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Encyclopedia > Polynomial

In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. is a polynomial. Note in particular that division by an expression containing a variable is not generally allowed in polynomials.  Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... An expression is a combination of numbers, operators, grouping symbols (such as brackets and parentheses) and/or free variables and bound variables arranged in a meaningful way which can be evaluated. ... In computer science and mathematics, a variable (IPA pronunciation: ) (sometimes called a pronumeral) is a symbolic representation denoting a quantity or expression. ... In computer science and mathematics, a variable (IPA pronunciation: ) (sometimes called a pronumeral) is a symbolic representation denoting a quantity or expression. ... In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ...

Polynomials are built from terms called monomials, which consist of a constant (called the coefficient) multiplied by one or more variables (these are usually represented by letters). Each variable may have a constant positive whole number exponent. The exponent on a variable in a monomial is equal to the degree of that variable in that monomial. Since x = x1, the degree of a variable without a written exponent is one. A monomial with no variables is called a constant monomial, or just a constant. The degree of a constant term is 0. The coefficient of a monomial may be any number, including fractions, irrational numbers, and negative numbers. A polynomial that is constructed from one variable is called univariate. In elementary mathematics, a term is either a single number or variable, or the product of several numbers and/or variables. ... In mathematics, a monomial (or mononomial) is a particular kind of polynomial, having just one term. ... For other senses of this word, see coefficient (disambiguation). ... In computer science and mathematics, a variable is a symbol denoting a quantity or symbolic representation. ... In mathematics, exponentiation is a process generalized from repeated multiplication, in much the same way that multiplication is a process generalized from repeated addition. ... The degree of a polynomial is the maximum of the degrees of all terms in the polynomial. ...

For example,

is a monomial. The coefficient is -5, the variables are x and y, the degree of x is two, and the degree of y is one.

The degree of the entire monomial 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 one or more monomial terms. For example, the following is a polynomial:

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

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

A polynomial of degree one is called linear, of degree two is called quadratic, and of degree three is called cubic. Less commonly used, degree four is called quartic and degree five quintic.

A polynomial with one term is called a monomial, two terms a binomial, and three terms a trinomial.

A polynomial whose term of highest degree has a coefficient of 1 is called monic.

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, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... 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 from elementary algebra. ...

For example

is considered a polynomial because it is equivalent to . The coefficient is .

But,

is not a polynomial because it includes division by a variable, and neither is, in general,

because it has a variable exponent.

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

A polynomial function is a function defined by evaluating a polynomial. For example, the function f defined by An expression is a combination of numbers, operators, grouping symbols (such as brackets and parentheses) and/or free variables and bound variables arranged in a meaningful way which can be evaluated. ...

is a polynomial function. Polynomial functions are an important class of smooth functions. The adjective smooth comes from calculus. It means that it is always possible to take the derivative of a polynomial function, repeatedly, as often as is desired. The word smooth is also descriptive of the appearance of the graph of a polynomial function. In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...

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 the same (or equivalent). ...

is a polynomial equation.

## Elementary properties of polynomials

1. A sum of polynomials is a polynomial
2. A product of polynomials is a polynomial
3. The derivative of a polynomial is a polynomial
4. A primitive or antiderivative of a polynomial is a polynomial

Polynomials serve to approximate other functions, such as sine, cosine, and exponential. Addition is one of the basic operations of arithmetic. ... For a non-technical overview of the subject, see Calculus. ... In calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose derivative is equal to f, i. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... The exponential function is one of the most important functions in mathematics. ...

All polynomials have an expanded form, in which the distributive law has been used to remove all parentheses. All polynomials also have a factored form in which the polynomial is written as a product of linear polynomials. For example, the polynomial In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ...

is the expanded form of, and is equal to, the polynomial

,

which is written in factored form. Note that the constants in the linear polynomials (like -3 and +1 in the above example) may be imaginary in certain cases.

In school algebra, students learn to move easily from one form to the other (see: factoring). ...

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

.

