The wordpolynomialjoins two diverse roots: the Greekpoly, meaning "many", and the Latinnomen, or "name". It was derived from the termbinomial by replacing the Latin rootbi- with the Greekpoly-. That is, it means a sum of many terms (manymonomials). The wordpolynomial was first used in the 17th century.[6]
The occurring in a polynomial is commonly called avariable or anindeterminate. When the polynomial is considered as an expression, is a fixed symbol which does not have any value (its value is "indeterminate"). However, when one considers thefunction defined by the polynomial, then represents the argument of the function, and is therefore called a "variable". Many authors use these two words interchangeably.
A polynomial in the indeterminate is commonly denoted either as or as. Formally, the name of the polynomial is, not, but the use of thefunctional notation dates from a time when the distinction between a polynomial and the associated function was unclear. Moreover, the functional notation is often useful for specifying, in a single phrase, a polynomial and its indeterminate. For example, "let be a polynomial" is a shorthand for "let be a polynomial in the indeterminate". On the other hand, when it is not necessary to emphasize the name of the indeterminate, many formulas are much simpler and easier to read if the name(s) of the indeterminate(s) do not appear at each occurrence of the polynomial.
The ambiguity of having two notations for a single mathematical object may be formally resolved by considering the general meaning of the functional notation for polynomials. If denotes a number, a variable, another polynomial, or, more generally, any expression, then denotes, by convention, the result of substituting for in. Thus, the polynomial defines the functionwhich is thepolynomial function associated to.Frequently, when using this notation, one supposes that is a number. However, one may use it over any domain where addition and multiplication are defined (that is, anyring). In particular, if is a polynomial then is also a polynomial.
More specifically, when is the indeterminate, then theimage of by this function is the polynomial itself (substituting for does not change anything). In other words,which justifies formally the existence of two notations for the same polynomial.
Apolynomial expression is anexpression that can be built fromconstants and symbols calledvariables orindeterminates by means ofaddition,multiplication andexponentiation to anon-negative integer power. The constants are generallynumbers, but may be any expression that do not involve the indeterminates, and representmathematical objects that can be added and multiplied. Two polynomial expressions are considered as defining the samepolynomial if they may be transformed, one to the other, by applying the usual properties ofcommutativity,associativity anddistributivity of addition and multiplication. For example and are two polynomial expressions that represent the same polynomial; so, one has theequality.
A polynomial in a single indeterminatex can always be written (or rewritten) in the formwhere are constants that are called thecoefficients of the polynomial, and is the indeterminate.[7] The word "indeterminate" means that represents no particular value, although any value may be substituted for it. The mapping that associates the result of this substitution to the substituted value is afunction, called apolynomial function.
This can be expressed more concisely by usingsummation notation:That is, a polynomial can either be zero or can be written as the sum of a finite number of non-zeroterms. Each term consists of the product of a number – called thecoefficient of the term[a] – and a finite number of indeterminates, raised to non-negative integer powers.
The exponent on an indeterminate in a term is called the degree of that indeterminate in that term; the degree of the term is the sum of the degrees of the indeterminates in that term, and the degree of a polynomial is the largest degree of any term with nonzero coefficient.[8] Because, the degree of an indeterminate without a written exponent is one.
A term with no indeterminates and a polynomial with no indeterminates are called, respectively, aconstant term and aconstant polynomial.[b] The degree of a constant term and of a nonzero constant polynomial is. The degree of the zero polynomial (which has no terms at all) is generally treated as not defined (but see below).[9]
For example:is a term. The coefficient is, the indeterminates are and, the degree of is two, while the degree of is one. The degree of the entire term is the sum of the degrees of each indeterminate in it, so in this example the degree is.
Forming a sum of several terms produces a polynomial. For example, the following is a polynomial:It consists of three terms: the first is degree two, the second is degree one, and the third is degree zero.
