{"history":[],"article":{"id":"779f9c83-e4b2-4338-9ce5-244d77b7d801","title":"Polynomial","slug":"polynomial","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), a **polynomial** is a [mathematical expression](https://bliptext.com/articles/expression-mathematics) consisting of [indeterminates](https://bliptext.com/articles/indeterminate-variable) (also called [variables](https://bliptext.com/articles/variable-mathematics)) and [coefficients](https://bliptext.com/articles/coefficient), that involves only the operations of [addition](https://bliptext.com/articles/addition), [subtraction](https://bliptext.com/articles/subtraction), [multiplication](https://bliptext.com/articles/multiplication) and [exponentiation](https://bliptext.com/articles/exponentiation) to [nonnegative integer](https://bliptext.com/articles/nonnegative-integer) powers, and has a finite number of terms.\nAn example of a polynomial of a single indeterminate\nx\nis\nx2 − 4x + 7\n. An example with three indeterminates is\nx3 + 2xyz2 − yz + 1\n.\nPolynomials appear in many areas of mathematics and science.\nFor example, they are used to form [polynomial equations](https://bliptext.com/articles/polynomial-equation), which encode a wide range of problems, from elementary [word problems](https://bliptext.com/articles/word-problem-mathematics-education) to complicated scientific problems; they are used to define **polynomial functions**, which appear in settings ranging from basic [chemistry](https://bliptext.com/articles/chemistry) and [physics](https://bliptext.com/articles/physics) to [economics](https://bliptext.com/articles/economics) and [social science](https://bliptext.com/articles/social-science); and they are used in [calculus](https://bliptext.com/articles/calculus) and [numerical analysis](https://bliptext.com/articles/numerical-analysis) to approximate other functions.\nIn advanced mathematics, polynomials are used to construct [polynomial rings](https://bliptext.com/articles/polynomial-ring) and [algebraic varieties](https://bliptext.com/articles/[algebra](https://bliptext.com/articles/algebra)ic-variety), which are central concepts in [algebra](https://bliptext.com/articles/algebra) and [algebra](https://bliptext.com/articles/algebra)ic geometry.\n\n# Etymology\nThe word polynomial [joins two diverse roots](https://bliptext.com/articles/hybrid-word): the Greek poly, meaning \"many\", and the Latin nomen, or \"name\".\nIt was derived from the term [binomial](https://bliptext.com/articles/binomial-polynomial) by replacing the Latin root bi- with the Greek poly-.\nThat is, it means a sum of many terms (many [monomials](https://bliptext.com/articles/monomial)).\nThe word polynomial was first used in the 17th century.\n\n# Notation and terminology\nThe x occurring in a polynomial is commonly called a variable or an indeterminate.\nWhen the polynomial is considered as an expression, x is a fixed symbol which does not have any value (its value is \"indeterminate\").\nHowever, when one considers the [function](https://bliptext.com/articles/function-mathematics) defined by the polynomial, then x represents the argument of the [function](https://bliptext.com/articles/function-mathematics), and is therefore called a \"variable\".\nMany authors use these two words interchangeably.\nA polynomial P in the indeterminate x is commonly denoted either as P or as P(x).\nFormally, the name of the polynomial is P, not P(x), but the use of the [functional notation](https://bliptext.com/articles/functional-notation) P(x) dates from a time when the distinction between a polynomial and the associated function was unclear.\nMoreover, the functional notation is often useful for specifying, in a single phrase, a polynomial and its indeterminate.\nFor example, \"let P(x) be a polynomial\" is a shorthand for \"let P be a polynomial in the indeterminate x\".\nOn 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.\nThe 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.\nIf 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.\nThus, the polynomial P defines the function\nwhich is the polynomial function associated to P.\nFrequently, when using this notation, one supposes that a is a number.\nHowever, one may use it over any domain where addition and multiplication are defined (that is, any [ring](https://bliptext.com/articles/ring-mathematics)).\nIn particular, if a is a polynomial then P(a) is also a polynomial.\nMore specifically, when a is the indeterminate x, then the [image](https://bliptext.com/articles/image-mathematics) of x by this function is the polynomial P itself (substituting x for x does not change anything).\nIn other words,\nP(x)=P,\nwhich justifies formally the existence of two notations for the same polynomial.\n# [The graph of a polynomial function of degree 3 | upload.wikimedia.org/wikipedia/commons/a/a3/Polynomialdeg3.svg]\n\n# Definition\nA polynomial [expression](https://bliptext.com/articles/expression-mathematics) is an [expression](https://bliptext.com/articles/expression-mathematics) that can be built from [constants](https://bliptext.com/articles/constant-mathematics) and symbols called variables or indeterminates by means of [addition](https://bliptext.com/articles/addition), [multiplication](https://bliptext.com/articles/multiplication) and [exponentiation](https://bliptext.com/articles/exponentiation) to a [non-negative integer](https://bliptext.com/articles/non-negative-integer) power.\nThe constants are generally [numbers](https://bliptext.com/articles/number), but may be any expression that do not involve the indeterminates, and represent [mathematical objects](https://bliptext.com/articles/mathematical-object) that can be added and multiplied.\nTwo polynomial expressions are considered as defining the same polynomial if they may be transformed, one to the other, by applying the usual properties of [commutativity](https://bliptext.com/articles/commutative-property), [associativity](https://bliptext.com/articles/associative-property) and [distributivity](https://bliptext.com/articles/distributive-property) of addition and multiplication.\nFor example (x-1)(x-2) and x^2-3x+2 are two polynomial expressions that represent the same polynomial; so, one has the [equality](https://bliptext.com/articles/equality-mathematics).\nA polynomial in a single indeterminate\nx\ncan always be written (or rewritten) in the form\nwhere are constants that are called the coefficients of the polynomial, and x is the indeterminate.\nThe word \"indeterminate\" means that x represents no particular value, although any value may be substituted for it.\nThe mapping that associates the result of this substitution to the substituted value is a [function](https://bliptext.com/articles/function-mathematics), called a polynomial [function](https://bliptext.com/articles/function-mathematics).\nThis can be expressed more concisely by using [summation notation](https://bliptext.com/articles/summation):\nThat is, a polynomial can either be zero or can be written as the sum of a finite number of non-zero [terms](https://bliptext.com/articles/summand).\nEach term consists of the product of a number – called the [coefficient](https://bliptext.com/articles/coefficient) of the term – and a finite number of indeterminates, raised to non-negative integer powers.\n\n# Classification\nThe 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.\nBecause\n, the degree of an indeterminate without a written exponent is one.\nA term with no indeterminates and a polynomial with no indeterminates are called, respectively, a [constant term](https://bliptext.com/articles/constant-term) and a **constant polynomial**.\nThe degree of a constant term and of a nonzero constant polynomial is 0.\nThe degree of the zero polynomial 0 (which has no terms at all) is generally treated as not defined (but see below).\nFor example:\n-5x^2y\nis a term.\nThe coefficient is\n−5\n, the indeterminates are\nx\nand\ny\n, the degree of\nx\nis two, while the degree of\ny\nis one.\nThe degree of the entire term is the sum of the degrees of each indeterminate in it, so in this example the degree is\n2 + 1 = 3\n.\nForming a sum of several terms produces a polynomial.\nFor example, the following is a polynomial:\nIt consists of three terms: the first is degree two, the second is degree one, and the third is degree zero.\nPolynomials of small degree have been given specific names.\nA polynomial of degree zero is a constant polynomial, or simply a constant.\nPolynomials of degree one, two or three are respectively linear polynomials, [quadratic polynomials](https://bliptext.com/articles/quadratic-polynomial) and cubic polynomials.\nFor higher degrees, the specific names are not commonly used, although quartic polynomial (for degree four) and quintic polynomial (for degree five) are sometimes used.\nThe names for the degrees may be applied to the polynomial or to its terms.\nFor example, the term\n2x\nin\nx2 + 2x + 1\nis a linear term in a quadratic polynomial.\nThe polynomial 0, which may be considered to have no terms at all, is called the **zero polynomial**.\nUnlike other constant polynomials, its degree is not zero.\nRather, the degree of the zero polynomial is either left explicitly undefined, or defined as negative (either −1 or −∞).\nThe zero polynomial is also unique in that it is the only polynomial in one indeterminate that has an infinite number of [roots](https://bliptext.com/articles/root-of-a-function).\nThe graph of the zero polynomial,\n, is the x-axis.\nIn the case of polynomials in more than one indeterminate, a polynomial is called homogeneous of degree\nn if all of its non-zero terms have degree\nn. The zero polynomial is homogeneous, and, as a homogeneous polynomial, its degree is undefined.\nFor example,\nx3y2 + 7x2y3 − 3x5\nis homogeneous of degree 5.\nFor more details, see [Homogeneous polynomial](https://bliptext.com/articles/homogeneous-polynomial).\nThe [commutative law](https://bliptext.com/articles/commutative-law) of addition can be used to rearrange terms into any preferred order.\nIn polynomials with one indeterminate, the terms are usually ordered according to degree, either in \"descending powers of\nx\n\", with the term of largest degree first, or in \"ascending powers of\nx\n\". The polynomial\n3x2 − 5x + 4\nis written in descending powers of\nx\n. The first term has coefficient\n3\n, indeterminate\nx\n, and exponent\n2\n. In the second term, the coefficient is\n−5. The third term is a constant.\nBecause the degree of a non-zero polynomial is the largest degree of any one term, this polynomial has degree two.