Polynomial

In mathematics, a polynomial is an expression of finite length constructed from variables (also known as indeterminates) and constants, using only the operations of addition, subtraction, multiplication, and non-negative, whole-number exponents. For example, x² − 4x + 7 is a polynomial, but x² − 4/x + 7x^3/2 is not, because its second term involves division by the variable x (4/x) and because its third term contains an exponent that is not a whole number (3/2). The term 'polynomial' indicates a simplified algebraic form such that all polynomials are similarly simple in complexity (cf. polynomial time).

Polynomials appear in a wide variety of 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 problems in the sciences; 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, a central concept in abstract algebra and algebraic geometry.

Overview

A polynomial is either zero, or can be written as the sum of one or more non-zero terms. The number of terms is finite. These terms consist of a constant (called the coefficient of the term) which may be multiplied by a finite number of variables (usually represented by letters). Each variable may have an exponent that is a non-negative integer, i.e., a natural number. The exponent on a variable in a term is called the degree of that variable in that term, the degree of the term is the sum of the degrees of the variables in that term, and the degree of a polynomial is the largest degree of any one term. Since x = x¹, the degree of a variable without a written exponent is one. A term with no variables is called a constant term, or just a constant. The degree of a constant term is 0. 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 of some prime number p. In most of the examples in this section, the coefficients are integers.

For example:

$-5x^2y\,$

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

The degree of the entire term is the sum of the degrees of each variable in it, so in this example the degree is 2 + 1 = 3.

A polynomial is a sum of terms. For example, the following is a polynomial:

$\underbrace{_\,3x^2}_{\begin{smallmatrix}\mathrm{term}\\\mathrm{1}\end{smallmatrix}} \underbrace{-_\,5x}_{\begin{smallmatrix}\mathrm{term}\\\mathrm{2}\end{smallmatrix}} \underbrace{+_\,4}_{\begin{smallmatrix}\mathrm{term}\\\mathrm{3}\end{smallmatrix}}.$

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

In polynomials in one variable, the terms are usually ordered according to degree, either in "descending powers of x", with the term of largest degree first, or in "ascending powers of x". The polynomial in the example above is written in descending powers of x. The first term has coefficient 3, variable x, and exponent 2. In the second term, the coefficient is –5. The third term is a constant. Since 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 variables raised to the same powers are called "like terms". Polynomials are added using the commutative, associative, and distributive laws, by combining like terms. For example if $P=3x^2-2x+5xy-2$ and $Q=-3x^2+3x+4y^2+8$ then $P+Q=3x^2-2x+5xy-2+-3x^2+3x+4y^2+8$ which can be rewritten as $P+Q=x+5xy+4y^2+6$ .

The same three laws are used to multiply polynomials, with each term of one polynomial multiplied by every term of another. For example if $P=2x+3y+5$ and $Q=2x+5y+xy+1$ then

$P\times Q=2x\times2x+2x\times5y+2x\times xy+2x\times1+3y\times2x+3y\times5y+3y\times xy+3y\times1+5\times2x+5\times5y+5\times xy+5\times1,$

which can be rewritten as

$PQ=4x^2+21xy+2x^2y+12x+15y^2+3xy^2+28y+5$ .

The sum or product of two polynomials is always a polynomial.

Alternative forms

In general any expression can be considered to be a polynomial if it is built up from variables and constants using only addition, subtraction, multiplication, and raising expressions to constant positive whole number powers. Such an expression can always be rewritten as a sum of terms. For example, (x + 1)³ is a polynomial; its standard form is x³ + 3x² + 3x + 1.

Division of one polynomial by another does not, in general, produce a polynomial, but rather produces a quotient and a remainder.^[1] A formal quotient of polynomials, that is, an expression in the form of a fraction where the numerator and denominator are polynomials, is called a "rational expression" or an "algebraic fraction" and is not, in general, a polynomial. Division of a polynomial by a number, however, does yield another polynomial. For example,

$\frac{x^3}{12}$

is considered a valid term in a polynomial (and a polynomial by itself) because it is equivalent to $\tfrac{1}{12}x^3$ and $\tfrac{1}{12}$ is just a constant. When this expression is used as a term, its coefficient is therefore $\tfrac{1}{12}$ . For similar reasons, if complex coefficients are allowed, one may have a single term like $(2+3i)x^3$ ; even though it looks like it should be expanded to two terms, the complex number 2 + 3i is one complex number, and is the coefficient of that term.