Polynomials of small degree have been given specific names. A polynomial of degree zero is aconstant polynomial, or simply aconstant. Polynomials of degree one, two or three are respectivelylinear polynomials,quadratic polynomials andcubic polynomials.[8] For higher degrees, the specific names are not commonly used, althoughquartic polynomial (for degree four) andquintic polynomial (for degree five) are sometimes used. The names for the degrees may be applied to the polynomial or to its terms. For example, the term in is a linear term in a quadratic polynomial.
The polynomial, which may be considered to have no terms at all, is called thezero polynomial. Unlike other constant polynomials, its degree is not zero. Rather, the degree of the zero polynomial is either left explicitly undefined, or defined as negative (either −1 or).[10] The zero polynomial is also unique in that it is the only polynomial in one indeterminate that has an infinite number ofroots. The graph of the zero polynomial,, is the-axis.
In the case of polynomials in more than one indeterminate, a polynomial is calledhomogeneous of degree ifall of its non-zero terms have degree. The zero polynomial is homogeneous, and, as a homogeneous polynomial, its degree is undefined.[c] For example, is homogeneous of degree. For more details, seehomogeneous polynomials.
Thecommutative law of addition can be used to rearrange terms into any preferred order. In polynomials with one indeterminate, the terms are usually ordered according to degree, either in "descending powers of", with the term of largest degree first, or in "ascending powers of". The polynomial is written in descending powers of. The first term has coefficient, indeterminate, and exponent. In the second term, the coefficient is. The third term is a constant. Because thedegree of a non-zero polynomial is the largest degree of any one term, this polynomial has degree two.[11]
Two terms with the same indeterminates raised to the same powers are called "similar terms" or "like terms", and they can be combined, using thedistributive law, into a single term whose coefficient is the sum of the coefficients of the terms that were combined. It may happen that this makes the coefficient.[12]
Polynomials can be classified by the number of terms with nonzero coefficients, so that a one-term polynomial is called amonomial,[d] a two-term polynomial is called abinomial, and a three-term polynomial is called atrinomial. A polynomial with two or more terms is also called amultinomial.[13][14]
Areal polynomial is a polynomial withreal coefficients. When it is used to define afunction, thedomain is not so restricted. However, areal polynomial function is a function from the reals to the reals that is defined by a real polynomial. Similarly, aninteger polynomial is a polynomial withinteger coefficients, and acomplex polynomial is a polynomial withcomplex coefficients.
A polynomial in one indeterminate is called aunivariate polynomial, a polynomial in more than one indeterminate is called amultivariate polynomial.[15] A polynomial with two indeterminates is called abivariate polynomial.[7] These notions refer more to the kind of polynomials one is generally working with than to individual polynomials; for instance, when working with univariate polynomials, one does not exclude constant polynomials (which may result from the subtraction of non-constant polynomials), although strictly speaking, constant polynomials do not contain any indeterminates at all. It is possible to further classify multivariate polynomials asbivariate,trivariate, and so on, according to the maximum number of indeterminates allowed. Again, so that the set of objects under consideration be closed under subtraction, a study of trivariate polynomials usually allows bivariate polynomials, and so on. It is also common to say simply "polynomials in, and", listing the indeterminates allowed.
Polynomials can be added using theassociative law of addition (grouping all their terms together into a single sum), possibly followed by reordering (using thecommutative law) and combining of like terms.[12][16] For example, if andthen the sumcan be reordered and regrouped asand then simplified toWhen polynomials are added together, the result is another polynomial.[17]
Polynomials can also be multiplied. To expand theproduct 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.[12] For example, ifthenCarrying out the multiplication in each term producesCombining similar terms yieldswhich can be simplified toAs in the example, the product of polynomials is always a polynomial.[17][9]
Given a polynomial of a single variable and another polynomial of any number of variables, thecomposition is obtained by substituting each copy of the variable of the first polynomial by the second polynomial.[9] For example, if and thenA composition may be expanded to a sum of terms using the rules for multiplication and division of polynomials. The composition of two polynomials is another polynomial.[18]
The division of one polynomial by another is not typically a polynomial. Instead, such ratios are a more general family of objects, calledrational fractions,rational expressions, orrational functions, depending on context.[19] This is analogous to the fact that the ratio of twointegers is arational number, not necessarily an integer.[20][21] For example, the fraction is not a polynomial, and it cannot be written as a finite sum of powers of the variable.
