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lved during the last fifty years are related to diophantine equations such as fermat s last theorem polynomial expressions edit polynomials where indeterminates are substituted for some other mathematical objects are often considered and sometimes have a special name trigonometric polynomials edit main article trigonometric polynomial a trigonometric polynomial is a finite linear combination of functions sin nx and cos nx with n taking on the values of one or more natural numbers 31 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 the multiple angle formulae conversely every polynomial in sin x and cos x may be converted with product to sum identities into a linear combination of functions sin nx and cos nx this equivalence explains why linear combinations are called polynomials for complex coefficients there is no difference between such a function and a finite fourier series trigonometric polynomials are widely used for example in trigonometric interpolation applied to the interpolation of periodic functions they are also used in the discrete fourier transform matrix polynomials edit main article matrix polynomial a matrix polynomial is a polynomial with square matrices as variables 32 given an ordinary scalar valued polynomial p x i 0 n a i x i a 0 a 1 x a 2 x 2 a n x n displaystyle p x sum _ i 0 n a_ i x i a_ 0 a_ 1 x a_ 2 x 2 cdots a_ n x n this polynomial evaluated at a matrix a is p a i 0 n a i a i a 0 i a 1 a a 2 a 2 a n a n displaystyle p a sum _ i 0 n a_ i a i a_ 0 i a_ 1 a a_ 2 a 2 cdots a_ n a n where i is the identity matrix 33 a matrix polynomial equation is an equality between two matrix polynomials which holds for the specific matrices in question a matrix polynomial identity is a matrix polynomial equation which holds for all matrices a in a specified matrix ring m n r exponential polynomials edit a bivariate polynomial where the second variable is substituted for an exponential function applied to the first variable for example p x e x may be called an exponential polynomial related concepts edit rational functions edit main article rational function a rational fraction is the quotient algebraic fraction of two polynomials any algebraic expression that can be rewritten as a rational fraction is a rational function 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 laurent polynomials edit main article laurent polynomial laurent polynomials are like polynomials but allow negative powers of the variable s to occur power series edit main article formal power series 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 like irrational numbers cannot but the rules for manipulating their terms are the same as for polynomials non formal power series also generalize polynomials but the multiplication of two power series may not converge polynomial ring edit main article polynomial ring a polynomial f over a commutative ring r is a polynomial all of whose coefficients belong to r it is straightforward to verify that the polynomials in a given set of indeterminates over r form a commutative ring called the polynomial ring in these indeterminates denoted r x displaystyle r x in the univariate case and r x 1 x n displaystyle r x_ 1 ldots x_ n in the multivariate case one has r x 1 x n r x 1 x n 1 x n displaystyle r x_ 1 ldots x_ n left r x_ 1 ldots x_ n 1 right x_ n so most of the theory of the multivariate case can be reduced to an iterated univariate case the map from r to r x sending r to itself considered as a constant polynomial is an injective ring homomorphism by which r is viewed as a subring of r x in particular r x is an algebra over r one can think of the ring r x as arising from r by adding one new element x to r and extending in a minimal way to a ring in which x satisfies no other relations than the obligatory ones plus commutation with all elements of r that is xr rx to do this one must add all powers of x and their linear combinations as well formation of the polynomial ring together with forming factor rings by factoring out ideals are important tools for constructing new rings out of known ones for instance the ring in fact field of complex numbers which can be constructed from the polynomial ring r x over the real numbers by factoring out the ideal of multiples of the polynomial x 2 1 another example is the construction of finite fields which proceeds similarly starting out with the field of integers modulo some prime number as the coefficient ring r see modular arithmetic if r is commutative then one can associate with every polynomial p in r x a polynomial function f with domain and range equal to r more generally one can take domain and range to be any same unital associative algebra over r one obtains the value f r by substitution of the value r for the symbol x in p one reason to distinguish between polynomials and polynomial functions is that over some rings different polynomials may give rise to the same polynomial function see fermat s little theorem for an example where r is the integers modulo p this is not the case when r is the real or complex numbers whence the two concepts are not always distinguished in analysis an even more important reason to distinguish between polynomials and polynomial functions is that many operations on polynomials like euclidean division require looking at what a polynomial is composed of as an expression rather than evaluating it at some constant value for x divisibility edit main articles polynomial greatest common divisor and factorization of polynomials if r is an integral domain and f and g are polynomials in r x it is said that f divides g or f is a divisor of g if there exists a polynomial q in r x such that f q g if a r displaystyle a in r then a is a root of f if and only x a displaystyle x a divides f in this case the quotient can be computed using the polynomial long division 34 35 if f is a field and f and g are polynomials in f x with g 0 then there exist unique polynomials q and r in f x with f q g r displaystyle f q g r and such that the degree of r is smaller than the degree of g using the convention that the polynomial 0 has a negative degree the polynomials q and r are uniquely determined by f and g this is called euclidean division division with remainder or polynomial long division and shows that the ring f x is a euclidean domain analogously prime polynomials more correctly irreducible polynomials can be defined as non zero polynomials which cannot be factorized into the product of two non constant polynomials 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 a unique factorization domain this decomposition is unique up to the order of the