${1 \over x^2 + 1} \,$

is not a polynomial because it includes division by a non-constant polynomial.

$( 5 + y ) ^ x ,\,$

is not a polynomial, because it contains a variable used as exponent.

Since subtraction can be replaced by addition of the opposite quantity, and since positive whole number exponents can be replaced by repeated multiplication, all polynomials can be constructed from constants and variables using only addition and multiplication.

Polynomial functions

A polynomial function is a function that can be defined by evaluating a polynomial. A function ƒ of one argument is called a polynomial function if it satisfies

$f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0 \,$

for all arguments x, where n is a non-negative integer and a₀, a₁,a₂, ..., a_n are constant coefficients.

For example, the function ƒ, taking real numbers to real numbers, defined by

$f(x) = x^3 - x\,$

is a polynomial function of one argument. Polynomial functions of multiple arguments can also be defined, using polynomials in multiple variables, as in

$f(x,y)= 2x^3+4x^2y+xy^5+y^2-7.\,$

An example is also the function $f(x)=\cos(2\arccos(x))$ which, although it doesn't look like a polynomial, is a polynomial function since for every x it is true that $f(x)=2x^2-1$ (see Chebyshev polynomials).

Polynomial functions are a class of functions having many important properties. They are all continuous, smooth, entire, computable, etc.

Polynomial equations

A polynomial equation is an equation in which a polynomial is set equal to another polynomial.

$3x^2 + 4x -5 = 0 \,$

is a polynomial equation. In case of a polynomial equation the variable is considered an unknown, and one seeks to find the possible values for which both members of the equation evaluate to the same value (in general more than one solution may exist). A polynomial equation is to be contrasted with a polynomial identity like (x + y)(x – y) = x² – y², where both members represent the same polynomial in different forms, and as a consequence any evaluation of both members will give a valid equality. This means that a polynomial identity is a polynomial equation for which all possible values of the unknowns are solutions.

Elementary properties of polynomials

A sum of polynomials is a polynomial.
A product of polynomials is a polynomial
The derivative of a polynomial function is a polynomial function.

The derivative of a_nxⁿ + a_n-1x^n-1 + ... + a₂x² + a₁x + a₀ is na_nx^n-1 + (n-1)a_n-1x^n-2 + ... + 2a₂x + a₁.

Any primitive or antiderivative of a polynomial function is a polynomial function.

The primitives of a_nxⁿ + a_n-1x^n-1 + ... + a₂x² + a₁x + a₀ are a_nxⁿ⁺¹/(n+1) + a_n-1xⁿ/n + ... + a₂x³/3 + a₁x²/2 + a₀x +c.

Polynomials serve to approximate other functions, such as sine, cosine, and exponential.

All polynomials have an expanded form, in which the distributive law has been used to remove all brackets. All polynomials with real or complex coefficients also have a factored form in which the polynomial is written as a product of linear complex polynomials. For example, the polynomial

$x^2 - 2x - 3 \,$

is the expanded form of the polynomial

$(x - 3)(x + 1),\,$

which is written in factored form. Note that the constants in the linear polynomials (like −3 and +1 in the above example) may be complex numbers in certain cases, even if all coefficients of the expanded form are real numbers. This is because the field of real numbers is not algebraically closed; however, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed.

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

$a_n x^n + a_{n-1}x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0.$

This form is sometimes taken as the definition of a polynomial in one variable.

Evaluation of a polynomial consists of assigning a number to each variable and carrying out the indicated multiplications and additions. Evaluation is sometimes performed more efficiently using the Horner scheme

$((\cdots(a_n x + a_{n-1})x + \cdots + a_2)x + a_1)x + a_0.\,$

In elementary algebra, methods are given for solving all first degree and second degree polynomial equations in one variable. In the case of polynomial equations, the variable is often called an unknown. The number of solutions may not exceed the degree, and will equal the degree when multiplicity of solutions and complex number solutions are counted. This fact is called the fundamental theorem of algebra.

A system of polynomial equations is a set of equations in which a given variable must take on the same value everywhere it appears in any of the equations. Systems of equations are usually grouped with a single open brace on the left. In elementary algebra, in particular in linear algebra, methods are given for solving a system of linear equations in several unknowns. If there are more unknowns than equations, the system is called underdetermined. If there are more equations than unknowns, the system is called overdetermined. Overdetermined systems are common in practical applications. For example, one U.S. mapping survey used computers to solve 2.5 million equations in 400,000 unknowns.^[2]

Viète's formulas relate the coefficients of a polynomial to symmetric polynomial functions of its roots.