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 In the mathematical subfield of numerical analysis, the Horner scheme or Horner algorithm, named after William George Horner, is an algorithm for the efficient evaluation of polynomials in monomial form. ...

.

In elementary algebra, methods are given for solving all first degree and second degree polynomial equations in one variable. 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. This fact is called the fundamental theorem of algebra. This article is about the branch of mathematics. ... In mathematics, the multiplicity of a member of a multiset is how many memberships in the multiset it has. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... In mathematics, the fundamental theorem of algebra states that every complex polynomial in one variable and of degree  â‰¥  has some complex root. ...

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. 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. If there are more equations than unknowns, the system is called overdetermined. This important subject is studied extensively in the area of mathematics known as linear algebra. 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.  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 arithmetic. ... In mathematics and linear algebra, a system of linear equations is a set of linear equations such as A standard problem is to decide if any assignment of values for the unknowns can satisfy all three equations simultaneously, and to find such an assignment if it exists. ... In mathematics and linear algebra, a system of linear equations is a set of linear equations such as A standard problem is to decide if any assignment of values for the unknowns can satisfy all three equations simultaneously, and to find such an assignment if it exists. ... In mathematics, a system of linear equations is considered overdetermined if there are more equations than unknowns. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ...

## 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 the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ... In linear algebra, one associates a polynomial to every square matrix, its characteristic polynomial. ... For the square matrix section, see square matrix. ... In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ...

In graph theory the chromatic polynomial of a graph encodes the different ways to vertex color the graph using x colors. A pictorial representation of a graph In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. ... In the mathematical field of graph theory the chromatic polynomial for a given graph is a polynomial which encodes the number of different ways to vertex color the graph using n colors. ... A 3-coloring suits this graph, but fewer colors would result in adjacent vertices of the same color. ...

In abstract algebra, one may define polynomials with coefficients in any ring. Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ...

In knot theory the Alexander polynomial, the Jones polynomial, and the HOMFLY polynomial are important knot invariants. Trefoil knot, the simplest non-trivial knot. ... In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. ... This article needs cleanup. ... In the mathematical field of knot theory, the HOMFLY polynomial, sometimes called the HOMFLY-PT polynomial or the generalized Jones polynomial, is a 2-variable knot polynomial, i. ... A knot invariant is a useful tool in knot theory. ...

## 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." We would write 3x + 2y + z = 29 The Nine Chapters on the Mathematical Art (ä¹ç« ç®—è¡“) is a Chinese mathematics book, probably composed in the 1st century AD, but perhaps as early as 200 BC. This book is the earliest surviving mathematical text from China that has come down to us by being copied by scribes and (centuries later...

The earliest known use of the equal sign is in Robert Recorde's The Whetstone of Witte, 1557. The signs + for addition, − for subtraction, and the use of a letter for an unknown appear in Michael Stifel's Arithemetica integra, 1544. René Descartes, in La geometrie, 1637, introduced the concept of the graph of a polynomial equation. He also 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.  Robert Recorde (c. ... Michael Stifel (1487 - 1567) was a German mathematician. ... â€œDescartesâ€ redirects here. ...

## 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. 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. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ... In mathematics, the fundamental theorem of algebra states that every complex polynomial in one variable and of degree  â‰¥  has some complex root. ...

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). 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). This result marked the start of Galois theory which engages in a detailed study of relationships among roots of polynomials. This article is about the term degree as used in mathematics. ... In mathematics, a quadratic equation is a polynomial equation of the second degree. ... (15th century - 16th century - 17th century - more centuries) As a means of recording the passage of time, the 16th century was that century which lasted from 1501 to 1600. ... Gerolamo Cardano. ... Niccolo Fontana Tartaglia. ... Niels Henrik Abel (August 5, 1802â€“April 6, 1829), Norwegian mathematician, was born in Nedstrand, near FinnÃ¸y where his father acted as rector. ... The Abel-Ruffini theorem states that there is no general solution in radicals to polynomial equations of degree five or higher. ... In mathematics, more specifically in abstract algebra, Galois theory, named after Ã‰variste Galois, provides a connection between field theory and group theory. ...