When the denominator ismonic and linear, that is, for some constant, then thepolynomial remainder theorem asserts that the remainder of the division of by is theevaluation.[21] In this case, the quotient may be computed byRuffini's rule, a special case of synthetic division.[23]
All polynomials with coefficients in aunique factorization domain (for example, the integers or afield) also have a factored form in which the polynomial is written as a product ofirreducible polynomials and a constant. This factored form is unique up to the order of the factors and their multiplication by an invertible constant. In the case of the field ofcomplex numbers, the irreducible factors are linear. Over thereal numbers, they have the degree either one or two. Over the integers and therational numbers the irreducible factors may have any degree.[24] For example, the factored form ofisover the integers and the reals, andover the complex numbers.
Calculatingderivatives and integrals of polynomials is particularly simple, compared to other kinds of functions.Thederivative of the polynomial with respect to is the polynomialSimilarly, the generalantiderivative (or indefinite integral) of iswhere is an arbitrary constant. For example, antiderivatives of have the form.
For polynomials whose coefficients come from more abstract settings (for example, if the coefficients are integersmodulo someprime number, or elements of an arbitrary ring), the formula for the derivative can still be interpreted formally, with the coefficient understood to mean the sum of copies of. For example, over the integers modulo, the derivative of the polynomial is the polynomial.[25]
Apolynomial function is a function that can be defined byevaluating a polynomial. More precisely, a function of oneargument from a given domain is a polynomial function if there exists a polynomialthat evaluates to for allx in thedomain of (here, is a non-negative integer and are constant coefficients).[26]Generally, unless otherwise specified, polynomial functions havecomplex coefficients, arguments, and values. In particular, a polynomial, restricted to have real coefficients, defines a function from the complex numbers to the complex numbers. If the domain of this function is alsorestricted to the reals, the resulting function is areal function that maps reals to reals.
For example, the function, defined byis a polynomial function of one variable. Polynomial functions of several variables are similarly defined, using polynomials in more than one indeterminate, as inAccording to the definition of polynomial functions, there may be expressions that obviously are not polynomials but nevertheless define polynomial functions. An example is the expression which takes the same values as the polynomial on the interval, and thus both expressions define the same polynomial function on this interval.
Theevaluation 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.
For polynomials in one indeterminate, the evaluation is usually more efficient (lower number of arithmetic operations to perform) usingHorner's method, which consists of rewriting the polynomial as
A non-constant polynomial functiontends to infinity when the variable increases indefinitely (inabsolute value). If the degree is higher than one, the graph does not have anyasymptote. It has twoparabolic branches with vertical direction (one branch for positivex and one for negativex).
Polynomial graphs are analyzed in calculus using intercepts, slopes, concavity, and end behavior.
Apolynomial equation, also called analgebraic equation, is anequation of the form[27]For example,is a polynomial equation.
When considering equations, the indeterminates (variables) of polynomials are also calledunknowns, and thesolutions are the possible values of the unknowns for which the equality is true (in general more than one solution may exist). A polynomial equation stands in contrast to apolynomialidentity like, where both expressions represent the same polynomial in different forms, and as a consequence any evaluation of both members gives a valid equality.
In elementaryalgebra, methods such as thequadratic formula are taught for solving all first degree and second degree polynomial equations in one variable. There are also formulas for thecubic andquartic equations. For higher degrees, theAbel–Ruffini theorem asserts that there can not exist a general formula in radicals. However,root-finding algorithms may be used to findnumerical approximations of the roots of a polynomial expression of any degree.
The number of solutions of a polynomial equation with real coefficients may not exceed the degree, and equals the degree when thecomplex solutions are counted with theirmultiplicity. This fact is called thefundamental theorem of algebra.
