Polynomial
In mathematics, a polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. An example of a polynomial of a single indeterminate,, is. An example in three variables is.
Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, central concepts in algebra and algebraic geometry.
Etymology
The word polynomial joins two diverse roots: the Greek poly, meaning "many," and the Latin nomen, or name. It was derived from the term binomial by replacing the Latin root bi- with the Greek poly-. The word polynomial was first used in the 17th century.Notation and terminology
The x occurring in a polynomial is commonly called either a variable or an indeterminate. When the polynomial is considered as an expression, x is a fixed symbol which does not have any value. However, when one considers the function defined by the polynomial, then x represents the argument of the function, and is therefore called a "variable". Many authors use these two words interchangeably.It is common to use uppercase letters for indeterminates and corresponding lowercase letters for the variables of the associated function.
A polynomial P in the indeterminate x is commonly denoted either as P or as P. Formally, the name of the polynomial is P, not P, but the use of the functional notation P date from the time where 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 P be a polynomial" is a shorthand for "let P be a polynomial in the indeterminate x". 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 of the indeterminate 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 a denotes a number, a variable, another polynomial, or, more generally any expression, then P denotes, by convention, the result of substituting a for x in P. Thus, the polynomial P defines the function
which is the polynomial function associated to P.
Frequently, when using this notation, one supposes that a is a number. However one may use it over any domain where addition and multiplication are defined. In particular, if a is a polynomial then P is also a polynomial.
More specifically, when a is the indeterminate x, then the image of x by this function is the polynomial P itself. In other words,
which justifies formally the existence of two notations for the same polynomial.
Definition
A polynomial is an expression that can be built from constants and symbols called indeterminates or variables by means of addition, multiplication and exponentiation to a non-negative integer power. Two such expressions that may be transformed, one to the other, by applying the usual properties of commutativity, associativity and distributivity of addition and multiplication are considered as defining the same polynomial.A polynomial in a single indeterminate x can always be written in the form
where are constants and is the indeterminate. 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 a function, called a polynomial function.
This can be expressed more concisely by using summation notation:
That is, a polynomial can either be zero or can be written as the sum of a finite number of non-zero terms. Each term consists of the product of a number called the coefficient of the term and a finite number of indeterminates, raised to nonnegative integer powers.
Classification
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 one term with nonzero coefficient. 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, a constant term and a constant polynomial. The degree of a constant term and of a nonzero constant polynomial is 0. The degree of the zero polynomial, 0, is generally treated as not defined.
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 a constant polynomial or simply a constant. Polynomials of degree one, two or three are respectively linear polynomials, quadratic polynomials and cubic polynomials. For higher degrees the specific names are not commonly used, although quartic polynomial and quintic polynomial are sometimes used. The names for the degrees may be applied to the polynomial or to its terms. For example, in the term is a linear term in a quadratic polynomial.
The polynomial 0, which may be considered to have no terms at all, is called the zero 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. The zero polynomial is also unique in that it is the only polynomial in one indeterminate having an infinite number of roots. The graph of the zero polynomial,, is the -axis.
In the case of polynomials in more than one indeterminate, a polynomial is called homogeneous of if all its non-zero terms have. The zero polynomial is homogeneous, and, as a homogeneous polynomial, its degree is undefined. For example, is homogeneous of degree 5. For more details, see Homogeneous polynomial.
The commutative 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 in the example above is written in descending powers of. The first term has coefficient, indeterminate, and exponent. In the second term, the coefficient. The third term is a constant. Because the degree of a non-zero polynomial is the largest degree of any one term, this polynomial has degree two.
Two terms with the same indeterminates raised to the same powers are called "similar terms" or "like terms", and they can be combined, using the distributive law, into a single term whose coefficient is the sum of the coefficients of the terms that were combined. It may happen that this makes the coefficient 0. Polynomials can be classified by the number of terms with nonzero coefficients, so that a one-term polynomial is called a monomial, a two-term polynomial is called a binomial, and a three-term polynomial is called a trinomial. The term "quadrinomial" is occasionally used for a four-term polynomial.
