In mathematics, a polynomial (from Greek poly, "many" and medieval Latin binomium, "binomial"  , the word has been introduced, in Latin, by Franciscus Vieta) 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 integer exponents. For example, x2 − 4x + 7 is a polynomial, but x2 − 4/x + 7x3/2 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 an integer (3/2). The term "polynomial" can also be used as an adjective, for quantities that can be expressed as a polynomial of some parameter, as in "polynomial time" which is used in computational complexity theory.
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.
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), also called indeterminates. 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 (nonzero) 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.
is a term. The coefficient is –5, the variables are x and y, the degree of x is in the term two, while 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.
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.
The commutative law of addition can be used to freely permute terms into any preferred order. In polynomials with 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", and they can be combined (after having been made adjacent) 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, in which case their combination just cancels out the terms. Polynomials can be added using the associative law of addition (which simply groups all their terms together into a single sum), possibly followed by reordering, and combining of like terms. For example, if
which can be simplified to
To work out the product of two polynomials into a sum of terms, the distributive law is repeatedly applied, with as result that each term of one polynomial multiplied by every term of the other. For example, if
which can be simplified to
The sum or product of two polynomials is always a polynomial.
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. A formal quotient of polynomials, that is, an algebraic fraction where the numerator and denominator are polynomials, is called a "rational expression" or "rational fraction" and is not, in general, a polynomial. Division of a polynomial by a number, however, does yield another polynomial. For example,
is considered a valid term in a polynomial (and a polynomial by itself) because it is equivalent to and is just a constant. When this expression is used as a term, its coefficient is therefore . For similar reasons, if complex coefficients are allowed, one may have a single term like (2 + 3i)x3; 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.
is not a polynomial because it includes division by a non-constant polynomial.
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.
for all arguments x, where n is a non-negative integer and a0, a1,a2, ..., an are constant coefficients.
For example, the function ƒ, taking real numbers to real numbers, defined by
is a polynomial function of one argument. Polynomial functions of multiple arguments can also be defined, using polynomials in multiple variables, as in
An example is also the function f(x) = cos(2arccos(x)) which, although it doesn't look like a polynomial, is a polynomial function since for every x it is true that f(x) = 2x2 − 1 (see Chebyshev polynomials).
is a polynomial equation. In case of a univariate 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.
- A composition of two polynomials is a polynomial, which is obtained by substituting a variable of the first polynomial by the second polynomial.
- The derivative of the polynomial anxn + an-1xn-1 + ... + a2x2 + a1x + a0 is the polynomial nanxn-1 + (n-1)an-1xn-2 + ... + 2a2x + a1. If the set of the coefficients does not contain the integers (for example if the coefficients are integers modulo some prime number p), then kak should be interpreted as the sum of ak with itself, k times. For example, over the integers modulo p, the derivative of the polynomial xp+1 is the polynomial 0.
- If the division by integers is allowed in the set of coefficients, a primitive or antiderivative of the polynomial anxn + an-1xn-1 + ... + a2x2 + a1x + a0 is anxn+1/(n+1) + an-1xn/n + ... + a2x3/3 + a1x2/2 + a0x +c, where c is an arbitrary constant. Thus x2+1 is a polynomial with integer coefficients whose primitives are not polynomials over the integers. If this polynomial is viewed as a polynomial over the integers modulo 3 it has no primitive at all.
All polynomials have an expanded form, in which the distributive and associative laws have been used to remove all brackets and commutative law has been used to make the like terms adjacent and combine them. All polynomials with coefficients in a unique factorization domain (for example, the integers or a field also have a factored form in which the polynomial is written as a product of irreducible polynomials and a constant. In the case of the field of complex numbers, the irreducible polynomials are linear. For example, the factored form of
over the integers and
over the complex numbers.
Every polynomial in one variable is equivalent to a polynomial with the form
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. Actual evaluation is usually more efficient using the Horner scheme:
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 each 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.
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.
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.