Approximate solutions to any polynomial equation can be found either by Newton's method or by one of the many more modern methods of approximating solutions. For a polynomial in Chebyshev form the Clenshaw algorithm can be used. As a practical matter, an approximate solution that is accurate to a desired precision may be as useful as an exact solution. In numerical analysis, Newtons method (also known as the Newtonâ€“Raphson method or the Newtonâ€“Fourier method) is an efficient algorithm for finding approximations to the zeros (or roots) of a real-valued function. ... In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev (&#1055;&#1072;&#1092;&#1085;&#1091;&#1090;&#1080;&#1081; &#1063;&#1077;&#1073;&#1099;&#1096;&#1105;&#1074;), are special polynomials. ... In the mathematical subfield of numerical analysis the Clenshaw algorithm (Invented by Charles William Clenshaw) is a recursive method to evaluate polynomials in Chebyshev form. ...

The difference engine of Charles Babbage was designed to create large tables of approximate values of logarithms and trigonometric functions automatically, by evaluating approximating polynomials at many points using Newton's method. Part of Babbages Difference engine, assembled after his death by Babbages son, using parts found in his laboratory. ... Babbage redirects here. ... Logarithms to various bases: is to base e, is to base 10, and is to base 1. ... In mathematics, the trigonometric functions (also called circular functions) are functions of an angle. ...

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. The reduction of equations in several unknowns to equations each in one unknown is discussed in the article on the Buchberger's algorithm. The special case where all the polynomials are of degree one is the subject of the article gaussian elimination. In numerical analysis, the Durand-Kerner method or method of Weierstrass is a root-finding algorithm for solving polynomial equations. ... A root-finding algorithm is a numerical method, or algorithm, for finding a value x such that f(x) = 0, for a given function f. ... In computational algebraic geometry and computational commutative algebra, Buchbergers algorithm is a method of transforming a given set of generators for a polynomial ideal into a Gröbner basis with respect to some monomial order. ... In mathematics, Gaussian elimination (not to be confused with Gaussâ€“Jordan elimination), named after Carl Friedrich Gauss, is an algorithm in linear algebra for determining the solutions of a system of linear equations, for determining the rank of a matrix, and for calculating the inverse of an invertible square 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. These characterizations of the roots of arbitrary polynomials are generalizations of the methods previously discovered to solve the quintic equation. In mathematics, a hypergeometric series could in principle be any formal power series in which the ratio of successive coefficients an/an-1 is a rational function of n. ... Carl Louis Ferdinand von Lindemann (April 12, 1852 - March 6, 1939) was a German mathematician, noted for his proof, published in 1882, that &#960; is a transcendental number, i. ... In mathematics, theta functions are special functions of several complex variables. ... In complex analysis, an elliptic function is, roughly speaking, a function defined on the complex plane which is periodic in two directions. ... Graph of a polynomial of degree 5, with 4 critical points. ...

## Graphs

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

• 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.
• The graph of a degree 2 polynomial
f(x) = a0 + a1x + a2x2, where a2 ≠ 0
is a parabola.
• The graph of any polynomial with degree 2 or greater
f(x) = a0 + a1x + a2x2 + . . . + anxn , where an ≠ 0 and n ≥ 2
is a continuous non-linear curve.

Polynomial graphs are analyzed in calculus using intercepts, slopes, concavity, and end behavior. Look up Slope in Wiktionary, the free dictionary. ... A parabola A graph showing the reflective property, the directrix (light blue), and the lines connecting the focus and directrix to the parabola (blue) In mathematics, the parabola (from the Greek: Ï€Î±ÏÎ±Î²Î¿Î»Î®) (IPA pronunciation: ) is a conic section generated by the intersection of a right circular conical surface and a plane...

