ruffini theorem § History Polynomial_sentence_248

Determining the roots of polynomials, or "solving algebraic equations", is among the oldest problems in mathematics. Polynomial_sentence_249

However, the elegant and practical notation we use today only developed beginning in the 15th century. Polynomial_sentence_250

Before that, equations were written out in words. Polynomial_sentence_251

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." Polynomial_sentence_252

We would write 3x + 2y + z = 29. Polynomial_sentence_253

History of the notation Polynomial_section_29

Main article: History of mathematical notation Polynomial_sentence_254

The earliest known use of the equal sign is in Robert Recorde's The Whetstone of Witte, 1557. Polynomial_sentence_255

The signs + for addition, − for subtraction, and the use of a letter for an unknown appear in Michael Stifel's Arithemetica integra, 1544. Polynomial_sentence_256

René Descartes, in La géometrie, 1637, introduced the concept of the graph of a polynomial equation. Polynomial_sentence_257

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. Polynomial_sentence_258

Descartes introduced the use of superscripts to denote exponents as well. Polynomial_sentence_259

See also Polynomial_section_30

Polynomial_unordered_list_13


Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Polynomial.