History

Determining the roots of polynomials, or "solving algebraic equations", is among the oldest problems in mathematics. However, the elegant and practical notation we use today only developed beginning in the 15th century. Before that, equations were written out in words. For example, an algebra problem from the Chinese Arithmetic in Nine Sections, circa 200 BCE, begins "Three sheafs of good crop, two sheafs of mediocre crop, and one sheaf of bad crop are sold for 29 dou." We would write 3x + 2y + z = 29.

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.^[3]

Solving polynomial equations

Every polynomial P in x corresponds to a function, ƒ(x) = P (where the occurrences of x in P are interpreted as the argument of ƒ), called the polynomial function of P; the equation in x setting f(x) = 0 is the polynomial equation corresponding to P. The solutions of this equation are called the roots of the polynomial; they are the zeroes of the function ƒ (corresponding to the points where the graph of ƒ meets the x-axis). A number a is a root of P if and only if the polynomial x − a (of degree one in x) divides P. It may happen that x − a divides P more than once: some power (x − a)^m divides P; the largest such power m is called the multiplicity of the root a in P. When P is the zero polynomial, the corresponding polynomial equation is trivial, and this case is usually excluded when considering roots: with the above definitions every number would be a root of the zero polynomial, with undefined (or infinite) multiplicity. With this exception made, the number of roots of P, even counted with their respective multiplicities, cannot exceed the degree of P.

Some polynomials, such as ƒ(x) = x² + 1, do not have any roots among the real numbers. If, however, the set of allowed candidates is expanded to the complex numbers, every non-constant polynomial has at least one root; this is the fundamental theorem of algebra.

There is a difference between approximating roots and finding exact expressions for roots. Formulas for expressing the roots of polynomials of degree 2 in terms of square roots have been known since ancient times (see quadratic equation), and for polynomials of degree 3 or 4 similar formulas (using cube roots in addition to square roots) were found in the 16th century (see Niccolo Fontana Tartaglia, Lodovico Ferrari, Gerolamo Cardano, and Vieta). But formulas for degree 5 eluded researchers. In 1824, Niels Henrik Abel proved the striking result that there can be no general (finite) formula, involving only arithmetic operations and radicals, that expresses the roots of a polynomial of degree 5 or greater in terms of its coefficients (see Abel-Ruffini theorem). This result marked the start of Galois theory which engages in a detailed study of relationships among roots of polynomials.

Numerical approximations of roots of polynomial equations in one unknown is easily done on a computer by the Durand–Kerner method or by some other root-finding algorithm. For polynomials in more than one variable the notion of root does not exist, and there are usually infinitely many combinations of values for the variables for which the polynomial function takes the value zero. However for certain sets of such polynomials it may happen that for only finitely many combinations all polynomial functions take the value zero. There exists a method of reduction of sets of equations in several unknowns to equations in fewer unknowns, which may be used to find such solutions; this is discussed in the article on the Buchberger's algorithm. The special case where all the polynomials are of degree one is called a system of linear equations, for which a range of different solution methods exist, including the classical gaussian elimination.

It has been shown by Richard Birkeland and Karl Meyr that the roots of any polynomial may be expressed in terms of multivariate hypergeometric functions. Ferdinand von Lindemann and Hiroshi Umemura showed that the roots may also be expressed in terms of Siegel modular functions, generalizations of the theta functions that appear in the theory of elliptic functions. These characterisations of the roots of arbitrary polynomials are generalisations of the methods previously discovered to solve the quintic equation.

Method based on eigenvalues computation

A numerical method for the progressive elimination of multiple roots, based on eigenvalue computation, has been proposed.^[4]^[5] The method transforms the problem into a sequence of eigenvalue problems involving tridiagonal matrices with only simple eigenvalues: such eigenvalues can easily be approximated by means of the QR algorithm.

Graphs

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

The graph of the zero polynomial

f(x) = 0

is the x-axis.

The graph of a degree 0 polynomial

f(x) = a₀, where a₀ ≠ 0,

is a horizontal line with y-intercept a₀

The graph of a degree 1 polynomial (or linear function)

f(x) = a₀ + a₁x , where a₁ ≠ 0,

is an oblique line with y-intercept a₀ and slope a₁.