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

Image File history File links Polynomial of degree 2 : f(x) = x2 - x - 2 File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Polynomial of degree 2 : f(x) = x2 - x - 2 File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Polynomial of degree 3: y = x3/5+4x2/5-7x/5-2=1/5 (x+5)(x+1)(x-2) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Polynomial of degree 3: y = x3/5+4x2/5-7x/5-2=1/5 (x+5)(x+1)(x-2) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Polynomial of degree 4: (x+4)(x+1)(x-1)(x-3)/14+0. ... Polynomial of degree 4: (x+4)(x+1)(x-1)(x-3)/14+0. ... Polynomial of degree 5: f(x) = (x+4)(x+2)(x+1)(x-1)(x-3)/20+2 File links The following pages link to this file: Polynomial Quintic function Categories: GFDL images ... Polynomial of degree 5: f(x) = (x+4)(x+2)(x+1)(x-1)(x-3)/20+2 File links The following pages link to this file: Polynomial Quintic function Categories: GFDL images ...

## Polynomials and calculus

One important aspect of calculus is the project of analyzing complicated functions by means of approximating them with polynomials. 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. Polynomials are also frequently used to interpolate functions. In mathematics, polynomials are perhaps the simplest functions with which to do calculus. ... In calculus, Taylors theorem, named after the mathematician Brook Taylor, who stated it in 1712, gives the approximation of a differentiable function near a point by a polynomial whose coefficients depend only on the derivatives of the function at that point. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on an interval [a,b] can be uniformly approximated as closely as desired by a polynomial function. ... 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, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... 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. Piecewise rationals are the only functions that can be evaluated directly on a computer, since typically only the operations of addition, multiplication, division and comparison are implemented in hardware. All the other functions that computers need to evaluate, such as trigonometric functions, logarithms and exponential functions, must then be approximated in software by suitable piecewise rational functions. In mathematics, a quotient is the end result of a division problem. ... In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ... In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ... In mathematics, a function f(x) of a real number variable x is defined piecewise, if f(x) is given by different expressions on various intervals. ... This article is about the machine. ... For other uses, see Hardware (disambiguation). ... In mathematics, the trigonometric functions (also called circular functions) are functions of an angle. ... Logarithms to various bases: is to base e, is to base 10, and is to base 1. ... The exponential function is one of the most important functions in mathematics. ... Computer software (or simply software) refers to one or more computer programs and data held in the storage of a computer for some purpose. ...

Calculating derivatives and integrals is particularly easy. For the polynomial

the derivative with respect to x is

and the indefinite integral is

.

## Abstract algebra

In abstract algebra, one must take care to distinguish between polynomials and polynomial functions. A polynomial f is defined to be a formal expression of the form In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring. ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ...

where the coefficients a0, ..., an are elements of some ring R and X is considered to be a formal symbol. Two polynomials are considered to be equal if and only if the sequences of their coefficients are equal. Polynomials with coefficients in R can be added by simply adding corresponding coefficients and multiplied using the distributive law and the rules In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ...

for all elements a of the ring R
for all natural numbers k and l.

One can then check that 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]. If R is commutative, then R[X] is an algebra over R. Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... 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 base field K is replaced by a commutative ring R. In this article, all rings and algebras are assumed to be unital and associative. ...

One can think of the ring R[X] as arising from R by adding one new element X to R and only requiring that X commute with all elements of R. In order for R[X] to form a ring, all sums of powers of X have to be included 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. For instance, the clean construction of finite fields involves the use of those operations, starting out with the field of integers modulo some prime number as the coefficient ring R (see modular arithmetic). In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ... In abstract algebra, a finite field or Galois field (so named in honor of Ã‰variste 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 and itself. ... Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic because of its use in the 24-hour clock system) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value â€” the modulus. ...

To every polynomial f in R[X], one can associate a polynomial function with domain and range equal to R. One obtains the value of this function for a given argument r by everywhere replacing the symbol X in f’s expression by r. The reason that algebraists have to distinguish between polynomials and polynomial functions is that over some rings R, two 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). This is not the case over the real or complex numbers and therefore many analysts often don't separate the two concepts. Fermats little theorem (not to be confused with Fermats last theorem) states that if p is a prime number, then for any integer a, This means that if you start with a number, initialized to 1, and repeatedly multiply, for a total of p multiplications, that number by...