\nTwo 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](https://bliptext.com/articles/distributive-law), into a single term whose coefficient is the sum of the coefficients of the terms that were combined.\nIt may happen that this makes the coefficient 0.\nPolynomials can be classified by the number of terms with nonzero coefficients, so that a one-term polynomial is called a [monomial](https://bliptext.com/articles/monomial), a two-term polynomial is called a [binomial](https://bliptext.com/articles/binomial-polynomial), and a three-term polynomial is called a [trinomial](https://bliptext.com/articles/trinomial).\nA **[real](https://bliptext.com/articles/real-number) polynomial** is a polynomial with [real](https://bliptext.com/articles/real-number) coefficients.\nWhen it is used to define a [function](https://bliptext.com/articles/function-mathematics), the [domain](https://bliptext.com/articles/domain-function) is not so restricted.\nHowever, a **real polynomial function** is a function from the reals to the reals that is defined by a real polynomial.\nSimilarly, an **[integer](https://bliptext.com/articles/integer) polynomial** is a polynomial with [integer](https://bliptext.com/articles/integer) coefficients, and a **[complex](https://bliptext.com/articles/complex-number) polynomial** is a polynomial with [complex](https://bliptext.com/articles/complex-number) coefficients.\nA polynomial in one indeterminate is called a **univariate polynomial**, a polynomial in more than one indeterminate is called a **multivariate polynomial**.\nA polynomial with two indeterminates is called a **bivariate polynomial**.\nThese 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.\nIt is possible to further classify multivariate polynomials as bivariate, trivariate, and so on, according to the maximum number of indeterminates allowed.\nAgain, so that the set of objects under consideration be closed under subtraction, a study of trivariate polynomials usually allows bivariate polynomials, and so on.\nIt is also common to say simply \"polynomials in\nx, y\n, and\nz\n\", listing the indeterminates allowed.\n\n# Operations\n\n## Addition and subtraction\nPolynomials can be added using the [associative law](https://bliptext.com/articles/associative-law) of addition (grouping all their terms together into a single sum), possibly followed by reordering (using the [commutative law](https://bliptext.com/articles/commutative-law)) and combining of like terms.\nFor example, if and\nthen the sum\ncan be reordered and regrouped as\nand then simplified to\nWhen polynomials are added together, the result is another polynomial.\nSubtraction of polynomials is similar.\n\n## Multiplication\nPolynomials can also be multiplied.\nTo expand the [product](https://bliptext.com/articles/product-mathematics) 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.\nFor example, if\nthen\nCarrying out the multiplication in each term produces\nCombining similar terms yields\nwhich can be simplified to\nAs in the example, the product of polynomials is always a polynomial.\n\n## Composition\nGiven a polynomial f of a single variable and another polynomial g of any number of variables, the [composition](https://bliptext.com/articles/function-composition) f \\circ g is obtained by substituting each copy of the variable of the first polynomial by the second polynomial.\nFor example, if and then\nA composition may be expanded to a sum of terms using the rules for multiplication and division of polynomials.\nThe composition of two polynomials is another polynomial.\n\n## Division\nThe division of one polynomial by another is not typically a polynomial.\nInstead, such ratios are a more general family of objects, called [rational fractions](https://bliptext.com/articles/rational-fraction), rational expressions, or [rational functions](https://bliptext.com/articles/rational-function), depending on context.\nThis is analogous to the fact that the ratio of two [integers](https://bliptext.com/articles/integer) is a [rational number](https://bliptext.com/articles/rational-number), not necessarily an integer.\nFor example, the fraction\n1/(x2 + 1)\nis not a polynomial, and it cannot be written as a finite sum of powers of the variable x.\nFor polynomials in one variable, there is a notion of [Euclidean division of polynomials](https://bliptext.com/articles/euclidean-division-of-polynomials), generalizing the [Euclidean division](https://bliptext.com/articles/euclidean-division) of integers.\nThis notion of the division\na(x)/b(x)\nresults in two polynomials, a quotient\nq(x)\nand a remainder\nr(x)\n, such that\nand\ndegree(r) < degree(b)\n. The quotient and remainder may be computed by any of several algorithms, including [polynomial long division](https://bliptext.com/articles/polynomial-long-division) and [synthetic division](https://bliptext.com/articles/synthetic-division).\nWhen the denominator\nb(x)\nis [monic](https://bliptext.com/articles/monic-polynomial) and linear, that is,\nb(x) = x − c\nfor some constant c, then the [polynomial remainder theorem](https://bliptext.com/articles/polynomial-remainder-theorem) asserts that the remainder of the division of\na(x)\nby\nb(x)\nis the [evaluation](https://bliptext.com/articles/)\na(c)\n. In this case, the quotient may be computed by [Ruffini's rule](https://bliptext.com/articles/ruffini-s-rule), a special case of synthetic division.\n\n## Factoring\nAll polynomials with coefficients in a [unique factorization domain](https://bliptext.com/articles/unique-factorization-domain) (for example, the integers or a [field](https://bliptext.com/articles/field-mathematics)) also have a factored form in which the polynomial is written as a product of [irreducible polynomials](https://bliptext.com/articles/irreducible-polynomial) and a constant.\nThis factored form is unique up to the order of the factors and their multiplication by an invertible constant.\nIn the case of the field of [complex numbers](https://bliptext.com/articles/complex-number), the irreducible factors are linear.\nOver the [real numbers](https://bliptext.com/articles/real-number), they have the degree either one or two.\nOver the integers and the [rational numbers](https://bliptext.com/articles/rational-number) the irreducible factors may have any degree.\nFor example, the factored form of 5x^3-5\nis\nover the integers and the reals, and\nover the complex numbers.\nThe computation of the factored form, called factorization is, in general, too difficult to be done by hand-written computation.\nHowever, efficient [polynomial factorization](https://bliptext.com/articles/factorization-of-polynomials) [algorithms](https://bliptext.com/articles/algorithm) are available in most [computer algebra systems](https://bliptext.com/articles/computer-algebra-system).\n\n## Calculus\nCalculating [derivatives](https://bliptext.com/articles/derivative) and integrals of polynomials is particularly simple, compared to other kinds of functions.\nThe [derivative](https://bliptext.com/articles/derivative) of the polynomial with respect to x is the polynomial\nSimilarly, the general [antiderivative](https://bliptext.com/articles/antiderivative) (or indefinite integral) of P is\nwhere c is an arbitrary constant.\nFor example, antiderivatives of\nx2 + 1\nhave the form\n1⁄3x3 + x + c\n.\nFor polynomials whose coefficients come from more abstract settings (for example, if the coefficients are integers [modulo](https://bliptext.com/articles/modular-arithmetic) some [prime number](https://bliptext.com/articles/prime-number)\np\n, or elements of an arbitrary ring), the formula for the derivative can still be interpreted formally, with the coefficient\nkak\nunderstood to mean the sum of k copies of\nak\n. For example, over the integers modulo\np\n, the derivative of the polynomial\nxp + x\nis the polynomial\n1\n.\n\n# Polynomial functions\nA polynomial function is a function that can be defined by [evaluating](https://bliptext.com/articles/) a polynomial.\nMore precisely, a function\nf\nof one [argument](https://bliptext.com/articles/argument-of-a-function) from a given domain is a polynomial function if there exists a polynomial\nthat evaluates to f(x) for all x in the [domain](https://bliptext.com/articles/domain-of-a-function) of f (here,\nn\nis a non-negative integer and\na0, a1, a2, ..., an\nare constant coefficients). Generally, unless otherwise specified, polynomial functions have [complex](https://bliptext.com/articles/complex-number) coefficients, arguments, and values. In particular, a polynomial, [restricted](https://bliptext.com/articles/restriction-of-a-function) to have real coefficients, defines a function from the [complex](https://bliptext.com/articles/complex-number) numbers to the [complex](https://bliptext.com/articles/complex-number) numbers. If the domain of this function is also [restricted](https://bliptext.com/articles/restriction-of-a-function) to the reals, the resulting function is a [real function](https://bliptext.com/articles/real-function) that maps reals to reals.\nFor example, the function\nf\n, defined by\nis a polynomial function of one variable.\nPolynomial functions of several variables are similarly defined, using polynomials in more than one indeterminate, as in\nAccording to the definition of polynomial functions, there may be expressions that obviously are not polynomials but nevertheless define polynomial functions.\nAn example is the expression which takes the same values as the polynomial 1-x^2 on the interval [-1,1], and thus both expressions define the same polynomial function on this interval.\nEvery polynomial function is [continuous](https://bliptext.com/articles/continuous-function), [smooth](https://bliptext.com/articles/smooth-function), and [entire](https://bliptext.com/articles/entire-function).\nThe **evaluation** of a polynomial is the computation of the corresponding polynomial function; that is, the **evaluation** consists of substituting a numerical value to each indeterminate and carrying out the indicated multiplications and additions.\nFor polynomials in one indeterminate, the evaluation is usually more efficient (lower number of arithmetic operations to perform) using [Horner's method](https://bliptext.com/articles/horner-s-method), which consists of rewriting the polynomial as\n\n## Graphs\nA polynomial function in one real variable can be represented by a [graph](https://bliptext.com/articles/graph-of-a-function).\n

\nThe graph of the zero polynomial\nis the\nx\n-axis.\n
\nThe graph of a degree 0 polynomial\nis a horizontal line with y\n-intercept\na0\n
\nThe graph of a degree 1 polynomial (or linear function)\nis an oblique line with y\n-intercept\na0 and [slope](https://bliptext.com/articles/slope)\na1\n.