The graph of a degree 2 polynomial

f(x) = a₀ + a₁x + a₂x², where a₂ ≠ 0

is a parabola.

The graph of a degree 3 polynomial

f(x) = a₀ + a₁x + a₂x², + a₃x³, where a₃ ≠ 0

is a cubic curve.

The graph of any polynomial with degree 2 or greater

f(x) = a₀ + a₁x + a₂x² + ... + a_nxⁿ , where a_n ≠ 0 and n ≥ 2

is a continuous non-linear curve.

The graph of a non-constant (univariate) polynomial always tends to infinity when the variable increases indefinitely (in absolute value).

Polynomial graphs are analyzed in calculus using intercepts, slopes, concavity, and end behavior.

The illustrations below show graphs of polynomials.

Polynomial of degree 2: f(x) = x² - x - 2 = (x+1)(x-2)	Polynomial of degree 3: f(x) = x³/4 + 3x²/4 - 3x/2 - 2 = 1/4 (x+4)(x+1)(x-2)	Polynomial of degree 4: f(x) = 1/14 (x+4)(x+1)(x-1)(x-3) + 0.5
Polynomial of degree 5: f(x) = 1/20 (x+4)(x+2)(x+1)(x-1)(x-3) + 2	Polynomial of degree 6: f(x) = 1/30 (x+3.5)(x+2)(x+1)(x-1)(x-3)(x-4) + 2

Polynomials and calculus

One important aspect of calculus is the project of analyzing complicated functions by means of approximating them with polynomials. The culmination of these efforts is Taylor's theorem, which roughly states that every differentiable function locally looks like a polynomial, and the Stone-Weierstrass theorem, which states that every continuous function defined on a compact interval of the real axis can be approximated on the whole interval as closely as desired by a polynomial. Polynomials are also frequently used to interpolate functions.

Calculating derivatives and integrals of polynomials is particularly simple. For the polynomial

$\sum_{i=0}^n a_i x^i$

the derivative with respect to x is

$\sum_{i=1}^n a_i i x^{i-1}$

and the indefinite integral is

$\sum_{i=0}^n {a_i\over i+1} x^{i+1}+c.$

Abstract algebra

In abstract algebra, one distinguishes between polynomials and polynomial functions. A polynomial f in one variable X over a ring R is defined to be a formal expression of the form

$f = a_n X^n + a_{n - 1} X^{n - 1} + \cdots + a_1 X^1 + a_0X^0$

where n is a natural number, the coefficients $a_0,\ldots,a_n$ are elements of R, and X is a formal symbol, whose powers Xⁱ are just placeholders for the corresponding coefficients a_i, so that the given formal expression is just a way to encode the sequence $(a_0, a_1, \ldots)$ , where there is an n such that a_i = 0 for all i > n. Two polynomials sharing the same value of n are considered to be equal if and only if the sequences of their coefficients are equal; furthermore any polynomial is equal to any polynomial with greater value of n obtained from it by adding terms in front whose coefficient is zero. These polynomials can be added by simply adding corresponding coefficients (the rule for extending by terms with zero coefficients can be used to make sure such coefficients exist). Thus each polynomial is actually equal to the sum of the terms used in its formal expression, if such a term a_iXⁱ is interpreted as a polynomial that has zero coefficients at all powers of X other than Xⁱ. 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

$a X^k \; b X^l = ab X^{k+l}$

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

Thus the set of all polynomials with coefficients in the ring R forms itself a ring, the ring of polynomials over R, which is denoted by R[X]. The map from R to R[X] sending r to rX⁰ is an injective homomorphism of rings, by which R is viewed as a subring of R[X]. If R is commutative, then 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 X² + 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 to 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 the 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. And it should be noted that if R is not commutative, there is no (well behaved) notion of polynomial function at all.

Divisibility

In commutative algebra, one major focus of study is divisibility among polynomials. If R is an integral domain and f and g are polynomials in R[X], it is said that f divides g if there exists a polynomial q in R[X] such that f q = g. One can then show that "every zero gives rise to a linear factor", or more formally: if f is a polynomial in R[X] and r is an element of R such that f(r) = 0, then the polynomial (X − r) divides f. The converse is also true. The quotient can be computed using the Horner scheme.

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

$f = q \, g + r$

and such that the degree of r is smaller than the degree of g. The polynomials q and r are uniquely determined by f and g. This is called "division with remainder" or "polynomial long division" and shows that the ring F[X] is a Euclidean domain.