Aroot of a nonzero univariate polynomialP is a valuea ofx such thatP(a) = 0. In other words, a root ofP is a solution of thepolynomial equationP(x) = 0 or azero of the polynomial function defined byP. In the case of the zero polynomial, every number is a zero of the corresponding function, and the concept of root is rarely considered.
A numbera is a root of a polynomialP if and only if thelinear polynomialx −a dividesP, that is if there is another polynomialQ such thatP = (x −a) Q. It may happen that a power (greater than1) ofx −a dividesP; in this case,a is amultiple root ofP, and otherwisea is asimple root ofP. IfP is a nonzero polynomial, there is a highest powerm such that(x −a)m dividesP, which is called themultiplicity ofa as a root ofP. The number of roots of a nonzero polynomialP, counted with their respective multiplicities, cannot exceed the degree ofP,[28] and equals this degree if allcomplex roots are considered (this is a consequence of thefundamental theorem of algebra).The coefficients of a polynomial and its roots are related byVieta's formulas.
Some polynomials, such asx2 + 1, do not have any roots among thereal numbers. If, however, the set of accepted solutions is expanded to thecomplex numbers, every non-constant polynomial has at least one root; this is thefundamental theorem of algebra. By successively dividing out factorsx −a, one sees that any polynomial with complex coefficients can be written as a constant (its leading coefficient) times a product of such polynomial factors of degree 1; as a consequence, the number of (complex) roots counted with their multiplicities is exactly equal to the degree of the polynomial.
There may be several meanings of"solving an equation". One may want to express the solutions as explicit numbers; for example, the unique solution of2x − 1 = 0 is1/2. This is, in general, impossible for equations of degree greater than one, and, since the ancient times, mathematicians have searched to express the solutions asalgebraic expressions; for example, thegolden ratio(1+√5)/2 is the unique positive solution ofx2 −x − 1 = 0 In the ancient times, they succeeded only for degrees one and two. Forquadratic equations, thequadratic formula provides such expressions of the solutions. Since 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 (seecubic equation andquartic equation). But formulas for degree 5 and higher eluded researchers for several centuries. In 1824,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 (seeAbel–Ruffini theorem). In 1830,Évariste 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. This result marked the start ofGalois theory andgroup theory, two important branches of modernalgebra. Galois himself noted that the computations implied by his method were impracticable. Nevertheless, formulas for solvable equations of degrees 5 and 6 have been published (seequintic function andsextic equation).
When there is no algebraic expression for the roots, and when such an algebraic expression exists but is too complicated to be useful, the unique way of solving it is to computenumerical approximations of the solutions.[29] There are many methods for that; some are restricted to polynomials and others may apply to anycontinuous function. The most efficientalgorithms allow solving easily (on acomputer) polynomial equations of degree higher than 1,000 (seeRoot-finding algorithm).
For polynomials with more than one indeterminate, the combinations of values for the variables for which the polynomial function takes the value zero are generally calledzeros instead of "roots". The study of the sets of zeros of polynomials is the object ofalgebraic geometry. For a set of polynomial equations with several unknowns, there arealgorithms to decide whether they have a finite number ofcomplex solutions, and, if this number is finite, for computing the solutions. SeeSystem of polynomial equations.
A polynomial equation for which one is interested only in the solutions which areintegers is called aDiophantine equation. Solving Diophantine equations is generally a very hard task. It has been proved that there cannot be any generalalgorithm for solving them, or even for deciding whether the set of solutions is empty (seeHilbert's tenth problem). Some of the most famous problems that have been solved during the last fifty years are related to Diophantine equations, such asFermat's Last Theorem.
Atrigonometric polynomial is a finitelinear combination offunctions sin(nx) and cos(nx) withn taking on the values of one or morenatural numbers.[30] The coefficients may be taken as real numbers, for real-valued functions.