\nThe graph of a degree 2 polynomial\nis a [parabola](https://bliptext.com/articles/parabola).\n
\nThe graph of a degree 3 polynomial\nis a [cubic curve](https://bliptext.com/articles/cubic-equation).\n
\nThe graph of any polynomial with degree 2 or greater\nis a continuous non-linear curve.\n

\nA non-constant polynomial function [tends to infinity](https://bliptext.com/articles/infinity) when the variable increases indefinitely (in [absolute value](https://bliptext.com/articles/absolute-value)).\nIf the degree is higher than one, the graph does not have any [asymptote](https://bliptext.com/articles/asymptote).\nIt has two [parabolic branches](https://bliptext.com/articles/parabolic-branch) with vertical direction (one branch for positive x and one for negative x).\nPolynomial graphs are analyzed in calculus using intercepts, slopes, concavity, and end behavior.\n\n# Equations\nA polynomial [equation](https://bliptext.com/articles/equation), also called an [algebraic equation](https://bliptext.com/articles/algebraic-[equation](https://bliptext.com/articles/equation)), is an [equation](https://bliptext.com/articles/equation) of the form\nFor example,\nis a polynomial equation.\nWhen considering equations, the indeterminates (variables) of polynomials are also called [unknowns](https://bliptext.com/articles/variable-mathematics), and the solutions are the possible values of the [unknowns](https://bliptext.com/articles/variable-mathematics) for which the equality is true (in general more than one solution may exist).\nA polynomial equation stands in contrast to a polynomial [identity](https://bliptext.com/articles/identity-mathematics) like\n(x + y)(x − y) = x2 − y2\n, where both expressions represent the same polynomial in different forms, and as a consequence any evaluation of both members gives a valid equality.\nIn elementary [algebra](https://bliptext.com/articles/algebra), methods such as the [quadratic formula](https://bliptext.com/articles/quadratic-formula) are taught for solving all first degree and second degree polynomial equations in one variable.\nThere are also formulas for the [cubic](https://bliptext.com/articles/cubic-equation) and [quartic equations](https://bliptext.com/articles/quartic-equation).\nFor higher degrees, the [Abel–Ruffini theorem](https://bliptext.com/articles/abel-ruffini-theorem) asserts that there can not exist a general formula in radicals.\nHowever, [root-finding algorithms](https://bliptext.com/articles/root-finding-algorithm) may be used to find [numerical approximations](https://bliptext.com/articles/numerical-approximation) of the roots of a polynomial expression of any degree.\nThe number of solutions of a polynomial equation with real coefficients may not exceed the degree, and equals the degree when the [complex](https://bliptext.com/articles/complex-number) solutions are counted with their [multiplicity](https://bliptext.com/articles/multiplicity-mathematics).\nThis fact is called the [fundamental theorem of algebra](https://bliptext.com/articles/fundamental-theorem-of-algebra).\n\n## Solving equations\nA root of a nonzero univariate polynomial\nP\nis a value a of x such that\n. In other words, a root of P is a solution of the [polynomial equation](https://bliptext.com/articles/polynomial-equation)\nor a [zero](https://bliptext.com/articles/zero-of-a-function) of the polynomial function defined by\nP\n. In the case of the zero polynomial, every number is a zero of the corresponding function, and the concept of root is rarely considered.\nA number\na\nis a root of a polynomial\nP\nif and only if the [linear polynomial](https://bliptext.com/articles/)\nx − a\ndivides\nP\n, that is if there is another polynomial\nQ\nsuch that\nP = (x − a) Q\n. It may happen that a power (greater than\n1\n) of\nx − a\ndivides\nP\na\nis a multiple root of\nP\n, and otherwise\na\nis a **simple root** of\nP\n. If\nP\nis a nonzero polynomial, there is a highest power\nm\nsuch that\n(x − a)m\ndivides\nP\n, which is called the multiplicity of\na\nas a root of\nP\n. The number of roots of a nonzero polynomial\nP\n, counted with their respective multiplicities, cannot exceed the degree of\nP\n, and equals this degree if all [complex](https://bliptext.com/articles/complex-number) roots are considered (this is a consequence of the [fundamental theorem of algebra](https://bliptext.com/articles/fundamental-theorem-of-algebra)).\nThe coefficients of a polynomial and its roots are related by [Vieta's formulas](https://bliptext.com/articles/vieta-s-formulas).\nSome polynomials, such as\nx2 + 1\n, do not have any roots among the [real numbers](https://bliptext.com/articles/real-number).\nIf, however, the set of accepted solutions is expanded to the [complex numbers](https://bliptext.com/articles/complex-number), every non-constant polynomial has at least one root; this is the [fundamental theorem of algebra](https://bliptext.com/articles/fundamental-theorem-of-algebra).\nBy successively dividing out factors\nx − a\n, 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.\nThere may be several meanings of [\"solving an equation\"](https://bliptext.com/articles/equation-solving).\nOne may want to express the solutions as explicit numbers; for example, the unique solution of\n2x − 1 = 0\nis\n1/2\n. This is, in general, impossible for equations of degree greater than one, and, since the ancient times, mathematicians have searched to express the solutions as [algebraic expressions](https://bliptext.com/articles/algebraic-expression); for example, the [golden ratio](https://bliptext.com/articles/golden-ratio) is the unique positive solution of x^2-x-1=0.\nIn the ancient times, they succeeded only for degrees one and two.\nFor [quadratic equations](https://bliptext.com/articles/quadratic-equation), the [quadratic formula](https://bliptext.com/articles/quadratic-formula) provides such expressions of the solutions.\nSince the 16th century, similar formulas (using cube roots in addition to square roots), although much more complicated, are known for equations of degree three and four (see [cubic equation](https://bliptext.com/articles/cubic-equation) and [quartic equation](https://bliptext.com/articles/quartic-equation)).\nBut formulas for degree 5 and higher eluded researchers for several centuries.\nIn 1824, [Niels Henrik Abel](https://bliptext.com/articles/niels-henrik-abel) proved the striking result that there are equations of degree 5 whose solutions cannot be expressed by a (finite) formula, involving only arithmetic operations and radicals (see [Abel–Ruffini theorem](https://bliptext.com/articles/abel-ruffini-theorem)).\nIn 1830, [Évariste Galois](https://bliptext.com/articles/variste-galois) proved that most equations of degree higher than four cannot be solved by radicals, and showed that for each equation, one may decide whether it is solvable by radicals, and, if it is, solve it.\nThis result marked the start of [Galois theory](https://bliptext.com/articles/galois-theory) and [group theory](https://bliptext.com/articles/group-theory), two important branches of modern [algebra](https://bliptext.com/articles/algebra).\nGalois himself noted that the computations implied by his method were impracticable.\nNevertheless, formulas for solvable equations of degrees 5 and 6 have been published (see [quintic function](https://bliptext.com/articles/quintic-function) and [sextic equation](https://bliptext.com/articles/sextic-equation)).\nWhen 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 it is to compute [numerical approximations](https://bliptext.com/articles/numerical-approximation) of the solutions.\nThere are many methods for that; some are restricted to polynomials and others may apply to any [continuous function](https://bliptext.com/articles/continuous-function).\nThe most efficient [algorithms](https://bliptext.com/articles/algorithm) allow solving easily (on a [computer](https://bliptext.com/articles/computer)) polynomial equations of degree higher than 1,000 (see [Root-finding algorithm](https://bliptext.com/articles/root-finding-algorithm)).\nFor polynomials with 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\".\nThe study of the sets of zeros of polynomials is the object of [algebraic geometry](https://bliptext.com/articles/algebraic-geometry).\nFor a set of polynomial equations with several unknowns, there are [algorithms](https://bliptext.com/articles/algorithm) to decide whether they have a finite number of [complex](https://bliptext.com/articles/complex-number) solutions, and, if this number is finite, for computing the solutions.\nSee [System of polynomial equations](https://bliptext.com/articles/system-of-polynomial-equations).\nThe special case where all the polynomials are of degree one is called a [system of linear equations](https://bliptext.com/articles/system-of-linear-equations), for which another range of different [solution methods](https://bliptext.com/articles/system-of-linear-equations) exist, including the classical [Gaussian elimination](https://bliptext.com/articles/gaussian-elimination).\nA polynomial equation for which one is interested only in the solutions which are [integers](https://bliptext.com/articles/integer) is called a [Diophantine equation](https://bliptext.com/articles/diophantine-equation).\nSolving Diophantine equations is generally a very hard task.\nIt has been proved that there cannot be any general [algorithm](https://bliptext.com/articles/algorithm) for solving them, or even for deciding whether the set of solutions is empty (see [Hilbert's tenth problem](https://bliptext.com/articles/hilbert-s-tenth-problem)).\nSome of the most famous problems that have been solved during the last fifty years are related to Diophantine equations, such as [Fermat's Last Theorem](https://bliptext.com/articles/fermat-s-last-theorem).\n\n# Polynomial expressions\nPolynomials where indeterminates are substituted for some other mathematical objects are often considered, and sometimes have a special name.\n\n## Trigonometric polynomials\nA **trigonometric polynomial** is a finite [linear combination](https://bliptext.com/articles/linear-combination) of [functions](https://bliptext.com/articles/function-mathematics) sin(nx) and cos(nx) with n taking on the values of one or more [natural numbers](https://bliptext.com/articles/natural-number).\nThe coefficients may be taken as real numbers, for real-valued functions.\nIf 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](https://bliptext.com/articles/list-of-trigonometric-identities)).