Analogously, polynomial "primes" (more correctly, irreducible polynomials) can be defined which cannot be factorized into the product of two polynomials of lesser degree. It is not easy to determine if a given polynomial is irreducible but algorithms exist for doing it (see Berlekamp's algorithm for the case in which the coefficients belong to a finite field or the Berlekamp–Zassenhaus algorithm when working over the rational numbers ^[6]). Eisenstein's criterion can also be used in some cases to determine irreducibility.

See also: Greatest common divisor of two polynomials.

Classifications

Polynomials are classified according to many different properties. One classification of polynomials is based on the number of distinct variables. A polynomial in one variable is called a univariate polynomial, a polynomial in more than one variable is called a multivariate 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 (which may result, for instance, from the subtraction of non-constant polynomials), although strictly speaking constant polynomials do not contain any variables at all. It is possible to further classify multivariate polynomials as bivariate, trivariate, and so on, according to the maximum number of variables 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 x, y, and z", listing the variables allowed. In this case, xy is allowed.

A second major way of classifying polynomials is by their degree. Recall that the degree of a term is the sum of the exponents on variables, and that the degree of a polynomial is the largest degree of any one term.

Polynomials classified by degree
Degree	Name	Example
−∞	zero	$0$
$0$	(non-zero) constant	$1$
$1$	linear	$x + 1$
$2$	quadratic	$x^2 + 1$
$3$	cubic	$x^3 + 1$
$4$	quartic (or biquadratic)	$x^4 + 1$
$5$	quintic	$x^5 + 1$
$6$	sextic (or hexic)	$x^6 + 1$
$7$	septic (or heptic)	$x^7 + 1$
$8$	octic	$x^8 + 1$
$9$	nonic	$x^9 + 1$
$10$	decic	$x^{10} + 1$

Usually, a polynomial of degree n, for n greater than 3, is called a polynomial of degree n, although the phrases quartic polynomial and quintic polynomial are sometimes used. The use of names for degrees greater than 5 is even less common. The names for the degrees may be applied to the polynomial or to its terms. For example, in $x^2 + 2x + 1$ the term $2x$ is a first degree term in a second degree polynomial.

In the context of polynomial interpolation there is some ambiguity when combining the two classifications above. For example, a bilinear interpolant, being the product of two univariate linear polynomials, is bivariate but is not linear; similar ambiguity affects the bicubic interpolant.

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 to be negative (either –1 or –∞).^[7] These conventions are important when defining Euclidean division of polynomials. The zero polynomial is also unique in that it is the only polynomial having an infinite number of roots.

Polynomials classified by number of non-zero terms
Number of non-zero terms	Name	Example
$0$	zero polynomial	$0$
$1$	monomial	$x^2$
$2$	binomial	$x^2 + 1$
$3$	trinomial	$x^2 + x + 1$

If a polynomial has only one variable, then the terms are usually written either from highest degree to lowest degree ("descending powers") or from lowest degree to highest degree ("ascending powers"). A univariate polynomial in x of degree n then takes the general form

$c_nx^n+c_{n-1}x^{n-1}+\cdots+c_2x^2+c_1x+c_0$

where

c_n ≠ 0, c_n-1, ..., c₂, c₁ and c₀ are constants, the coefficients of this polynomial.

Here the term c_nxⁿ is called the leading term and its coefficient c_n the leading coefficient; if the leading coefficient is 1, the univariate polynomial is called monic.

Note that apart from the leading coefficient c_n (which must be non-zero or else the polynomial would not be of degree n) this general form allows for coefficients to be zero; when this happens the corresponding term is zero and may be removed from the sum without changing the polynomial. It is nevertheless common to refer to c_i as the coefficient of xⁱ, even when c_i happens to be 0, so that xⁱ does not really occur in any term; for instance one can speak of the constant term of the polynomial, meaning c₀ even if it is zero.

In the case of polynomials in more than one variable, a polynomial is called homogeneous of degree n if all its terms have degree n. For example, $x^3y^2 + 7x^2y^3 - 3x^5$ is homogeneous.

Another classification of polynomials is by the kind of constant values allowed as coefficients. One can work with polynomials with integer, rational, real, or complex coefficients, and in abstract algebra polynomials with many other types of coefficients can be defined, such as integers modulo p. As in the classification by number of variables, when working with coefficients for a given set, such as the complex numbers, coefficients from any subset are allowed. Thus $x^2 + 3x -5$ is a polynomial with integer coefficients, but it is also a polynomial with complex coefficients, because the integers are a subset of the complex numbers.