A real polynomial is a polynomial with real coefficients. When it is used to define a function, the domain is not so restricted. However, a real polynomial function is a function from the reals to the reals that is defined by a real polynomial. Similarly, an integer polynomial is a polynomial with integer coefficients, and a complex polynomial is a polynomial with complex coefficients.
A polynomial in one indeterminate is called a univariate polynomial, a polynomial in more than one indeterminate is called a multivariate polynomial. A polynomial with two indeterminates is called a bivariate polynomial. 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, although strictly speaking constant polynomials do not contain any indeterminates at all. It is possible to further classify multivariate polynomials as bivariate, 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 common, also, to say simply "polynomials in, and ", listing the indeterminates allowed.
The [|evaluation] of a polynomial 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 using Horner's method:
Arithmetic
Addition
Polynomials can be added using the associative law of addition, possibly followed by reordering and combining of like terms. For example, ifthen the sum
can be reordered and regrouped as
and then simplified to
More generally, a sum of polynomials is always another polynomial.
Multiplication
Polynomials can also be multiplied. To expand the product of two polynomials into a sum of terms, the distributive law is repeatedly applied, which results in each term of one polynomial being multiplied by every term of the other. For example, ifthen
Carrying out the multiplication in each term produces
Combining similar terms yields
which can be simplified to
As in the example, the product of polynomials is always a polynomial.
Composition
Given a polynomial of a single variable and another polynomial of any number of variables, the composition is obtained by substituting each copy of the variable of the first polynomial by the second polynomial. 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 always another polynomial.
Division
The division of one polynomial by another is not typically a polynomial. Instead, such ratios are a more general family of objects, called rational fractions, rational expressions, or rational functions, depending on context. This is analogous to the fact that the ratio of two integers is a rational number, not necessarily an integer. For example, the fraction is not a polynomial, and it cannot be written as a finite sum of powers of the variable.For polynomials in one variable, there is a notion of Euclidean division of polynomials, generalizing the Euclidean division of integers. This notion of the division results in two polynomials, a quotient and a remainder, such that and. The quotient and remainder may be computed by any of several algorithms, including polynomial long division and synthetic division.
When the denominator is monic and linear, that is, for some constant, then the polynomial remainder theorem asserts that the remainder of the division of by is the evaluation. In this case, the quotient may be computed by Ruffini's rule, a special case of synthetic division.
Factoring
All polynomials with coefficients in a unique factorization domain also have a factored form in which the polynomial is written as a product of irreducible 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 of complex numbers, the irreducible factors are linear. Over the real numbers, they have the degree either one or two. Over the integers and the rational numbers the irreducible factors may have any degree. For example, the factored form ofis
over the integers and the reals and
over the complex numbers.
The computation of the factored form, called factorization is, in general, too difficult to be done by hand-written computation. However, efficient polynomial factorization algorithms are available in most computer algebra systems.
Calculus
Calculating derivatives and integrals of polynomials is particularly simple, compared to other kinds of functions.The derivative of the polynomial with respect to is the polynomial
Similarly, the general antiderivative of is
where is an arbitrary constant. For example, antiderivatives of have the form.
For polynomials whose coefficients come from more abstract settings, 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.
Polynomial functions
A polynomial function is a function that can be defined by [|evaluating] a polynomial. More precisely, a function of one argument from a given domain is a polynomial function if there exists a polynomialthat evaluates to for all in the domain of .
Generally, unless otherwise specified, polynomial functions have complex 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 also restricted to the reals, the resulting function is a real function that maps reals to reals.
For example, the function, defined by
is a polynomial function of one variable. Polynomial functions of several variables are similarly defined, using polynomials in more than one indeterminate, as in
According 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.
Every polynomial function is continuous, smooth, and entire.