\nConversely, every polynomial in sin(x) and cos(x) may be converted, with [Product-to-sum identities](https://bliptext.com/articles/list-of-trigonometric-identities), into a linear combination of functions sin(nx) and cos(nx).\nThis equivalence explains why linear combinations are called polynomials.\nFor [complex coefficients](https://bliptext.com/articles/complex-number), there is no difference between such a function and a finite [Fourier series](https://bliptext.com/articles/fourier-series).\nTrigonometric polynomials are widely used, for example in [trigonometric interpolation](https://bliptext.com/articles/trigonometric-[interpolation](https://bliptext.com/articles/interpolation)) applied to the [interpolation](https://bliptext.com/articles/interpolation) of [periodic functions](https://bliptext.com/articles/periodic-function).\nThey are also used in the [discrete Fourier transform](https://bliptext.com/articles/discrete-fourier-transform).\n\n## Matrix polynomials\nA [matrix polynomial](https://bliptext.com/articles/matrix-polynomial) is a polynomial with [square matrices](https://bliptext.com/articles/square-matrix) as variables.\nGiven an ordinary, scalar-valued polynomial\nthis polynomial evaluated at a matrix A is\nwhere I is the [identity matrix](https://bliptext.com/articles/identity-matrix).\nA **matrix polynomial equation** is an equality between two matrix polynomials, which holds for the specific matrices in question.\nA **matrix polynomial identity** is a matrix polynomial equation which holds for all matrices A in a specified [matrix ring](https://bliptext.com/articles/matrix-ring) Mn(R).\n\n## Exponential polynomials\nA bivariate polynomial where the second variable is substituted for an exponential function applied to the first variable, for example\nP(x, ex)\n, may be called an [exponential polynomial](https://bliptext.com/articles/exponential-polynomial).\n\n# Related concepts\n\n## Rational functions\nA [rational fraction](https://bliptext.com/articles/rational-fraction) is the [quotient](https://bliptext.com/articles/quotient) ([algebraic fraction](https://bliptext.com/articles/algebraic-fraction)) of two polynomials.\nAny [algebraic expression](https://bliptext.com/articles/algebraic-expression) that can be rewritten as a rational fraction is a [rational function](https://bliptext.com/articles/rational-function).\nWhile 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.\nThe rational fractions include the Laurent polynomials, but do not limit denominators to powers of an indeterminate.\n\n## Laurent polynomials\n[Laurent polynomials](https://bliptext.com/articles/laurent-polynomial) are like polynomials, but allow negative powers of the variable(s) to occur.\n\n## Power series\n[Formal power series](https://bliptext.com/articles/formal-power-series) are like polynomials, but allow infinitely many non-zero terms to occur, so that they do not have finite degree.\nUnlike polynomials they cannot in general be explicitly and fully written down (just like [irrational numbers](https://bliptext.com/articles/irrational-number) cannot), but the rules for manipulating their terms are the same as for polynomials.\nNon-formal [power series](https://bliptext.com/articles/power-series) also generalize polynomials, but the multiplication of two [power series](https://bliptext.com/articles/power-series) may not converge.\n\n# Polynomial ring\nA polynomial\nf\nover a [commutative ring](https://bliptext.com/articles/commutative-ring)\nR\nis a polynomial all of whose coefficients belong to\nR\n. It is straightforward to verify that the polynomials in a given set of indeterminates over\nR\nform a commutative ring, called the polynomial ring in these indeterminates, denoted R[x] in the univariate case and in the multivariate case.\nOne has\nSo, most of the theory of the multivariate case can be reduced to an iterated univariate case.\nThe map from\nR\nto\nR[x]\nsending\nr\nto itself considered as a constant polynomial is an injective [ring homomorphism](https://bliptext.com/articles/ring-homomorphism), by which\nR\nis viewed as a subring of\nR[x]\n. In particular,\nR[x]\nis an [algebra](https://bliptext.com/articles/algebra-ring-theory) over\nR\n.\nOne can think of the ring\nR[x]\nas arising from\nR\nby adding one new element x to R, and extending in a minimal way to a ring in which\nx\nsatisfies no other relations than the obligatory ones, plus commutation with all elements of\nR\n(that is\n). To do this, one must add all powers of\nx\nand their linear combinations as well.\nFormation of the polynomial ring, together with forming factor rings by factoring out [ideals](https://bliptext.com/articles/ideal-ring-theory), are important tools for constructing new rings out of known ones.\nFor instance, the ring (in fact field) of complex numbers, which can be constructed from the polynomial ring\nR[x]\nover the real numbers by factoring out the ideal of multiples of the polynomial\nx2 + 1\n. Another example is the construction of [finite fields](https://bliptext.com/articles/finite-field), which proceeds similarly, starting out with the field of integers modulo some [prime number](https://bliptext.com/articles/prime-number) as the coefficient ring\nR\n(see [modular arithmetic](https://bliptext.com/articles/modular-arithmetic)).\nIf\nR\nis commutative, then one can associate with every polynomial\nP\nin\nR[x]\na polynomial function\nf\nwith domain and range equal to\nR\n. (More generally, one can take domain and range to be any same [unital](https://bliptext.com/articles/unital-algebra) [associative algebra](https://bliptext.com/articles/associative-algebra) over\nR\n.) One obtains the value\nf(r)\nby [substitution](https://bliptext.com/articles/substitution-algebra) of the value\nr\nfor the symbol\nx\nin\nP\n. 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](https://bliptext.com/articles/fermat-s-little-theorem) for an example where\nR\nis the integers modulo\np\n). This is not the case when\nR\nis the real or complex numbers, whence the two concepts are not always distinguished in [analysis](https://bliptext.com/articles/analysis-mathematics).\nAn even more important reason to distinguish between polynomials and polynomial functions is that many operations on polynomials (like [Euclidean division](https://bliptext.com/articles/euclidean-division)) require looking at what a polynomial is composed of as an expression rather than evaluating it at some constant value for\nx\n.\n\n## Divisibility\nIf\nR\nis an [integral domain](https://bliptext.com/articles/integral-domain) and\nf\nand\ng\nare polynomials in\nR[x]\n, it is said that\nf\ndivides\ng\nor\nf\nis a divisor of\ng\nif there exists a polynomial\nq\nin\nR[x]\nsuch that\n. If a\\in R, then a is a root of f if and only x-a divides f. In this case, the quotient can be computed using the [polynomial long division](https://bliptext.com/articles/polynomial-long-division).\nIf\nF\nis a [field](https://bliptext.com/articles/field-mathematics) and\nf\nand\ng\nare polynomials in\nF[x]\nwith\ng ≠ 0\n, then there exist unique polynomials\nq\nand\nr\nin\nF[x]\nwith\nand such that the degree of\nr\nis smaller than the degree of\ng\n(using the convention that the polynomial 0 has a negative degree).\nThe polynomials\nq\nand\nr\nare uniquely determined by\nf\nand\ng\n. This is called [Euclidean division](https://bliptext.com/articles/euclidean-division-of-polynomials), division with remainder or polynomial long division and shows that the ring\nF[x]\nis a [Euclidean domain](https://bliptext.com/articles/euclidean-domain).\nAnalogously, prime polynomials (more correctly, [irreducible polynomials](https://bliptext.com/articles/irreducible-polynomial)) can be defined as non-zero polynomials which cannot be factorized into the product of two non-constant polynomials.\nIn the case of coefficients in a ring, \"non-constant\" must be replaced by \"non-constant or non-[unit](https://bliptext.com/articles/unit-ring-theory)\" (both definitions agree in the case of coefficients in a field).\nAny polynomial may be decomposed into the product of an invertible constant by a product of irreducible polynomials.\nIf the coefficients belong to a field or a [unique factorization domain](https://bliptext.com/articles/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).\nWhen 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](https://bliptext.com/articles/factorization-of-polynomials)).\nThese algorithms are not practicable for hand-written computation, but are available in any [computer algebra system](https://bliptext.com/articles/computer-algebra-system).\n[Eisenstein's criterion](https://bliptext.com/articles/eisenstein-s-criterion) can also be used in some cases to determine irreducibility.\n\n# Applications\n\n## Positional notation\nIn modern positional numbers systems, such as the [decimal system](https://bliptext.com/articles/decimal), the digits and their positions in the representation of an integer, for example, 45, are a shorthand notation for a polynomial in the [radix](https://bliptext.com/articles/radix) or base, in this case, 4 × 101 + 5 × 100.\nAs another example, in radix 5, a string of digits such as 132 denotes the (decimal) number 1 × 52 + 3 × 51 + 2 × 50 = 42.\nThis representation is unique.\nLet b be a positive integer greater than 1.\nThen every positive integer a can be expressed uniquely in the form\nwhere m is a nonnegative integer and the r's are integers such that\n0 < rm < b\nand\n0 ≤ ri < b\nfor\ni = 0, 1,.\n. ., m − 1\n.\n\n## Interpolation and approximation\nThe simple structure of polynomial functions makes them quite useful in analyzing general functions using polynomial approximations.\nAn important example in [calculus](https://bliptext.com/articles/calculus) is [Taylor's theorem](https://bliptext.com/articles/taylor-s-theorem), which roughly states that every [differentiable function](https://bliptext.com/articles/differentiable-function) locally looks like a polynomial function, and the [Stone–Weierstrass theorem](https://bliptext.com/articles/stone-weierstrass-theorem), which states that every [continuous function](https://bliptext.com/articles/continuous-function) defined on a [compact](https://bliptext.com/articles/compact-space) [interval](https://bliptext.com/articles/interval-mathematics) of the real axis can be approximated on the whole [interval](https://bliptext.com/articles/interval-mathematics) as closely as desired by a polynomial function.\nPractical methods of approximation include [polynomial interpolation](https://bliptext.com/articles/polynomial-interpolation) and the use of [splines](https://bliptext.com/articles/spline-mathematics).