polynomial function

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In , a polynomial is an consisting of (also called ) and s, that involves only the operations of , , , and non-negative 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 s, which encode a wide range of problems, from elementary to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic and to and ; they are used in and to approximate other functions. In advanced mathematics, polynomials are used to construct s and , which are central concepts in and .

Etymology

The word ''polynomial'' : the Greek ''poly'', meaning "many", and the Latin ''nomen'', or name. It was derived from the term ' by replacing the Latin root ''bi-'' with the Greek ''poly-''. That is, it means a sum of many terms (many s). The word ''polynomial'' was first used in the 17th century.

Notation and terminology

The ''x'' occurring in a polynomial is commonly called a ''variable'' or an ''indeterminate''. When the polynomial is considered as an expression, ''x'' is a fixed symbol which does not have any value (its value is "indeterminate"). However, when one considers the 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 (or arguments) of the associated function. A polynomial ''P'' in the indeterminate ''x'' is commonly denoted either as ''P'' or as ''P''(''x''). Formally, the name of the polynomial is ''P'', not ''P''(''x''), but the use of the ''P''(''x'') dates from a time when the distinction between a polynomial and the associated function was unclear. Moreover, the functional notation is often useful for specifying, in a single phrase, a polynomial and its indeterminate. For example, "let ''P''(''x'') 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(s) of the indeterminate(s) do not appear at each occurrence of the polynomial. The ambiguity of having two notations for a single mathematical object may be formally resolved by considering the general meaning of the functional notation for polynomials. If ''a'' denotes a number, a variable, another polynomial, or, more generally, any expression, then ''P''(''a'') denotes, by convention, the result of substituting ''a'' for ''x'' in ''P''. Thus, the polynomial ''P'' defines the function :$a\mapsto P\left(a\right),$ 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 (that is, any ). In particular, if ''a'' is a polynomial then ''P''(''a'') is also a polynomial. More specifically, when ''a'' is the indeterminate ''x'', then the of ''x'' by this function is the polynomial ''P'' itself (substituting ''x'' for ''x'' does not change anything). In other words, :$P\left(x\right)=P,$ which justifies formally the existence of two notations for the same polynomial.

Definition

A ''polynomial expression'' is an that can be built from and symbols called ''variables'' or ''indeterminates'' by means of , and to a power. The constants are generally s, but may be any expression that do not involve the indeterminates, and represent s that can be added and multiplied. Two polynomial expressions are considered as defining the same ''polynomial'' if they may be transformed, one to the other, by applying the usual properties of , and of addition and multiplication. For example $\left(x-1\right)\left(x-2\right)$ and $x^2-3x+2$ are two polynomial expressions that represent the same polynomial; so, one writes $\left(x-1\right)\left(x-2\right)=x^2-3x+2.$ A polynomial in a single indeterminate can always be written (or rewritten) in the form :$a_n x^n + a_x^ + \dotsb + a_2 x^2 + a_1 x + a_0,$ where $a_0, \ldots, a_n$ are constants that are called the ''coefficients'' of the polynomial, and $x$ is the indeterminate. The word "indeterminate" means that $x$ 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 , called a ''polynomial function''. This can be expressed more concisely by using : :$\sum_^n a_k x^k$ That is, a polynomial can either be zero or can be written as the sum of a finite number of non-zero . Each term consists of the product of a number called the 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 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 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 (which has no terms at all) is generally treated as not defined (but see below). For example: :$-5x^2y$ 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: :$\underbrace_ \underbrace_ \underbrace_.$ 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,'' ''s'' and ''cubic polynomials''. For higher degrees, the specific names are not commonly used, although ''quartic polynomial'' (for degree four) and ''quintic polynomial'' (for degree five) are sometimes used. The names for the degrees may be applied to the polynomial or to its terms. For example, the term in is a linear term in a quadratic polynomial. The polynomial 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 (either −1 or −∞). The zero polynomial is also unique in that it is the only polynomial in one indeterminate that has an infinite number of . The graph of the zero polynomial, , is the ''x''-axis. In the case of polynomials in more than one indeterminate, a polynomial is called ''homogeneous'' of if ''all'' of 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 . The of addition can be used to rearrange terms into any preferred order. In polynomials with one indeterminate, the terms are usually ordered according to degree, either in "descending powers of ", with the term of largest degree first, or in "ascending powers of ". The polynomial is written in descending powers of . The first term has coefficient , indeterminate , and exponent . In the second term, the coefficient . 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 , 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 , a two-term polynomial is called a , 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 coefficients. When it is used to define a , the 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 coefficients, and a complex polynomial is a polynomial with coefficients. A polynomial in one indeterminate is called a '' 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 (which may result from the subtraction of non-constant polynomials), although strictly speaking, constant polynomials do not contain any indeterminates at all. It is possible to further classify multivariate polynomials 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 also common to say simply "polynomials in , and ", listing the indeterminates allowed. The consists of substituting a numerical value to each indeterminate and carrying out the indicated multiplications and additions. For polynomials in one indeterminate, the evaluation is usually more efficient (lower number of arithmetic operations to perform) using : :$\left(\left(\left(\left(\left(a_n x + a_\right)x + a_\right)x + \dotsb + a_3\right)x + a_2\right)x + a_1\right)x + a_0.$