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: if (x − a)2 divides P then a is called a multiple root of P, and otherwise a is called a simple root of P. If P is a nonzero polynomial, there is a highest power m such that (x − a)m divides P, which 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 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. By successively dividing out factors x − 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 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 cubic function and quartic function for the formulas and Niccolo Fontana Tartaglia, Lodovico Ferrari, Gerolamo Cardano, and Vieta for historical details). 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). In 1830, Évariste Galois, studying the permutations of the roots of a polynomial, extended Abel-Ruffini theorem by showing that, given a polynomial equation, one may decide if 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 mathematics. 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 (see quintic function and sextic equation).
Numerical approximations of roots of polynomial equations in one unknown is easily done on a computer by the Jenkins-Traub method, Laguerre's method, 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.
For a set of polynomial equations in several unknowns, there are algorithms to decide if they have a finite number of complex solutions. If the number of solutions is finite, there are algorithms to compute the solutions. The methods underlying these algorithms are described in the article systems 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.
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. 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.
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 + ... + anxn , where an ≠ 0 and n ≥ 2
- is a continuous non-linear curve.
Polynomial graphs are analyzed in calculus using intercepts, slopes, concavity, and end behavior.
The illustrations below show graphs of polynomials.
Polynomials and calculus
One important aspect of calculus is the project of analyzing complicated functions by means of approximating them with polynomial functions. The culmination of these efforts 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. Polynomial functions are also frequently used to interpolate functions.
Calculating derivatives and integrals of polynomial functions is particularly simple. For the polynomial function
the derivative with respect to x is
and the indefinite integral is
where n is a natural number, the coefficients are elements of R, and X is a formal symbol, whose powers Xi are just placeholders for the corresponding coefficients ai, so that the given formal expression is just a way to encode the sequence , where there is an n such that ai = 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 aiXi 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
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.
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 or f is a divisor of g if there exists a polynomial q in R[X] such that f q = g. One can show that every zero gives rise to a linear divisor, 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
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 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 polynomials which cannot be factorized into the product of two non constant polynomials. Any polynomial may be decomposed into the product of a constant by a product of irreducible polynomials. This decomposition is unique up to the order of the factors and the multiplication of any constant factors by a constant (and division of the constant factor by the same constant. When the coefficients belong to a finite field or are rational numbers, 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 (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 ). Eisenstein's criterion can also be used in some cases to determine irreducibility.
See also: Greatest common divisor of two polynomials.
Polynomials are classified according to many different properties.
Number of variables
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 x2 + 1 3 cubic x3 + 1 4 quartic (or biquadratic) x4 + 1 5 quintic x5 + 1 6 sextic (or hexic) x6 + 1 7 septic (or heptic) x7 + 1 8 octic x8 + 1 9 nonic x9 + 1 10 decic x10 + 1 100 hectic x100 + 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 x2 + 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 –∞). 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 x2 2 binomial x2 + 1 3 trinomial x2 + 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
- cn ≠ 0, cn-1, ..., c2, c1 and c0 are constants, the coefficients of this polynomial.
Here the term cnxn is called the leading term and its coefficient cn the leading coefficient; if the leading coefficient is 1, the univariate polynomial is called monic.
Note that apart from the leading coefficient cn (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 ci as the coefficient of xi, even when ci 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, x3y2 + 7x2y3 − 3x5 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 x2 + 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.
Number of non-zero terms
Polynomials may also 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 so on. (Some authors use "monomial" to mean "monic monomial".)
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,eX ), 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 (or rational fractions), and functions that evaluate rational expressions are called rational functions. Rational fractions 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 function defined by a rational fraction is the quotient of the polynomial functions defined by the numerator and the denominator of the rational fraction. The rational fractions contain the Laurent polynomials, but do not limit denominators to be powers of a variable. 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 null. A rational function produces rational output for any rational input for which it is defined; this is not true of 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.
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- List of Calculators for Quadratic through Sextic equations
- Euler's work on Imaginary Roots of Polynomials at Convergence
- Characteristics of polynomials
- Free online polynomial root finder for both real and complex coefficients
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