\n\n## Other applications\nPolynomials are frequently used to encode information about some other object.\nThe [characteristic polynomial](https://bliptext.com/articles/characteristic-polynomial) of a matrix or linear operator contains information about the operator's [eigenvalues](https://bliptext.com/articles/eigenvalue).\nThe [minimal polynomial](https://bliptext.com/articles/minimal-polynomial-field-theory) of an [algebraic element](https://bliptext.com/articles/algebraic-element) records the simplest algebraic relation satisfied by that element.\nThe [chromatic polynomial](https://bliptext.com/articles/chromatic-polynomial) of a [graph](https://bliptext.com/articles/graph-discrete-mathematics) counts the number of proper colourings of that [graph](https://bliptext.com/articles/graph-discrete-mathematics).\nThe term \"polynomial\", as an adjective, can also be used for quantities or functions that can be written in polynomial form.\nFor example, in [computational complexity theory](https://bliptext.com/articles/computational-complexity-theory) the phrase [polynomial time](https://bliptext.com/articles/polynomial-time) means that the time it takes to complete an [algorithm](https://bliptext.com/articles/algorithm) is bounded by a polynomial function of some variable, such as the size of the input.\n\n# History\nDetermining the roots of polynomials, or \"solving algebraic equations\", is among the oldest problems in mathematics.\nHowever, the elegant and practical notation we use today only developed beginning in the 15th century.\nBefore that, equations were written out in words.\nFor example, an algebra problem from the Chinese [Arithmetic in Nine Sections](https://bliptext.com/articles/the-nine-chapters-on-the-mathematical-art), c. 200 BCE, begins \"Three sheafs of good crop, two sheafs of mediocre crop, and one sheaf of bad crop are sold for 29 dou.\"\nWe would write\n3x + 2y + z = 29\n.\n\n## History of the notation\nThe earliest known use of the equal sign is in [Robert Recorde](https://bliptext.com/articles/robert-recorde)'s [The Whetstone of Witte](https://bliptext.com/articles/the-whetstone-of-witte), 1557.\nThe signs + for addition, − for subtraction, and the use of a letter for an unknown appear in [Michael Stifel](https://bliptext.com/articles/michael-stifel)'s Arithemetica integra, 1544.\n[René Descartes](https://bliptext.com/articles/ren-descartes), in La géometrie, 1637, introduced the concept of the graph of a polynomial equation.\nHe 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\na\ns denote constants and\nx\ndenotes a variable.\nDescartes introduced the use of superscripts to denote exponents as well.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T17:22:15.194Z","updated_at":"2025-01-25T17:22:15.194Z","search_vector":"'polynomi':1","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), a **polynomial** is a [mathematical expression](https://bliptext.com/articles/expression-mathematics) consisting of [indeterminates](https://bliptext.com/articles/indeterminate-variable) (also called [variables](https://bliptext.com/articles/variable-mathematics)) and [coefficients](https://bliptext.com/articles/coefficient), that involves only the operations of [addition](https://bliptext.com/articles/addition), [subtraction](https://bliptext.com/articles/subtraction), [multiplication](https://bliptext.com/articles/multiplication) and [exponentiation](https://bliptext.com/articles/exponentiation) to [nonnegative integer](https://bliptext.com/articles/nonnegative-integer) powers, and has a finite number of terms.\nAn example of a polynomial of a single indeterminate\nx\nis\nx2 − 4x + 7\n. An example with three indeterminates is\nx3 + 2xyz2 − yz + 1\n.\nPolynomials appear in many areas of mathematics and science.\nFor example, they are used to form [polynomial equations](https://bliptext.com/articles/polynomial-equation), which encode a wide range of problems, from elementary [word problems](https://bliptext.com/articles/word-problem-mathematics-education) to complicated scientific problems; they are used to define **polynomial functions**, which appear in settings ranging from basic [chemistry](https://bliptext.com/articles/chemistry) and [physics](https://bliptext.com/articles/physics) to [economics](https://bliptext.com/articles/economics) and [social science](https://bliptext.com/articles/social-science); and they are used in [calculus](https://bliptext.com/articles/calculus) and [numerical analysis](https://bliptext.com/articles/numerical-analysis) to approximate other functions.\nIn advanced mathematics, polynomials are used to construct [polynomial rings](https://bliptext.com/articles/polynomial-ring) and [[algebra](https://bliptext.com/articles/algebra)ic varieties](https://bliptext.com/articles/[algebra](https://bliptext.com/articles/algebra)ic-variety), which are central concepts in [algebra](https://bliptext.com/articles/algebra) and [algebra](https://bliptext.com/articles/algebra)ic geometry.\n\n# Etymology\nThe word polynomial [joins two diverse roots](https://bliptext.com/articles/hybrid-word): the Greek poly, meaning \"many\", and the Latin nomen, or \"name\".\nIt was derived from the term [binomial](https://bliptext.com/articles/binomial-polynomial) by replacing the Latin root bi- with the Greek poly-.\nThat is, it means a sum of many terms (many [monomials](https://bliptext.com/articles/monomial)).\nThe word polynomial was first used in the 17th century.\n\n# Notation and terminology\nThe x occurring in a polynomial is commonly called a variable or an indeterminate.\nWhen the polynomial is considered as an expression, x is a fixed symbol which does not have any value (its value is \"indeterminate\").\nHowever, when one considers the [function](https://bliptext.com/articles/function-mathematics) defined by the polynomial, then x represents the argument of the [function](https://bliptext.com/articles/function-mathematics), and is therefore called a \"variable\".\nMany authors use these two words interchangeably.\nA polynomial P in the indeterminate x is commonly denoted either as P or as P(x).\nFormally, the name of the polynomial is P, not P(x), but the use of the [functional notation](https://bliptext.com/articles/functional-notation) P(x) dates from a time when the distinction between a polynomial and the associated function was unclear.\nMoreover, the functional notation is often useful for specifying, in a single phrase, a polynomial and its indeterminate.\nFor example, \"let P(x) be a polynomial\" is a shorthand for \"let P be a polynomial in the indeterminate x\".\nOn 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.\nThe 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.\nIf 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.\nThus, the polynomial P defines the function\nwhich is the polynomial function associated to P.\nFrequently, when using this notation, one supposes that a is a number.\nHowever, one may use it over any domain where addition and multiplication are defined (that is, any [ring](https://bliptext.com/articles/ring-mathematics)).\nIn particular, if a is a polynomial then P(a) is also a polynomial.\nMore specifically, when a is the indeterminate x, then the [image](https://bliptext.com/articles/image-mathematics) of x by this function is the polynomial P itself (substituting x for x does not change anything).\nIn other words,\nP(x)=P,\nwhich justifies formally the existence of two notations for the same polynomial.\n# [The graph of a polynomial function of degree 3 | upload.wikimedia.org/wikipedia/commons/a/a3/Polynomialdeg3.svg]\n\n# Definition\nA polynomial [expression](https://bliptext.com/articles/expression-mathematics) is an [expression](https://bliptext.com/articles/expression-mathematics) that can be built from [constants](https://bliptext.com/articles/constant-mathematics) and symbols called variables or indeterminates by means of [addition](https://bliptext.com/articles/addition), [multiplication](https://bliptext.com/articles/multiplication) and [exponentiation](https://bliptext.com/articles/exponentiation) to a [non-negative integer](https://bliptext.com/articles/non-negative-integer) power.\nThe constants are generally [numbers](https://bliptext.com/articles/number), but may be any expression that do not involve the indeterminates, and represent [mathematical objects](https://bliptext.com/articles/mathematical-object) that can be added and multiplied.\nTwo polynomial expressions are considered as defining the same polynomial if they may be transformed, one to the other, by applying the usual properties of [commutativity](https://bliptext.com/articles/commutative-property), [associativity](https://bliptext.com/articles/associative-property) and [distributivity](https://bliptext.com/articles/distributive-property) of addition and multiplication.\nFor example (x-1)(x-2) and x^2-3x+2 are two polynomial expressions that represent the same polynomial; so, one has the [equality](https://bliptext.com/articles/equality-mathematics).\nA polynomial in a single indeterminate\nx\ncan always be written (or rewritten) in the form\nwhere are constants that are called the coefficients of the polynomial, and x is the indeterminate.\nThe word \"indeterminate\" means that x represents no particular value, although any value may be substituted for it.\nThe mapping that associates the result of this substitution to the substituted value is a [function](https://bliptext.com/articles/function-mathematics), called a polynomial [function](https://bliptext.com/articles/function-mathematics).\nThis can be expressed more concisely by using [summation notation](https://bliptext.com/articles/summation):\nThat is, a polynomial can either be zero or can be written as the sum of a finite number of non-zero [terms](https://bliptext.com/articles/summand).\nEach term consists of the product of a number – called the [coefficient](https://bliptext.com/articles/coefficient) of the term – and a finite number of indeterminates, raised to non-negative integer powers.\n\n# Classification\nThe 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.\nBecause\n, the degree of an indeterminate without a written exponent is one.\nA term with no indeterminates and a polynomial with no indeterminates are called, respectively, a [constant term](https://bliptext.com/articles/constant-term) and a **constant polynomial**.\nThe degree of a constant term and of a nonzero constant polynomial is 0.\nThe degree of the zero polynomial 0 (which has no terms at all) is generally treated as not defined (but see below).