Arithmetic

Polynomials can be added using the of addition (grouping all their terms together into a single sum), possibly followed by reordering (using the ) and combining of like terms. For example, if :$P = 3x^2 - 2x + 5xy - 2$ and $Q = -3x^2 + 3x + 4y^2 + 8$ then the sum :$P + Q = 3x^2 - 2x + 5xy - 2 - 3x^2 + 3x + 4y^2 + 8$ can be reordered and regrouped as :$P + Q = \left(3x^2 - 3x^2\right) + \left(- 2x + 3x\right) + 5xy + 4y^2 + \left(8 - 2\right)$ and then simplified to :$P + Q = x + 5xy + 4y^2 + 6.$ When polynomials are added together, the result is another polynomial. Subtraction of polynomials is similar.

Multiplication

Polynomials can also be multiplied. To expand the 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, if :$\begin \color P &\color \\ \color Q &\color \end$ then :$\begin & &&\left(\cdot\right) &+&\left(\cdot\right)&+&\left(\cdot \right)&+&\left(\cdot\right) \\&&+&\left(\cdot\right)&+&\left(\cdot\right)&+&\left(\cdot \right)&+& \left(\cdot\right) \\&&+&\left(\cdot\right)&+&\left(\cdot\right)&+& \left(\cdot \right)&+&\left(\cdot\right) \end$ Carrying out the multiplication in each term produces :$\begin PQ & = && 4x^2 &+& 10xy &+& 2x^2y &+& 2x \\ &&+& 6xy &+& 15y^2 &+& 3xy^2 &+& 3y \\ &&+& 10x &+& 25y &+& 5xy &+& 5. \end$ Combining similar terms yields :$\begin PQ & = && 4x^2 &+&\left( 10xy + 6xy + 5xy \right) &+& 2x^2y &+& \left( 2x + 10x \right) \\ && + & 15y^2 &+& 3xy^2 &+&\left( 3y + 25y \right)&+&5 \end$ which can be simplified to :$PQ = 4x^2 + 21xy + 2x^2y + 12x + 15y^2 + 3xy^2 + 28y + 5.$ As in the example, the product of polynomials is always a polynomial.

Composition

Given a polynomial $f$ of a single variable and another polynomial of any number of variables, the $f \circ g$ is obtained by substituting each copy of the variable of the first polynomial by the second polynomial. For example, if $f\left(x\right) = x^2 + 2x$ and $g\left(x\right) = 3x + 2$ then $(f\circ g)(x) = f(g(x)) = (3x + 2)^2 + 2(3x + 2).$ A composition may be expanded to a sum of terms using the rules for multiplication and division of polynomials. The composition of two polynomials is another polynomial.