factors and the multiplication of any non unit factor by a unit and division of the unit factor by the same unit when the coefficients belong to integers rational numbers or a finite field there are algorithms to test irreducibility and to compute the factorization into irreducible polynomials see factorization of polynomials these algorithms are not practicable for hand written computation but are available in any computer algebra system eisenstein s criterion can also be used in some cases to determine irreducibility applications edit positional notation edit main article positional notation in modern positional numbers systems such as the decimal system the digits and their positions in the representation of an integer for example 45 are a shorthand notation for a polynomial in the radix or base in this case 4 10 1 5 10 0 as another example in radix 5 a string of digits such as 132 denotes the decimal number 1 5 2 3 5 1 2 5 0 42 this representation is unique let b be a positive integer greater than 1 then every positive integer a can be expressed uniquely in the form a r m b m r m 1 b m 1 r 1 b r 0 displaystyle a r_ m b m r_ m 1 b m 1 dotsb r_ 1 b r_ 0 where m is a nonnegative integer and the r s are integers such that 0 r m b and 0 r i b for i 0 1 m 1 36 interpolation and approximation edit see also polynomial interpolation orthogonal polynomials b spline and spline interpolation the simple structure of polynomial functions makes them quite useful in analyzing general functions using polynomial approximations an important example in calculus is taylor s theorem which roughly states that every differentiable function locally looks like a polynomial function and the stone weierstrass theorem which states that every continuous function defined on a compact interval of the real axis can be approximated on the whole interval as closely as desired by a polynomial function practical methods of approximation include polynomial interpolation and the use of splines 37 in other mathematical fields edit polynomials are frequently used to encode information about some other object in linear algebra the characteristic polynomial of a matrix or linear operator contains information about the operator s eigenvalues in field theory the minimal polynomial of an algebraic element records the simplest algebraic relation satisfied by that element 38 in algebraic graph theory the chromatic polynomial of a graph counts the number of proper colourings of that graph 39 the term polynomial as an adjective can also be used for quantities or functions that can be written in polynomial form for example in computational complexity theory the phrase polynomial time means that the time it takes to complete an algorithm is bounded by a polynomial function of some variable such as the size of the input history edit main articles cubic function history quartic function history and abel ruffini theorem history determining the roots of polynomials or solving algebraic equations is among the oldest problems in mathematics however the notation we use today only developed beginning in the 15th century before that equations were written out in words for example an algebra problem from the chinese arithmetic in nine sections 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 we would write 3 x 2 y z 29 history of the notation edit main article history of mathematical notation the earliest known use of the equal sign is in robert recorde s the whetstone of witte 1557 the signs for addition for subtraction and the use of a letter for an unknown appear in michael stifel s arithemetica integra 1544 rené descartes in la 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 the a s denote constants and x denotes a variable descartes introduced the use of superscripts to denote exponents as well 40 see also edit hilbert s seventeenth problem list of polynomial topics footnotes edit 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 integers modulo some prime number p displaystyle p 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 define constant functions citation needed in fact as a homogeneous function it is homogeneous of every degree citation needed some authors use monomial to mean monic monomial see knapp anthony w 2007 advanced algebra along with a companion volume basic algebra springer p 457 isbn 978 0 8176 4522 9 this paragraph assumes that the polynomials have coefficients in a field notes edit see polynomial and binomial compact oxford english dictionary birkhoff lane 1997 p 72 1 2 3 sahai bist 2002 p 20 1 2 young 2022 p 346 1 2 beauregard fraleigh 1973 p 153 barbeau 2003 pp 1 2 polynomials brilliant math science wiki brilliant org retrieved 2020 08 28 beauregard fraleigh 1973 p 154 weisstein eric w zero polynomial mathworld edwards 1995 p 78 1 2 3 edwards harold m 1995 linear algebra springer p 47 isbn 978 0 8176 3731 6 weisstein eric w multinomial mathworld wolfram com retrieved 2025 08 26 clapham christopher nicholson james 2009 the concise oxford dictionary of mathematics 4th ed united states oxford university press p 303 isbn 9780199235940 weisstein eric w multivariate polynomial mathworld wolfram com retrieved 2025 08 26 geddes czapor labahn 2007 p 46 geddes czapor labahn 2007 p 47 salomon david 2006 coding for data and computer communications springer p 459 isbn 978 0 387 23804 3 1 2 introduction 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 1 2 barbeau 2003 pp 1 2 kriete hartje 1998 05 20 progress in holomorphic dynamics crc press p 159 isbn 978 0 582 32388 9 this class of endomorphisms is closed under composition marecek lynn mathis andrea honeycutt 6 may 2020 intermediate algebra 2e openstax 7 1 haylock derek cockburn anne d 2008 10 14 understanding mathematics for young children a guide for foundation stage and lower primary teachers sage p 49 isbn 978 1 4462 0497 9 we find that the set of integers is not closed under this operation of division 1 2 marecek mathis 2020 5 4 selby peter h slavin steve 1991 practical algebra a self teaching guide 2nd ed wiley isbn 978 0 471 53012 1 weisstein eric w ruffini s rule mathworld wolfram com retrieved 2020 07 25 barbeau 2003 pp 80 2 barbeau 2003 pp 64 5 proskuryakov i v 1994 algebraic equation in hazewinkel michiel ed encyclopaedia of mathematics vol 1 springer isbn 978 1 55608 010 4 leung kam tim et al 1992 polynomials and equations hong kong university press p 134 isbn 9789622092716 mcnamee j m 2007 numerical methods for roots of polynomials part 1 elsevier isbn 978 0 08 048947 6 powell michael j d 1981 approximation theory and methods cambridge university press isbn 978 0 521 29514 7 gohberg israel lancaster peter rodman leiba 2009 1982 matrix polynomial...