\nFor example:\n-5x^2y\nis a term.\nThe coefficient is\n−5\n, the indeterminates are\nx\nand\ny\n, the degree of\nx\nis two, while the degree of\ny\nis one.\nThe degree of the entire term is the sum of the degrees of each indeterminate in it, so in this example the degree is\n2 + 1 = 3\n.\nForming a sum of several terms produces a polynomial.\nFor example, the following is a polynomial:\nIt consists of three terms: the first is degree two, the second is degree one, and the third is degree zero.\nPolynomials of small degree have been given specific names.\nA polynomial of degree zero is a constant polynomial, or simply a constant.\nPolynomials of degree one, two or three are respectively linear polynomials, [quadratic polynomials](https://bliptext.com/articles/quadratic-polynomial) and cubic polynomials.\nFor higher degrees, the specific names are not commonly used, although quartic polynomial (for degree four) and quintic polynomial (for degree five) are sometimes used.\nThe names for the degrees may be applied to the polynomial or to its terms.\nFor example, the term\n2x\nin\nx2 + 2x + 1\nis a linear term in a quadratic polynomial.\nThe polynomial 0, which may be considered to have no terms at all, is called the **zero polynomial**.\nUnlike other constant polynomials, its degree is not zero.\nRather, the degree of the zero polynomial is either left explicitly undefined, or defined as negative (either −1 or −∞).\nThe zero polynomial is also unique in that it is the only polynomial in one indeterminate that has an infinite number of [roots](https://bliptext.com/articles/root-of-a-function).\nThe graph of the zero polynomial,\n, is the x-axis.\nIn the case of polynomials in more than one indeterminate, a polynomial is called homogeneous of degree\nn if all of its non-zero terms have degree\nn. The zero polynomial is homogeneous, and, as a homogeneous polynomial, its degree is undefined.\nFor example,\nx3y2 + 7x2y3 − 3x5\nis homogeneous of degree 5.\nFor more details, see [Homogeneous polynomial](https://bliptext.com/articles/homogeneous-polynomial).\nThe [commutative law](https://bliptext.com/articles/commutative-law) of addition can be used to rearrange terms into any preferred order.\nIn polynomials with one indeterminate, the terms are usually ordered according to degree, either in \"descending powers of\nx\n\", with the term of largest degree first, or in \"ascending powers of\nx\n\". The polynomial\n3x2 − 5x + 4\nis written in descending powers of\nx\n. The first term has coefficient\n3\n, indeterminate\nx\n, and exponent\n2\n. In the second term, the coefficient is\n−5. The third term is a constant.\nBecause the degree of a non-zero polynomial is the largest degree of any one term, this polynomial has degree two.\nTwo 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](https://bliptext.com/articles/distributive-law), into a single term whose coefficient is the sum of the coefficients of the terms that were combined.\nIt may happen that this makes the coefficient 0.\nPolynomials can be classified by the number of terms with nonzero coefficients, so that a one-term polynomial is called a [monomial](https://bliptext.com/articles/monomial), a two-term polynomial is called a [binomial](https://bliptext.com/articles/binomial-polynomial), and a three-term polynomial is called a [trinomial](https://bliptext.com/articles/trinomial).\nA **[real](https://bliptext.com/articles/real-number) polynomial** is a polynomial with [real](https://bliptext.com/articles/real-number) coefficients.\nWhen it is used to define a [function](https://bliptext.com/articles/function-mathematics), the [domain](https://bliptext.com/articles/domain-function) is not so restricted.\nHowever, a **real polynomial function** is a function from the reals to the reals that is defined by a real polynomial.\nSimilarly, an **[integer](https://bliptext.com/articles/integer) polynomial** is a polynomial with [integer](https://bliptext.com/articles/integer) coefficients, and a **[complex](https://bliptext.com/articles/complex-number) polynomial** is a polynomial with [complex](https://bliptext.com/articles/complex-number) coefficients.\nA polynomial in one indeterminate is called a **univariate polynomial**, a polynomial in more than one indeterminate is called a **multivariate polynomial**.\nA polynomial with two indeterminates is called a **bivariate polynomial**.\nThese 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.\nIt is possible to further classify multivariate polynomials as bivariate, trivariate, and so on, according to the maximum number of indeterminates allowed.\nAgain, so that the set of objects under consideration be closed under subtraction, a study of trivariate polynomials usually allows bivariate polynomials, and so on.\nIt is also common to say simply \"polynomials in\nx, y\n, and\nz\n\", listing the indeterminates allowed.\n\n# Operations\n\n## Addition and subtraction\nPolynomials can be added using the [associative law](https://bliptext.com/articles/associative-law) of addition (grouping all their terms together into a single sum), possibly followed by reordering (using the [commutative law](https://bliptext.com/articles/commutative-law)) and combining of like terms.\nFor example, if and\nthen the sum\ncan be reordered and regrouped as\nand then simplified to\nWhen polynomials are added together, the result is another polynomial.\nSubtraction of polynomials is similar.\n\n## Multiplication\nPolynomials can also be multiplied.\nTo expand the [product](https://bliptext.com/articles/product-mathematics) 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.\nFor example, if\nthen\nCarrying out the multiplication in each term produces\nCombining similar terms yields\nwhich can be simplified to\nAs in the example, the product of polynomials is always a polynomial.\n\n## Composition\nGiven a polynomial f of a single variable and another polynomial g of any number of variables, the [composition](https://bliptext.com/articles/function-composition) f \\circ g is obtained by substituting each copy of the variable of the first polynomial by the second polynomial.\nFor example, if and then\nA composition may be expanded to a sum of terms using the rules for multiplication and division of polynomials.\nThe composition of two polynomials is another polynomial.\n\n## Division\nThe division of one polynomial by another is not typically a polynomial.\nInstead, such ratios are a more general family of objects, called [rational fractions](https://bliptext.com/articles/rational-fraction), rational expressions, or [rational functions](https://bliptext.com/articles/rational-function), depending on context.\nThis is analogous to the fact that the ratio of two [integers](https://bliptext.com/articles/integer) is a [rational number](https://bliptext.com/articles/rational-number), not necessarily an integer.\nFor example, the fraction\n1/(x2 + 1)\nis not a polynomial, and it cannot be written as a finite sum of powers of the variable x.\nFor polynomials in one variable, there is a notion of [[Euclidean division](https://bliptext.com/articles/euclidean-division) of polynomials](https://bliptext.com/articles/euclidean-division-of-polynomials), generalizing the [Euclidean division](https://bliptext.com/articles/euclidean-division) of integers.\nThis notion of the division\na(x)/b(x)\nresults in two polynomials, a quotient\nq(x)\nand a remainder\nr(x)\n, such that\nand\ndegree(r) \n \nThe graph of the zero polynomial\nis the\nx\n-axis.\n \n \nThe graph of a degree 0 polynomial\nis a horizontal line with y\n-intercept\na0\n \n \nThe graph of a degree 1 polynomial (or linear function)\nis an oblique line with y\n-intercept\na0 and [slope](https://bliptext.com/articles/slope)\na1\n. \n \nThe graph of a degree 2 polynomial\nis a [parabola](https://bliptext.com/articles/parabola).\n \n \nThe graph of a degree 3 polynomial\nis a [cubic curve](https://bliptext.com/articles/cubic-equation).\n \n \nThe graph of any polynomial with degree 2 or greater\nis a continuous non-linear curve.\n \n \nA non-constant polynomial function [tends to infinity](https://bliptext.com/articles/infinity) when the variable increases indefinitely (in [absolute value](https://bliptext.com/articles/absolute-value)).\nIf the degree is higher than one, the graph does not have any [asymptote](https://bliptext.com/articles/asymptote).\nIt has two [parabolic branches](https://bliptext.com/articles/parabolic-branch) with vertical direction (one branch for positive x and one for negative x).\nPolynomial graphs are analyzed in calculus using intercepts, slopes, concavity, and end behavior.\n\n# Equations\nA polynomial [equation](https://bliptext.com/articles/equation), also called an [algebraic [equation](https://bliptext.com/articles/equation)](https://bliptext.com/articles/algebraic-[equation](https://bliptext.com/articles/equation)), is an [equation](https://bliptext.com/articles/equation) of the form\nFor example,\nis a polynomial equation.\nWhen considering equations, the indeterminates (variables) of polynomials are also called [unknowns](https://bliptext.com/articles/variable-mathematics), and the solutions are the possible values of the [unknowns](https://bliptext.com/articles/variable-mathematics) for which the equality is true (in general more than one solution may exist).\nA polynomial equation stands in contrast to a polynomial [identity](https://bliptext.com/articles/identity-mathematics) like\n(x + y)(x − y) = x2 − y2\n, where both expressions represent the same polynomial in different forms, and as a consequence any evaluation of both members gives a valid equality.\nIn elementary [algebra](https://bliptext.com/articles/algebra), methods such as the [quadratic formula](https://bliptext.com/articles/quadratic-formula) are taught for solving all first degree and second degree polynomial equations in one variable.\nThere are also formulas for the [cubic](https://bliptext.com/articles/cubic-equation) and [quartic equations](https://bliptext.com/articles/quartic-equation).\nFor higher degrees, the [Abel–Ruffini theorem](https://bliptext.com/articles/abel-ruffini-theorem) asserts that there can not exist a general formula in radicals.\nHowever, [root-finding algorithms](https://bliptext.com/articles/root-finding-algorithm) may be used to find [numerical approximations](https://bliptext.com/articles/numerical-approximation) of the roots of a polynomial expression of any degree.