Division

The division of one polynomial by another is not typically a polynomial. Instead, such ratios are a more general family of objects, called ''s'', ''rational expressions'', or ''s'', depending on context. This is analogous to the fact that the ratio of two s is a , 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 , generalizing the 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 and . When the denominator is and linear, that is, for some constant , then the asserts that the remainder of the division of by is the . In this case, the quotient may be computed by , a special case of synthetic division.

Factoring

All polynomials with coefficients in a (for example, the integers or a ) also have a factored form in which the polynomial is written as a product of s 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 s, the irreducible factors are linear. Over the s, they have the degree either one or two. Over the integers and the s the irreducible factors may have any degree. For example, the factored form of :$5x^3-5$ is :$5\left(x - 1\right)\left\left(x^2 + x + 1\right\right)$ over the integers and the reals, and :$5\left(x - 1\right)\left\left(x + \frac\right\right)\left\left(x + \frac\right\right)$ 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 s are available in most s.

Calculus

Calculating s and integrals of polynomials is particularly simple, compared to other kinds of functions. The of the polynomial $P = a_n x^n + a_ x^ + \dots + a_2 x^2 + a_1 x + a_0 = \sum_^n a_i x^i$ with respect to is the polynomial $n a_n x^ + (n - 1)a_ x^ + \dots + 2 a_2 x + a_1 = \sum_^n i a_i x^.$ Similarly, the general (or indefinite integral) of $P$ is $\frac + \frac + \dots + \frac + \frac + a_0 x + c = c + \sum_^n \frac$ where is an arbitrary constant. For example, antiderivatives of have the form . For polynomials whose coefficients come from more abstract settings (for example, if the coefficients are integers some , or elements of an arbitrary ring), the formula for the derivative can still be interpreted formally, with the coefficient understood to mean the sum of copies of . For example, over the integers modulo , the derivative of the polynomial is the polynomial .

Polynomial functions

A ''polynomial function'' is a function that can be defined by a polynomial. More precisely, a function of one from a given domain is a polynomial function if there exists a polynomial :$a_n x^n + a_ x^ + \cdots + a_2 x^2 + a_1 x + a_0$ that evaluates to $f\left(x\right)$ for all in the of (here, is a non-negative integer and are constant coefficients). Generally, unless otherwise specified, polynomial functions have 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 to the reals, the resulting function is a that maps reals to reals. For example, the function , defined by :$f\left(x\right) = x^3 - x,$ is a polynomial function of one variable. Polynomial functions of several variables are similarly defined, using polynomials in more than one indeterminate, as in :$f\left(x,y\right)= 2x^3+4x^2y+xy^5+y^2-7.$ 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 $\left\left(\sqrt\right\right)^2,$ which takes the same values as the polynomial $1-x^2$ on the interval

Graphs

File:Algebra1 fnz fig037 pc.svg, Polynomial of degree 0:
File:Fonction de Sophie Germain.png, Polynomial of degree 1:
File:Polynomialdeg2.svg, Polynomial of degree 2:

File:Polynomialdeg3.svg, Polynomial of degree 3:

File:Polynomialdeg4.svg, Polynomial of degree 4:
File:Quintic polynomial.svg, Polynomial of degree 5:
File:Sextic Graph.svg, Polynomial of degree 6:

File:Septic graph.svg, Polynomial of degree 7:

A polynomial function in one real variable can be represented by a .
• The graph of the zero polynomial is the -axis.
• The graph of a degree 0 polynomial is a horizontal line with
• The graph of a degree 1 polynomial (or linear function) is an oblique line with and .
• The graph of a degree 2 polynomial is a .
• The graph of a degree 3 polynomial is a .
• The graph of any polynomial with degree 2 or greater is a continuous non-linear curve.
A non-constant polynomial function when the variable increases indefinitely (in ). If the degree is higher than one, the graph does not have any . It has two es with vertical direction (one branch for positive ''x'' and one for negative ''x''). Polynomial graphs are analyzed in calculus using intercepts, slopes, concavity, and end behavior.