If sin(nx) and cos(nx) are expanded in terms of sin(x) and cos(x), a trigonometric polynomial becomes a polynomial in the two variables sin(x) and cos(x) (using themultiple-angle formulae). Conversely, every polynomial in sin(x) and cos(x) may be converted, withProduct-to-sum identities, into a linear combination of functions sin(nx) and cos(nx). This equivalence explains why linear combinations are called polynomials.
Amatrix polynomial equation is an equality between two matrix polynomials, which holds for the specific matrices in question. Amatrix polynomial identity is a matrix polynomial equation which holds for all matricesA in a specifiedmatrix ringMn(R).
A bivariate polynomial where the second variable is substituted for an exponential function applied to the first variable, for exampleP(x,ex), may be called anexponential polynomial.
While polynomial functions are defined for all values of the variables, a rational function is defined only for the values of the variables for which the denominator is not zero.
The rational fractions include the Laurent polynomials, but do not limit denominators to powers of an indeterminate.
Formal power series are like polynomials, but allow infinitely many non-zero terms to occur, so that they do not have finite degree. Unlike polynomials they cannot in general be explicitly and fully written down (just likeirrational numbers cannot), but the rules for manipulating their terms are the same as for polynomials. Non-formalpower series also generalize polynomials, but the multiplication of two power series may not converge.
Apolynomialf over acommutative ringR is a polynomial all of whose coefficients belong toR. It is straightforward to verify that the polynomials in a given set of indeterminates overR form a commutative ring, called thepolynomial ring in these indeterminates, denoted in the univariate case and in the multivariate case.
One hasSo, most of the theory of the multivariate case can be reduced to an iterated univariate case.
The map fromR toR[x] sendingr to itself considered as a constant polynomial is an injectivering homomorphism, by whichR is viewed as a subring ofR[x]. In particular,R[x] is analgebra overR.
One can think of the ringR[x] as arising fromR by adding one new elementx toR, and extending in a minimal way to a ring in whichx satisfies no other relations than the obligatory ones, plus commutation with all elements ofR (that isxr =rx). To do this, one must add all powers ofx and their linear combinations as well.
Formation of the polynomial ring, together with forming factor rings by factoring outideals, are important tools for constructing new rings out of known ones. For instance, the ring (in fact field) of complex numbers, which can be constructed from the polynomial ringR[x] over the real numbers by factoring out the ideal of multiples of the polynomialx2 + 1. Another example is the construction offinite fields, which proceeds similarly, starting out with the field of integers modulo someprime number as the coefficient ringR (seemodular arithmetic).
IfR is commutative, then one can associate with every polynomialP inR[x] apolynomial functionf with domain and range equal toR. (More generally, one can take domain and range to be any sameunitalassociative algebra overR.) One obtains the valuef(r) bysubstitution of the valuer for the symbolx inP. One reason to distinguish between polynomials and polynomial functions is that, over some rings, different polynomials may give rise to the same polynomial function (seeFermat's little theorem for an example whereR is the integers modulop). This is not the case whenR is the real or complex numbers, whence the two concepts are not always distinguished inanalysis. An even more important reason to distinguish between polynomials and polynomial functions is that many operations on polynomials (likeEuclidean division) require looking at what a polynomial is composed of as an expression rather than evaluating it at some constant value forx.
IfR is anintegral domain andf andg are polynomials inR[x], it is said thatfdividesg orf is a divisor ofg if there exists a polynomialq inR[x] such thatfq =g. If thena is a root off if and only dividesf. In this case, the quotient can be computed using thepolynomial long division.[33][34]
IfF is afield andf andg are polynomials inF[x] withg ≠ 0, then there exist unique polynomialsq andr inF[x] withand such that the degree ofr is smaller than the degree ofg (using the convention that the polynomial 0 has a negative degree). The polynomialsq andr are uniquely determined byf andg. This is calledEuclidean division, division with remainder orpolynomial long division and shows that the ringF[x] is aEuclidean domain.