And polynomials may be classified by the number of non-zero terms (after like terms are combined), so that a one-term polynomial is called a monomial, a two-term polynomial is called a binomial, and so on. (Some authors use "monomial" to mean "monic monomial".^[8])

Polynomials associated to other objects

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.

Extensions of the concept of a polynomial

Polynomials can involve more than one variable, in which they are called multivariate. Rings of polynomials in a finite number of variables are of fundamental importance in algebraic geometry which studies the simultaneous zero sets of several such multivariate polynomials. These rings can alternatively be constructed by repeating the construction of univariate polynomials with as coefficient ring another ring of polynomials: thus the ring R[X,Y] of polynomials in X and Y can be viewed as the ring (R[X])[Y] of polynomials in Y with as coefficients polynomials in X, or as the ring (R[Y])[X] of polynomials in X with as coefficients polynomials in Y. These identifications are compatible with arithmetic operations (they are isomorphisms of rings), but some notions such as degree or whether a polynomial is considered monic can change between these points of view. One can construct rings of polynomials in infinitely many variables, but since polynomials are (finite) expressions, any individual polynomial can only contain finitely many variables.

A binary polynomial where the second variable takes the form of an exponential function applied to the first variable, for example P(X,e^X), may be called an exponential polynomial.

Laurent polynomials are like polynomials, but allow negative powers of the variable(s) to occur.

Quotients of polynomials are called rational expressions, and functions that evaluate rational expressions are called rational functions. Rational functions are formal quotients of polynomials (they are formed from polynomials just as rational numbers are formed from integers, writing a fraction of two of them; fractions related by the canceling of common factors are identified with each other). The rational functions contain the Laurent polynomials, but do not limit denominators to be powers of a variable. A rational function produces rational output for any rational input; this is not true other functions such as trigonometric functions, logarithms and exponential functions.

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 real numbers cannot), but the rules for manipulating their terms are the same as for polynomials.

Notes

↑ Peter H. Selby, Steve Slavin, Practical Algebra: A Self-Teaching Guide, 2nd Edition, Wiley, ISBN 0471530123 ISBN 978-0471530121
↑ Gilbert Strang, Linear Algebra and its Applications, Fourth Edition, Thompson Brooks/Cole, ISBN 0030105676.
↑ Howard Eves, An Introduction to the History of Mathematics, Sixth Edition, Saunders, ISBN 0030295580
↑ L.Brugnano, D.Trigiante. Polynomial Roots: the Ultimate Answer?, Linear Algebra and its Applications 225 (1995) 207-219
↑ L.Brugnano. Numerical Implementation of a New Algorithm for Polynomials with Multiple Roots, Journal of Difference Equations and Applications 1 (1995) 187-207
↑ http://mathworld.wolfram.com/Berlekamp-ZassenhausAlgorithm.html
↑ Weisstein, Eric W., "Zero Polynomial" from MathWorld.
↑ Anthony W. Knapp (2007). Advanced Algebra: Along with a Companion Volume Basic Algebra. Springer. p. 457. ISBN 0817645225.

References

R. Birkeland. Über die Auflösung algebraischer Gleichungen durch hypergeometrische Funktionen. Mathematische Zeitschrift vol. 26, (1927) pp. 565–578. Shows that the roots of any polynomial may be written in terms of multivariate hypergeometric functions.
F. von Lindemann. Über die Auflösung der algebraischen Gleichungen durch transcendente Functionen. Nachrichten von der Königl. Gesellschaft der Wissenschaften, vol. 7, 1884. Polynomial solutions in terms of theta functions.
F. von Lindemann. Über die Auflösung der algebraischen Gleichungen durch transcendente Functionen II. Nachrichten von der Königl. Gesellschaft der Wissenschaften und der Georg-Augusts-Universität zu Göttingen, 1892 edition.
K. Mayr. Über die Auflösung algebraischer Gleichungssysteme durch hypergeometrische Funktionen. Monatshefte für Mathematik und Physik vol. 45, (1937) pp. 280–313.
H. Umemura. Solution of algebraic equations in terms of theta constants. In D. Mumford, Tata Lectures on Theta II, Progress in Mathematics 43, Birkhäuser, Boston, 1984.