## Divisibility

In commutative algebra, one major focus of study is divisibility among polynomials. 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. 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 (Xr) divides f. The converse is also true. The quotient can be computed using the Horner scheme. In abstract algebra, commutative algebra studies commutative rings, their ideals, and modules over such rings. ... In abstract algebra, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 â‰  1, in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. ...

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 In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

and such that the degree of r is smaller than the degree of g. 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 lower degree, a generalized version of the familiar arithmetic technique called long division. ... In abstract algebra, a Euclidean domain (also called a Euclidean ring) is a type of ring in which the Euclidean algorithm can be used. ...

Analogously, polynomial "primes" (more correctly, irreducible polynomials) can be defined which cannot be factorized into the product of two polynomials of lesser degree. 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, the adjective irreducible means that an object cannot be expressed as a product of at least two non-trivial factors in a given ring. ... This article is about the term degree as used in mathematics. ... In mathematics, Eisensteins criterion gives sufficient conditions for a polynomial to be irreducible over Q (or equivalently, over Z). ...

See also: Greatest common divisor of two polynomials. This article or section is in need of attention from an expert on the subject. ...

## 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 the language and the problematics of geometry. ...

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. The minimal polynomial of an algebraic element records the simplest algebraic relation satisfied by that element. In linear algebra, one associates a polynomial to every square matrix, its characteristic polynomial. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... In mathematics, the minimal polynomial of an object Î± is the monic polynomial p of least degree such that p(Î±)=0. ... 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 have natural notions of convergence. They are also useful to compactly describe sequences and to find closed formulas for recursively defined sequences; this is... In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ...

Look up polynomial in Wiktionary, the free dictionary.

Wikipedia does not have an article with this exact name. ... Wiktionary (a portmanteau of wiki and dictionary) is a multilingual, Web-based project to create a free content dictionary, available in over 150 languages. ... In mathematics, a monomial (or mononomial) is a particular kind of polynomial, having just one term. ... In the mathematical subfield of numerical analysis, a trigonometric polynomial is a finite linear linear combination of sin(nx) and cos(nx) with n a natural number. ... In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial. ... In mathematics, integral polytopes have associated Ehrhart polynomials which encode some geometrical information about them. ... In mathematics, a homogeneous polynomial is a polynomial whose terms are monomials all having the same total degree; or are elements of the same dimension. ... A Hurwitz polynomial is a polynomial whose coefficients are positive real numbers and whose zeros are located in the left half-plane of the complex plane, that is, the real part of every zero is negative. ... In the mathematical subfield of numerical analysis, polynomial interpolation is the interpolation of a given data set by a polynomial. ... Definition In mathematics, a polynomial sequence, i. ... In mathematics, a polynomial sequence, i. ... One type of spline, a bÃ©zier curve In the mathematical subfield of numerical analysis, a spline is a special function defined piecewise by polynomials. ... In mathematics, a symmetric polynomial is a polynomial in n variables , such that if some of the variables are interchanged, the polynomial stays the same. ... In linear algebra, one associates a polynomial to every square matrix, its characteristic polynomial. ... In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. ... This is a list of polynomial topics, by Wikipedia page. ... Results from FactBites:

 Polynomial - Wikipedia, the free encyclopedia (2422 words) Because of their simple structure, polynomials are easy to evaluate, and are used extensively in numerical analysis for polynomial interpolation or to numerically integrate more complex functions. The degree of a term in a polynomial is the sum of all of the exponents on the variables in that term, where a variable with no exponent is understood to have an exponent of 1. In knot theory the Alexander polynomial, the Jones polynomial, and the HOMFLY polynomial are important knot invariants.
 Encyclopedia4U - Knot polynomial - Encyclopedia Article (1116 words) Technically, an Alexander polynomial is a generator of a principal Alexander ideal related to the homology of the infinitely cyclic cover of a knot complement—where all the emphasised phrases have particular mathematical meanings. (this is an Alexander polynomial of the knot). HOMFLYPT is a binary (two-variable) polynomial, with as with the predecessors.
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