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, and has a finite number of terms.

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.

The word polynomial joins two diverse roots: the Greek poly, meaning "many", and the Latin nomen, or "name".

It was derived from the term binomial by replacing the Latin root bi- with the Greek poly-.

The 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.

That is, a polynomial can either be zero or can be written as the sum of a finite number of non-zero terms.

The degree of the zero polynomial 0 (which has no terms at all) is generally treated as not defined (but see below).

Rather, the degree of the zero polynomial is either left explicitly undefined, or defined as negative (either −1 or −∞).

The commutative law of addition can be used to rearrange terms into any preferred order.

It is possible to further classify multivariate polynomials as bivariate, trivariate, and so on, according to the maximum number of indeterminates allowed.

A composition may be expanded to a sum of terms using the rules for multiplication and division of polynomials.

Calculating derivatives and integrals of polynomials is particularly simple, compared to other kinds of functions.

A non-constant polynomial function tends to infinity when the variable increases indefinitely (in absolute value).

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

For higher degrees, the Abel–Ruffini theorem asserts that there can not exist a general formula in radicals.

However, root-finding algorithms may be used to find numerical approximations of the roots of a polynomial expression of any degree.

If, however, the set of accepted solutions is expanded to the complex numbers, every non-constant polynomial has at least one root; this is the fundamental theorem of algebra.

In 1824, Niels Henrik Abel 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 Abel–Ruffini theorem).

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 proved that there cannot be any general algorithm for solving them, or even for deciding whether the set of solutions is empty (see Hilbert's tenth problem).

Some of the most famous problems that have been solved during the last fifty years are related to Diophantine equations, such as Fermat's Last Theorem.

Polynomials where indeterminates are substituted for some other mathematical objects are often considered, and sometimes have a special name.

The rational fractions include the Laurent polynomials, but do not limit denominators to powers of an indeterminate.

Formal power series are like polynomials, but allow infinitely many non-zero terms to occur, so that they do not have finite degree.

The map from R to R[x] sending r to itself considered as a constant polynomial is an injective ring homomorphism, by which R is viewed as a subring of R[x].

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 Factorization of polynomials).

For example, in computational complexity theory the phrase polynomial time means that the time it takes to complete an algorithm is bounded by a polynomial function of some variable, such as the size of the input.

Determining the roots of polynomials, or "solving algebraic equations", is among the oldest problems in mathematics.

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.