Equations

A ''polynomial equation'', also called an ', is an of the form :$a_n x^n + a_x^ + \dotsb + a_2 x^2 + a_1 x + a_0 = 0.$ For example, :$3x^2 + 4x -5 = 0$ is a polynomial equation. When considering equations, the indeterminates (variables) of polynomials are also called s, and the ''solutions'' are the possible values of the unknowns for which the equality is true (in general more than one solution may exist). A polynomial equation stands in contrast to a ''polynomial '' 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 , methods such as the are taught for solving all first degree and second degree polynomial equations in one variable. There are also formulas for the and s. For higher degrees, the asserts that there can not exist a general formula in radicals. However, s may be used to find s 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 solutions are counted with their . This fact is called the .

Solving equations

A ''root'' of a nonzero univariate polynomial is a value of such that . In other words, a root of is a solutions of the or a of the polynomial function defined by . In the case of the zero polynomial, every number is ia zero of the corresponding function, and the concept of root is rarely cosidered. A number is a root of a polynomial if and only if the divides , that is if there is another polynomial such that . It may happen that a power (greater than ) of divides ; in this case, is a ''multiple root'' of , and otherwise is simple root of . If is a nonzero polynomial, there is a highest power such that divides , which is called the ''multiplicity'' of as a root of . The number of roots of a nonzero polynomial , counted with their respective multiplicities, cannot exceed the degree of , and equals this degree if all roots are considered (this is a consequence of the . The the coefficients of a polynomial and its roots are related by . Some polynomials, such as , do not have any roots among the s. If, however, the set of accepted solutions is expanded to the s, every non-constant polynomial has at least one root; this is the . By successively dividing out factors , one sees that any polynomial with complex coefficients can be written as a constant (its leading coefficient) times a product of such polynomial factors of degree 1; as a consequence, the number of (complex) roots counted with their multiplicities is exactly equal to the degree of the polynomial. There may be several meanings of "solving an equation". One may want to express the solutions as explicit numbers; for example, the unique solution 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 ; for example the $\left(1+\sqrt 5\right)/2$ is the unique positive solution of $x^2-x-1=0.$ In the ancient times, they succeeded only for degrees one and two. For s, the provides such expressions of the solutions. Since the 16th century, similar formulas (using cube roots in addition to square roots), but much more complicated are known for equations of degree three and four (see and ). But formulas for degree 5 and higher eluded researchers for several centuries. In 1824, proved the striking result that there are equations of degree 5 whose solutions cannot be expressed by a (finite) formula, involving only arithmetic operations and radicals (see ). In 1830, 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 and , two important branches of modern . 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 and ). 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 s of the solutions. There are many methods for that; some are restricted to polynomials and others may apply to any . The most efficient s allow solving easily (on a ) polynomial equations of degree higher than 1,000 (see ). 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 . For a set of polynomial equations in several unknowns, there are s to decide whether they have a finite number of solutions, and, if this number is finite, for computing the solutions. See . The special case where all the polynomials are of degree one is called a , for which another range of different exist, including the classical . A polynomial equation for which one is interested only in the solutions which are s is called a . Solving Diophantine equations is generally a very hard task. It has been proved that there cannot be any general for solving them, and even for deciding whether the set of solutions is empty (see ). Some of the most famous problems that have been solved during the fifty last years are related to Diophantine equations, such as .

Generalizations

There are several generalizations of the concept of polynomials.