\nThe number of solutions of a polynomial equation with real coefficients may not exceed the degree, and equals the degree when the [complex](https://bliptext.com/articles/complex-number) solutions are counted with their [multiplicity](https://bliptext.com/articles/multiplicity-mathematics).\nThis fact is called the [fundamental theorem of algebra](https://bliptext.com/articles/fundamental-theorem-of-algebra).\n\n## Solving equations\nA root of a nonzero univariate polynomial\nP\nis a value a of x such that\n. In other words, a root of P is a solution of the [polynomial equation](https://bliptext.com/articles/polynomial-equation)\nor a [zero](https://bliptext.com/articles/zero-of-a-function) of the polynomial function defined by\nP\n. In the case of the zero polynomial, every number is a zero of the corresponding function, and the concept of root is rarely considered.\nA number\na\nis a root of a polynomial\nP\nif and only if the [linear polynomial](https://bliptext.com/articles/)\nx − a\ndivides\nP\n, that is if there is another polynomial\nQ\nsuch that\nP = (x − a) Q\n. It may happen that a power (greater than\n1\n) of\nx − a\ndivides\nP\na\nis a multiple root of\nP\n, and otherwise\na\nis a **simple root** of\nP\n. If\nP\nis a nonzero polynomial, there is a highest power\nm\nsuch that\n(x − a)m\ndivides\nP\n, which is called the multiplicity of\na\nas a root of\nP\n. The number of roots of a nonzero polynomial\nP\n, counted with their respective multiplicities, cannot exceed the degree of\nP\n, and equals this degree if all [complex](https://bliptext.com/articles/complex-number) roots are considered (this is a consequence of the [fundamental theorem of algebra](https://bliptext.com/articles/fundamental-theorem-of-algebra)).\nThe coefficients of a polynomial and its roots are related by [Vieta's formulas](https://bliptext.com/articles/vieta-s-formulas).\nSome polynomials, such as\nx2 + 1\n, do not have any roots among the [real numbers](https://bliptext.com/articles/real-number).\nIf, however, the set of accepted solutions is expanded to the [complex numbers](https://bliptext.com/articles/complex-number), every non-constant polynomial has at least one root; this is the [fundamental theorem of algebra](https://bliptext.com/articles/fundamental-theorem-of-algebra).\nBy successively dividing out factors\nx − a\n, 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.\nThere may be several meanings of [\"solving an equation\"](https://bliptext.com/articles/equation-solving).\nOne may want to express the solutions as explicit numbers; for example, the unique solution of\n2x − 1 = 0\nis\n1/2\n. This is, in general, impossible for equations of degree greater than one, and, since the ancient times, mathematicians have searched to express the solutions as [algebraic expressions](https://bliptext.com/articles/algebraic-expression); for example, the [golden ratio](https://bliptext.com/articles/golden-ratio) is the unique positive solution of x^2-x-1=0.\nIn the ancient times, they succeeded only for degrees one and two.\nFor [quadratic equations](https://bliptext.com/articles/quadratic-equation), the [quadratic formula](https://bliptext.com/articles/quadratic-formula) provides such expressions of the solutions.\nSince the 16th century, similar formulas (using cube roots in addition to square roots), although much more complicated, are known for equations of degree three and four (see [cubic equation](https://bliptext.com/articles/cubic-equation) and [quartic equation](https://bliptext.com/articles/quartic-equation)).\nBut formulas for degree 5 and higher eluded researchers for several centuries.\nIn 1824, [Niels Henrik Abel](https://bliptext.com/articles/niels-henrik-abel) proved the striking result that there are equations of degree 5 whose solutions cannot be expressed by a (finite) formula, involving only arithmetic operations and radicals (see [Abel–Ruffini theorem](https://bliptext.com/articles/abel-ruffini-theorem)).\nIn 1830, [Évariste Galois](https://bliptext.com/articles/variste-galois) proved that most equations of degree higher than four cannot be solved by radicals, and showed that for each equation, one may decide whether it is solvable by radicals, and, if it is, solve it.\nThis result marked the start of [Galois theory](https://bliptext.com/articles/galois-theory) and [group theory](https://bliptext.com/articles/group-theory), two important branches of modern [algebra](https://bliptext.com/articles/algebra).\nGalois himself noted that the computations implied by his method were impracticable.\nNevertheless, formulas for solvable equations of degrees 5 and 6 have been published (see [quintic function](https://bliptext.com/articles/quintic-function) and [sextic equation](https://bliptext.com/articles/sextic-equation)).\nWhen 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 it is to compute [numerical approximations](https://bliptext.com/articles/numerical-approximation) of the solutions.\nThere are many methods for that; some are restricted to polynomials and others may apply to any [continuous function](https://bliptext.com/articles/continuous-function).\nThe most efficient [algorithms](https://bliptext.com/articles/algorithm) allow solving easily (on a [computer](https://bliptext.com/articles/computer)) polynomial equations of degree higher than 1,000 (see [Root-finding algorithm](https://bliptext.com/articles/root-finding-algorithm)).\nFor polynomials with 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\".\nThe study of the sets of zeros of polynomials is the object of [algebraic geometry](https://bliptext.com/articles/algebraic-geometry).\nFor a set of polynomial equations with several unknowns, there are [algorithms](https://bliptext.com/articles/algorithm) to decide whether they have a finite number of [complex](https://bliptext.com/articles/complex-number) solutions, and, if this number is finite, for computing the solutions.\nSee [System of polynomial equations](https://bliptext.com/articles/system-of-polynomial-equations).\nThe special case where all the polynomials are of degree one is called a [system of linear equations](https://bliptext.com/articles/system-of-linear-equations), for which another range of different [solution methods](https://bliptext.com/articles/system-of-linear-equations) exist, including the classical [Gaussian elimination](https://bliptext.com/articles/gaussian-elimination).\nA polynomial equation for which one is interested only in the solutions which are [integers](https://bliptext.com/articles/integer) is called a [Diophantine equation](https://bliptext.com/articles/diophantine-equation).\nSolving Diophantine equations is generally a very hard task.\nIt has been proved that there cannot be any general [algorithm](https://bliptext.com/articles/algorithm) for solving them, or even for deciding whether the set of solutions is empty (see [Hilbert's tenth problem](https://bliptext.com/articles/hilbert-s-tenth-problem)).\nSome of the most famous problems that have been solved during the last fifty years are related to Diophantine equations, such as [Fermat's Last Theorem](https://bliptext.com/articles/fermat-s-last-theorem).\n\n# Polynomial expressions\nPolynomials where indeterminates are substituted for some other mathematical objects are often considered, and sometimes have a special name.\n\n## Trigonometric polynomials\nA **trigonometric polynomial** is a finite [linear combination](https://bliptext.com/articles/linear-combination) of [functions](https://bliptext.com/articles/function-mathematics) sin(nx) and cos(nx) with n taking on the values of one or more [natural numbers](https://bliptext.com/articles/natural-number).\nThe coefficients may be taken as real numbers, for real-valued functions.\nIf 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](https://bliptext.com/articles/list-of-trigonometric-identities)).\nConversely, every polynomial in sin(x) and cos(x) may be converted, with [Product-to-sum identities](https://bliptext.com/articles/list-of-trigonometric-identities), into a linear combination of functions sin(nx) and cos(nx).\nThis equivalence explains why linear combinations are called polynomials.\nFor [complex coefficients](https://bliptext.com/articles/complex-number), there is no difference between such a function and a finite [Fourier series](https://bliptext.com/articles/fourier-series).\nTrigonometric polynomials are widely used, for example in [trigonometric [interpolation](https://bliptext.com/articles/interpolation)](https://bliptext.com/articles/trigonometric-[interpolation](https://bliptext.com/articles/interpolation)) applied to the [interpolation](https://bliptext.com/articles/interpolation) of [periodic functions](https://bliptext.com/articles/periodic-function).\nThey are also used in the [discrete Fourier transform](https://bliptext.com/articles/discrete-fourier-transform).\n\n## Matrix polynomials\nA [matrix polynomial](https://bliptext.com/articles/matrix-polynomial) is a polynomial with [square matrices](https://bliptext.com/articles/square-matrix) as variables.\nGiven an ordinary, scalar-valued polynomial\nthis polynomial evaluated at a matrix A is\nwhere I is the [identity matrix](https://bliptext.com/articles/identity-matrix).\nA **matrix polynomial equation** is an equality between two matrix polynomials, which holds for the specific matrices in question.\nA **matrix polynomial identity** is a matrix polynomial equation which holds for all matrices A in a specified [matrix ring](https://bliptext.com/articles/matrix-ring) Mn(R).\n\n## Exponential polynomials\nA bivariate polynomial where the second variable is substituted for an exponential function applied to the first variable, for example\nP(x, ex)\n, may be called an [exponential polynomial](https://bliptext.com/articles/exponential-polynomial).\n\n# Related concepts\n\n## Rational functions\nA [rational fraction](https://bliptext.com/articles/rational-fraction) is the [quotient](https://bliptext.com/articles/quotient) ([algebraic fraction](https://bliptext.com/articles/algebraic-fraction)) of two polynomials.\nAny [algebraic expression](https://bliptext.com/articles/algebraic-expression) that can be rewritten as a rational fraction is a [rational function](https://bliptext.com/articles/rational-function).\nWhile 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.\nThe rational fractions include the Laurent polynomials, but do not limit denominators to powers of an indeterminate.\n\n## Laurent polynomials\n[Laurent polynomials](https://bliptext.com/articles/laurent-polynomial) are like polynomials, but allow negative powers of the variable(s) to occur.