Polynomial
In mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers. An example of a polynomial of a single indeterminate x is x2 − 4x + 7. An example with three indeterminates is x3 + 2xyz2 − yz + 1.
For less elementary aspects of the subject, see Polynomial ring.Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; and they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry.
Etymology[edit]
The word polynomial joins two diverse roots: the Greek poly, meaning "many", and the Latin nomen, or "name". It was derived from the term binomial by replacing the Latin root bi- with the Greek poly-. That is, it means a sum of many terms (many monomials). The word polynomial was first used in the 17th century.[1]
Definition[edit]
A polynomial expression is an expression that can be built from constants and symbols called variables or indeterminates by means of addition, multiplication and exponentiation to a non-negative integer power. The constants are generally numbers, but may be any expression that do not involve the indeterminates, and represent mathematical objects 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 commutativity, associativity and distributivity of addition and multiplication. For example and are two polynomial expressions that represent the same polynomial; so, one has the equality .
A polynomial in a single indeterminate x can always be written (or rewritten) in the form
where are constants that are called the coefficients of the polynomial, and is the indeterminate.[2] The word "indeterminate" means that represents no particular value, although any value may be substituted for it. The mapping that associates the result of this substitution to the substituted value is a function, called a polynomial function.
This can be expressed more concisely by using summation notation:
That is, a polynomial can either be zero or can be written as the sum of a finite number of non-zero terms. Each term consists of the product of a number – called the coefficient of the term[a] – and a finite number of indeterminates, raised to non-negative integer powers.
Operations[edit]
Addition and subtraction[edit]
Polynomials can be added using the associative law of addition (grouping all their terms together into a single sum), possibly followed by reordering (using the commutative law) and combining of like terms.[7][8] For example, if
and
then the sum
can be reordered and regrouped as
and then simplified to
When polynomials are added together, the result is another polynomial.[9]
Subtraction of polynomials is similar.
Multiplication[edit]
Polynomials can also be multiplied. To expand the product of two polynomials into a sum of terms, the distributive law is repeatedly applied, which results in each term of one polynomial being multiplied by every term of the other.[7] For example, if then Carrying out the multiplication in each term produces Combining similar terms yields which can be simplified to As in the example, the product of polynomials is always a polynomial.[9][4]
Composition[edit]
Given a polynomial of a single variable and another polynomial g of any number of variables, the composition is obtained by substituting each copy of the variable of the first polynomial by the second polynomial.[4] For example, if and then 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.[10]
Division[edit]
The division of one polynomial by another is not typically a polynomial. Instead, such ratios are a more general family of objects, called rational fractions, rational expressions, or rational functions, depending on context.[11] This is analogous to the fact that the ratio of two integers is a rational number, not necessarily an integer.[12][13] For example, the fraction 1/(x2 + 1) is not a polynomial, and it cannot be written as a finite sum of powers of the variable x.
For polynomials in one variable, there is a notion of Euclidean division of polynomials, generalizing the Euclidean division of integers.[e] This notion of the division a(x)/b(x) results in two polynomials, a quotient q(x) and a remainder r(x), such that a = b q + r and degree(r) < degree(b). The quotient and remainder may be computed by any of several algorithms, including polynomial long division and synthetic division.[14]
When the denominator b(x) is monic and linear, that is, b(x) = x − c for some constant c, then the polynomial remainder theorem asserts that the remainder of the division of a(x) by b(x) is the evaluation a(c).[13] In this case, the quotient may be computed by Ruffini's rule, a special case of synthetic division.[15]
Factoring[edit]
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. This factored form is unique up to the order of the factors and their multiplication by an invertible constant. In the case of the field of complex numbers, the irreducible factors are linear. Over the real numbers, they have the degree either one or two. Over the integers and the rational numbers the irreducible factors may have any degree.[16] For example, the factored form of
is
over the integers and the reals, and
over the complex numbers.
The computation of the factored form, called factorization is, in general, too difficult to be done by hand-written computation. However, efficient polynomial factorization algorithms are available in most computer algebra systems.