Trigonometric polynomials

A trigonometric polynomial is a finite of sin(''nx'') and cos(''nx'') with ''n'' taking on the values of one or more s. The coefficients may be taken as real numbers, for real-valued functions. If sin(''nx'') and cos(''nx'') are expanded in terms of sin(''x'') and cos(''x''), a trigonometric polynomial becomes a polynomial in the two variables sin(''x'') and cos(''x'') (using ). Conversely, every polynomial in sin(''x'') and cos(''x'') may be converted, with , into a linear combination of functions sin(''nx'') and cos(''nx''). This equivalence explains why linear combinations are called polynomials. For , there is no difference between such a function and a finite . Trigonometric polynomials are widely used, for example in applied to the of s. They are used also in the .

Matrix polynomials

A is a polynomial with as variables. Given an ordinary, scalar-valued polynomial :$P\left(x\right) = \sum_^n =a_0 + a_1 x+ a_2 x^2 + \cdots + a_n x^n,$ this polynomial evaluated at a matrix ''A'' is :$P\left(A\right) = \sum_^n =a_0 I + a_1 A + a_2 A^2 + \cdots + a_n A^n,$ where ''I'' is the . 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 ''Mn''(''R'').

Laurent polynomials

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

Rational functions

A is the () of two polynomials. Any that can be rewritten as a rational fraction is a . 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 (just like s cannot), but the rules for manipulating their terms are the same as for polynomials. Non-formal 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 .

Polynomial ring

A ''polynomial'' over a is a polynomial whose all coefficients belong to . It is straightforward to verify that the polynomials in a given set of indeterminates over form a commutative ring, called the ''polynomial ring'' in these indeterminates, denoted

Divisibility

If is an and and are polynomials in , it is said that ''divides'' or is a divisor of if there exists a polynomial in such that . If $a\in R,$ then is a root of if and only $x-a$ divides . In this case, the quotient can be computed using the . If is a and and are polynomials in with , then there exist unique polynomials and in with :$f = q \, g + r$ and such that the degree of is smaller than the degree of (using the convention that the polynomial 0 has a negative degree). The polynomials and are uniquely determined by and . This is called '', division with remainder'' or ''polynomial long division'' and shows that the ring is a . Analogously, ''prime polynomials'' (more correctly, ''s'') 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-"'' (both definitions agree in the case of coefficients in a field). Any polynomial may be decomposed into the product of an invertible constant by a product of irreducible polynomials. If the coefficients belong to a field or a this decomposition is unique up to the order of the factors and the multiplication of any non-unit factor by a unit (and division of the unit factor by the same unit). When the coefficients belong to integers, rational numbers or a finite field, there are algorithms to test irreducibility and to compute the factorization into irreducible polynomials (see ). These algorithms are not practicable for hand-written computation, but are available in any . can also be used in some cases to determine irreducibility.

Applications

Positional notation

In modern positional numbers systems, such as the , the digits and their positions in the representation of an integer, for example, 45, are a shorthand notation for a polynomial in the or base, in this case, . As another example, in radix 5, a string of digits such as 132 denotes the (decimal) 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 form :$a = r_m b^m + r_ b^ + \dotsb + r_1 b + r_0,$ where ''m'' is a nonnegative integer and the ''rs are integers such that : and for .

Interpolation and approximation

The simple structure of polynomial functions makes them quite useful in analyzing general functions using polynomial approximations. An important example in is , which roughly states that every locally looks like a polynomial function, and the , which states that every defined on a of the real axis can be approximated on the whole interval as closely as desired by a polynomial function. Practical methods of approximation include and the use of .

Other applications

Polynomials are frequently used to encode information about some other object. The of a matrix or linear operator contains information about the operator's s. The of an records the simplest algebraic relation satisfied by that element. The of a 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 the phrase ' means that the time it takes to complete an is bounded by a polynomial function of some variable, such as the size of the input.

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 , 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 .

History of the notation

The earliest known use of the equal sign is in 's ', 1557. The signs + for addition, − for subtraction, and the use of a letter for an unknown appear in 's ''Arithemetica integra'', 1544. , 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 's denote constants and denotes a variable. Descartes introduced the use of superscripts to denote exponents as well.