Graphs
A polynomial function in one real variable can be represented by a graph.
- The graph of the zero polynomial
- The graph of a degree 0 polynomial
- The graph of a degree 1 polynomial
- The graph of a degree 2 polynomial
- The graph of a degree 3 polynomial
- The graph of any polynomial with degree 2 or greater
Polynomial graphs are analyzed in calculus using intercepts, slopes, concavity, and end behavior.
Equations
A polynomial equation, also called an algebraic equation, is an equation of the formFor example,
is a polynomial equation.
When considering equations, the indeterminates of polynomials are also called unknowns, and the solutions are the possible values of the unknowns for which the equality is true. A polynomial equation stands in contrast to a polynomial identity 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 elementary algebra, methods such as the quadratic formula are taught for solving all first degree and second degree polynomial equations in one variable. There are also formulas for the cubic and quartic equations. For higher degrees, the Abel–Ruffini theorem asserts that there can not exist a general formula in radicals. However, root-finding algorithms may be used to find numerical 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 the complex solutions are counted with their multiplicity. This fact is called the fundamental theorem of algebra.
Solving equations
Every polynomial in defines a function called the polynomial function associated to ; the equation is the polynomial equation associated to. The solutions of this equation are called the roots of the polynomial, or the zeros of the associated function.A number is a root of a polynomial if and only if the linear polynomial divides, that is if there is another polynomial such that. It may happen that divides more than once: if divides then is called a multiple root of, and otherwise is called a simple root of. If is a nonzero polynomial, there is a highest power such that divides, which is called the multiplicity of the root in. When is the zero polynomial, the corresponding polynomial equation is trivial, and this case is usually excluded when considering roots, as, with the above definitions, every number is a root of the zero polynomial, with an undefined multiplicity. With this exception made, the number of roots of, even counted with their respective multiplicities, cannot exceed the degree of.
The relation between the coefficients of a polynomial and its roots is described by Vieta's formulas.
Some polynomials, such as, do not have any roots among the real numbers. If, however, the set of accepted solutions is expanded to the complex numbers, every non-constant polynomial has at least one root; this is the fundamental theorem of algebra. By successively dividing out factors, one sees that any polynomial with complex coefficients can be written as a constant times a product of such polynomial factors of degree 1; as a consequence, the number of 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 of is. Unfortunately, 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 expression; for example the golden ratio is the unique positive solution of In the ancient times, they succeeded only for degrees one and two. For quadratic equations, the quadratic formula provides such expressions of the solutions. Since the 16th century, similar formulas, but much more complicated are known for equations of degree three and four. 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 formula, involving only arithmetic operations and radicals. 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 of Galois theory and group theory, two important branches of modern algebra. 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.
When there is no algebraic expression for the roots, and when such an algebraic expression exists but is too complicated to be useful, the unique way of solving is to compute numerical approximations of the solutions. There are many methods for that; some are restricted to polynomials and others may apply to any continuous function. The most efficient algorithms allow solving easily polynomial equations of degree higher than 1,000.
For polynomials in more than one indeterminate, the combinations of values for the variables for which the polynomial function takes the value zero are generally called zeros instead of "roots". The study of the sets of zeros of polynomials is the object of algebraic geometry. For a set of polynomial equations in several unknowns, there are algorithms to decide whether they have a finite number of complex solutions, and, if this number is finite, for computing the solutions. See System of polynomial equations.
The special case where all the polynomials are of degree one is called a system of linear equations, for which another range of different solution methods exist, including the classical Gaussian elimination.
A polynomial equation for which one is interested only in the solutions which are integers is called a Diophantine equation. Solving Diophantine equations is generally a very hard task. It has been proved that there cannot be any general algorithm for solving them, and even for deciding whether the set of solutions is empty. Some of the most famous problems that have been solved during the fifty last years are related to Diophantine equations, such as Fermat's Last Theorem.