Analogously,prime polynomials (more correctly,irreducible polynomials) can be defined asnon-zero polynomials which cannot be factorized into the product of two non-constant polynomials. In the case of coefficients in a ring,"non-constant" must be replaced by"non-constant or non-unit" (both definitions agree in the case of coefficients in a field). Any polynomial may be decomposed into the product of an invertible constant by a product of irreducible polynomials. If the coefficients belong to a field or aunique 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). When the coefficients belong to integers, rational numbers or a finite field, there are algorithms to test irreducibility and to compute the factorization into irreducible polynomials (seeFactorization of polynomials). These algorithms are not practicable for hand-written computation, but are available in anycomputer algebra system.Eisenstein's criterion can also be used in some cases to determine irreducibility.
In modern positional numbers systems, such as thedecimal system, the digits and their positions in the representation of an integer, for example, 45, are a shorthand notation for a polynomial in theradix or base, in this case,4 × 101 + 5 × 100. As another example, in radix 5, a string of digits such as 132 denotes the (decimal) number1 × 52 + 3 × 51 + 2 × 50 = 42. This representation is unique. Letb be a positive integer greater than 1. Then every positive integera can be expressed uniquely in the form
wherem is a nonnegative integer and ther's are integers such that
The simple structure of polynomial functions makes them quite useful in analyzing general functions using polynomial approximations. An important example incalculus isTaylor's theorem, which roughly states that everydifferentiable function locally looks like a polynomial function, and theStone–Weierstrass theorem, which states that everycontinuous function defined on acompactinterval of the real axis can be approximated on the whole interval as closely as desired by a polynomial function. Practical methods of approximation includepolynomial interpolation and the use ofsplines.[36]
Polynomials are frequently used to encode information about some other object. Thecharacteristic polynomial of a matrix or linear operator contains information about the operator'seigenvalues. Theminimal polynomial of analgebraic element records the simplest algebraic relation satisfied by that element. Thechromatic polynomial of agraph counts the number of proper colourings of that graph.
The term "polynomial", as an adjective, can also be used for quantities or functions that can be written in polynomial form. For example, incomputational complexity theory the phrasepolynomial time means that the time it takes to complete analgorithm is bounded by a polynomial function of some variable, such as the size of the input.
The earliest known use of the equal sign is inRobert Recorde'sThe Whetstone of Witte, 1557. The signs + for addition, − for subtraction, and the use of a letter for an unknown appear inMichael Stifel'sArithemetica integra, 1544.René Descartes, inLa géometrie, 1637, introduced the concept of the graph of a polynomial equation. He popularized the use of letters from the beginning of the alphabet to denote constants and letters from the end of the alphabet to denote variables, as can be seen above, in the general formula for a polynomial in one variable, where theas denote constants andx denotes a variable. Descartes introduced the use of superscripts to denote exponents as well.[37]
^The coefficient of a term may be any number from a specified set. If that set is the set of real numbers, we speak of "polynomials over the reals". Other common kinds of polynomials are polynomials with integer coefficients, polynomials with complex coefficients, and polynomials with coefficients that are integersmodulo someprime number.
^This terminology dates from the time when the distinction was not clear between a polynomial and the function that it defines: a constant term and a constant polynomial defineconstant functions.[citation needed]
^Some authors use "monomial" to mean "monic monomial". SeeKnapp, Anthony W. (2007).Advanced Algebra: Along with a Companion Volume Basic Algebra. Springer. p. 457.ISBN978-0-8176-4522-9.
^This paragraph assumes that the polynomials have coefficients in afield.
^abIntroduction to Algebra. Yale University Press. 1965. p. 621.Any two such polynomials can be added, subtracted, or multiplied. Furthermore, the result in each case is another polynomial
Beauregard, Raymond A.; Fraleigh, John B. (1973),A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields, Boston:Houghton Mifflin Company,ISBN0-395-14017-X
Mayr, K. (1937). "Über die Auflösung algebraischer Gleichungssysteme durch hypergeometrische Funktionen".Monatshefte für Mathematik und Physik.45:280–313.doi:10.1007/BF01707992.S2CID197662587.