\n\n## Power series\n[Formal power series](https://bliptext.com/articles/formal-power-series) are like polynomials, but allow infinitely many non-zero terms to occur, so that they do not have finite degree.\nUnlike polynomials they cannot in general be explicitly and fully written down (just like [irrational numbers](https://bliptext.com/articles/irrational-number) cannot), but the rules for manipulating their terms are the same as for polynomials.\nNon-formal [power series](https://bliptext.com/articles/power-series) also generalize polynomials, but the multiplication of two [power series](https://bliptext.com/articles/power-series) may not converge.\n\n# Polynomial ring\nA polynomial\nf\nover a [commutative ring](https://bliptext.com/articles/commutative-ring)\nR\nis a polynomial all of whose coefficients belong to\nR\n. It is straightforward to verify that the polynomials in a given set of indeterminates over\nR\nform a commutative ring, called the polynomial ring in these indeterminates, denoted R[x] in the univariate case and in the multivariate case.\nOne has\nSo, most of the theory of the multivariate case can be reduced to an iterated univariate case.\nThe map from\nR\nto\nR[x]\nsending\nr\nto itself considered as a constant polynomial is an injective [ring homomorphism](https://bliptext.com/articles/ring-homomorphism), by which\nR\nis viewed as a subring of\nR[x]\n. In particular,\nR[x]\nis an [algebra](https://bliptext.com/articles/algebra-ring-theory) over\nR\n.\nOne can think of the ring\nR[x]\nas arising from\nR\nby adding one new element x to R, and extending in a minimal way to a ring in which\nx\nsatisfies no other relations than the obligatory ones, plus commutation with all elements of\nR\n(that is\n). To do this, one must add all powers of\nx\nand their linear combinations as well.\nFormation of the polynomial ring, together with forming factor rings by factoring out [ideals](https://bliptext.com/articles/ideal-ring-theory), are important tools for constructing new rings out of known ones.\nFor instance, the ring (in fact field) of complex numbers, which can be constructed from the polynomial ring\nR[x]\nover the real numbers by factoring out the ideal of multiples of the polynomial\nx2 + 1\n. Another example is the construction of [finite fields](https://bliptext.com/articles/finite-field), which proceeds similarly, starting out with the field of integers modulo some [prime number](https://bliptext.com/articles/prime-number) as the coefficient ring\nR\n(see [modular arithmetic](https://bliptext.com/articles/modular-arithmetic)).\nIf\nR\nis commutative, then one can associate with every polynomial\nP\nin\nR[x]\na polynomial function\nf\nwith domain and range equal to\nR\n. (More generally, one can take domain and range to be any same [unital](https://bliptext.com/articles/unital-algebra) [associative algebra](https://bliptext.com/articles/associative-algebra) over\nR\n.) One obtains the value\nf(r)\nby [substitution](https://bliptext.com/articles/substitution-algebra) of the value\nr\nfor the symbol\nx\nin\nP\n. 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](https://bliptext.com/articles/fermat-s-little-theorem) for an example where\nR\nis the integers modulo\np\n). This is not the case when\nR\nis the real or complex numbers, whence the two concepts are not always distinguished in [analysis](https://bliptext.com/articles/analysis-mathematics).\nAn even more important reason to distinguish between polynomials and polynomial functions is that many operations on polynomials (like [Euclidean division](https://bliptext.com/articles/euclidean-division)) require looking at what a polynomial is composed of as an expression rather than evaluating it at some constant value for\nx\n.\n\n## Divisibility\nIf\nR\nis an [integral domain](https://bliptext.com/articles/integral-domain) and\nf\nand\ng\nare polynomials in\nR[x]\n, it is said that\nf\ndivides\ng\nor\nf\nis a divisor of\ng\nif there exists a polynomial\nq\nin\nR[x]\nsuch that\n. If a\\in R, then a is a root of f if and only x-a divides f. In this case, the quotient can be computed using the [polynomial long division](https://bliptext.com/articles/polynomial-long-division).\nIf\nF\nis a [field](https://bliptext.com/articles/field-mathematics) and\nf\nand\ng\nare polynomials in\nF[x]\nwith\ng ≠ 0\n, then there exist unique polynomials\nq\nand\nr\nin\nF[x]\nwith\nand such that the degree of\nr\nis smaller than the degree of\ng\n(using the convention that the polynomial 0 has a negative degree).\nThe polynomials\nq\nand\nr\nare uniquely determined by\nf\nand\ng\n. This is called [Euclidean division](https://bliptext.com/articles/euclidean-division-of-polynomials), division with remainder or polynomial long division and shows that the ring\nF[x]\nis a [Euclidean domain](https://bliptext.com/articles/euclidean-domain).\nAnalogously, prime polynomials (more correctly, [irreducible polynomials](https://bliptext.com/articles/irreducible-polynomial)) can be defined as non-zero polynomials which cannot be factorized into the product of two non-constant polynomials.\nIn the case of coefficients in a ring, \"non-constant\" must be replaced by \"non-constant or non-[unit](https://bliptext.com/articles/unit-ring-theory)\" (both definitions agree in the case of coefficients in a field).\nAny polynomial may be decomposed into the product of an invertible constant by a product of irreducible polynomials.\nIf the coefficients belong to a field or a [unique factorization domain](https://bliptext.com/articles/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).\nWhen 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](https://bliptext.com/articles/factorization-of-polynomials)).\nThese algorithms are not practicable for hand-written computation, but are available in any [computer algebra system](https://bliptext.com/articles/computer-algebra-system).\n[Eisenstein's criterion](https://bliptext.com/articles/eisenstein-s-criterion) can also be used in some cases to determine irreducibility.\n\n# Applications\n\n## Positional notation\nIn modern positional numbers systems, such as the [decimal system](https://bliptext.com/articles/decimal), the digits and their positions in the representation of an integer, for example, 45, are a shorthand notation for a polynomial in the [radix](https://bliptext.com/articles/radix) or base, in this case, 4 × 101 + 5 × 100.\nAs another example, in radix 5, a string of digits such as 132 denotes the (decimal) number 1 × 52 + 3 × 51 + 2 × 50 = 42.\nThis representation is unique.\nLet b be a positive integer greater than 1.\nThen every positive integer a can be expressed uniquely in the form\nwhere m is a nonnegative integer and the r's are integers such that\n0 < rm < b\nand\n0 ≤ ri < b\nfor\ni = 0, 1,.\n. ., m − 1\n.\n\n## Interpolation and approximation\nThe simple structure of polynomial functions makes them quite useful in analyzing general functions using polynomial approximations.\nAn important example in [calculus](https://bliptext.com/articles/calculus) is [Taylor's theorem](https://bliptext.com/articles/taylor-s-theorem), which roughly states that every [differentiable function](https://bliptext.com/articles/differentiable-function) locally looks like a polynomial function, and the [Stone–Weierstrass theorem](https://bliptext.com/articles/stone-weierstrass-theorem), which states that every [continuous function](https://bliptext.com/articles/continuous-function) defined on a [compact](https://bliptext.com/articles/compact-space) [interval](https://bliptext.com/articles/interval-mathematics) of the real axis can be approximated on the whole [interval](https://bliptext.com/articles/interval-mathematics) as closely as desired by a polynomial function.\nPractical methods of approximation include [polynomial interpolation](https://bliptext.com/articles/polynomial-interpolation) and the use of [splines](https://bliptext.com/articles/spline-mathematics).\n\n## Other applications\nPolynomials are frequently used to encode information about some other object.\nThe [characteristic polynomial](https://bliptext.com/articles/characteristic-polynomial) of a matrix or linear operator contains information about the operator's [eigenvalues](https://bliptext.com/articles/eigenvalue).\nThe [minimal polynomial](https://bliptext.com/articles/minimal-polynomial-field-theory) of an [algebraic element](https://bliptext.com/articles/algebraic-element) records the simplest algebraic relation satisfied by that element.\nThe [chromatic polynomial](https://bliptext.com/articles/chromatic-polynomial) of a [graph](https://bliptext.com/articles/graph-discrete-mathematics) counts the number of proper colourings of that [graph](https://bliptext.com/articles/graph-discrete-mathematics).\nThe term \"polynomial\", as an adjective, can also be used for quantities or functions that can be written in polynomial form.\nFor example, in [computational complexity theory](https://bliptext.com/articles/computational-complexity-theory) the phrase [polynomial time](https://bliptext.com/articles/polynomial-time) means that the time it takes to complete an [algorithm](https://bliptext.com/articles/algorithm) is bounded by a polynomial function of some variable, such as the size of the input.\n\n# History\nDetermining the roots of polynomials, or \"solving algebraic equations\", is among the oldest problems in mathematics.\nHowever, the elegant and practical notation we use today only developed beginning in the 15th century.\nBefore that, equations were written out in words.\nFor example, an algebra problem from the Chinese [Arithmetic in Nine Sections](https://bliptext.com/articles/the-nine-chapters-on-the-mathematical-art), c. 200 BCE, begins \"Three sheafs of good crop, two sheafs of mediocre crop, and one sheaf of bad crop are sold for 29 dou.\"\nWe would write\n3x + 2y + z = 29\n.\n\n## History of the notation\nThe earliest known use of the equal sign is in [Robert Recorde](https://bliptext.com/articles/robert-recorde)'s [The Whetstone of Witte](https://bliptext.com/articles/the-whetstone-of-witte), 1557.\nThe signs + for addition, − for subtraction, and the use of a letter for an unknown appear in [Michael Stifel](https://bliptext.com/articles/michael-stifel)'s Arithemetica integra, 1544.\n[René Descartes](https://bliptext.com/articles/ren-descartes), in La géometrie, 1637, introduced the concept of the graph of a polynomial equation.\nHe 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\na\ns denote constants and\nx\ndenotes a variable.\nDescartes introduced the use of superscripts to denote exponents as well.","revisionCount":0},"hasMore":false,"nextCursor":null}