Generalizations
There are several generalizations of the concept of polynomials.Trigonometric polynomials
A trigonometric polynomial is a finite linear combination of functions sin and cos with n taking on the values of one or more natural numbers. The coefficients may be taken as real numbers, for real-valued functions.If sin and cos are expanded in terms of sin and cos, a trigonometric polynomial becomes a polynomial in the two variables sin and cos. Conversely, every polynomial in sin and cos may be converted, with Product-to-sum identities, into a linear combination of functions sin and cos. 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 used also in the discrete Fourier transform.
Matrix polynomials
A matrix polynomial is a polynomial with square matrices as variables. Given an ordinary, scalar-valued polynomialthis polynomial evaluated at a matrix A is
where I is the identity matrix.
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 Mn.
Laurent polynomials
s are like polynomials, but allow negative powers of the variable to occur.Rational functions
A rational fraction is the quotient 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.
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, 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.Other examples
A bivariate polynomial where the second variable is substituted by an exponential function applied to the first variable, for example, may be called an exponential polynomial.Applications
Abstract algebra
In abstract algebra, one distinguishes between polynomials and polynomial functions. A polynomial in one indeterminate over a ring is defined as a formal expression of the formwhere is a natural number, the coefficients are elements of, and is a formal symbol, whose powers are just placeholders for the corresponding coefficients, so that the given formal expression is just a way to encode the sequence, where there is an such that for all. Two polynomials sharing the same value of n are considered equal if and only if the sequences of their coefficients are equal; furthermore any polynomial is equal to any polynomial with greater value of obtained from it by adding terms in front whose coefficient is zero. These polynomials can be added by simply adding corresponding coefficients. Thus each polynomial is actually equal to the sum of the terms used in its formal expression, if such a term is interpreted as a polynomial that has zero coefficients at all powers of other than. Then to define multiplication, it suffices by the distributive law to describe the product of any two such terms, which is given by the rule
Thus the set of all polynomials with coefficients in the ring forms itself a ring, the ring of polynomials over, which is denoted by. The map from to sending to is an injective homomorphism of rings, by which is viewed as a subring of. If is commutative, then is an algebra over.
One can think of the ring as arising from by adding one new element x to R, and extending in a minimal way to a ring in which satisfies no other relations than the obligatory ones, plus commutation with all elements of . To do this, one must add all powers of 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 of complex numbers, which can be constructed from the polynomial ring over the real numbers by factoring out the ideal of multiples of the polynomial. 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 .
If is commutative, then one can associate to every polynomial in, a polynomial function with domain and range equal to . One obtains the value by substitution of the value for the symbol in. One reason to distinguish between polynomials and polynomial functions is that over some rings different polynomials may give rise to the same polynomial function. This is not the case when 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 require looking at what a polynomial is composed of as an expression rather than evaluating it at some constant value for.
Divisibility
In commutative algebra, one major focus of study is divisibility among polynomials. If is an integral domain and and are polynomials in, it is said that divides or is a divisor of if there exists a polynomial in such that. One can show that every zero gives rise to a linear divisor, or more formally, if is a polynomial in and is an element of such that, then the polynomial divides. The converse is also true. The quotient can be computed using the polynomial long division.If is a field and and are polynomials in with, then there exist unique polynomials and in with
and such that the degree of is smaller than the degree of . The polynomials and are uniquely determined by and. This is called Euclidean division, division with remainder or polynomial long division and shows that the ring is a Euclidean domain.
Analogously, prime 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". 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. 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. 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.
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,. As another example, in radix 5, a string of digits such as 132 denotes the number = 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 formwhere m is a nonnegative integer and the r's are integers such that
Interpolation and approximation
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.Other applications
Polynomials are frequently used to encode information about some other object. The characteristic polynomial of a matrix or linear operator contains information about the operator's eigenvalues. The minimal polynomial of an algebraic element records the simplest algebraic relation satisfied by that element. The chromatic polynomial of a graph 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, 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.