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) multiplied by zero or more variables (which are usually represented by letters). Each variable may have an exponent which is a non-negative integer. The exponent on a variable in a term is equal to the degree of that variable in that 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, including fractions, irrational numbers, negative numbers, and complex numbers.For example,

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. In the example above, the degree is 2 + 1 = 3.

A polynomial is a sum of terms. 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. Here "− 5x" stands for "+ (−5)x", so the coefficient of the middle term is −5.

When a polynomial in one variable is arranged in the traditional order, the terms of higher degree come before the terms of lower degree. In the first term above, the coefficient is 3, the variable is x, and the exponent is 2. In the second term, the coefficient is –5. The third term is a constant. The degree of a non-zero polynomial is the largest degree of any one term. In the example, the polynomial has degree two.

Alternative forms

An expression that can be converted to polynomial form through a sequence of applications of the commutative, associative, and distributive laws is usually considered to be a polynomial. For instance (x + 1)3

is a polynomial because it can be worked out to x3 + 3x2 + 3x + 1. Similarly

is considered a valid term in a polynomial, even though it involves a division, because it is equivalent to x^3/12 and 1/12 is just a constant.The coefficient of this term is therefore 1/12.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 worked out to two terms, the complex number 2+3i is in fact just a single coefficient in this case that happens to require a "+" to be written down.

Division by an expression containing a variable is not generally allowed in polynomials.[1] For example,

is not a polynomial because it includes division by a variable. Similarly,

is not a polynomial, because it has a variable exponent.

Since subtraction can be treated as addition of the additive opposite, and since exponentiation to a constant positive whole number power can be treated as repeated multiplication, polynomials can be constructed from constants and variables with just the two operations addition and multiplication.

Polynomial functions

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

ƒ(x) = anxn + an−1xn−1 + ... + a2x2 + a1x + a0

for all arguments x, where n is a nonnegative integer and a0, a1,a2, ..., an are constant coefficients.
For example, the function f, 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

Polynomial functions are an important class of smooth functions.

Polynomial equations

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

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.

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.

Elementary properties of polynomials

1.A sum of polynomials is a polynomial
2.A product of polynomials is a polynomial
3.The derivative of a polynomial function is a polynomial function
4.Any primitive or antiderivative of a polynomial function is a polynomial function

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 parentheses. All polynomials with real or complex coefficients also have a factored form in which the polynomial is written as a product of linear polynomials. For example, the polynomial

is the expanded form of the polynomial

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.
In school algebra, students learn to move easily from one form to the other (see: factoring).

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. Evaluation is sometimes performed more efficiently 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 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, methods are given for solving a system of linear equations in several unknowns. To get a unique solution, the number of equations should equal the number of 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. This important subject is studied extensively in the area of mathematics known as linear algebra. 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]

Solving polynomial equations

Every polynomial corresponds to a polynomial function, where f(x) is set equal to the polynomial, and to a polynomial equation, where the polynomial is set equal to zero. The solutions to the equation are called the roots of the polynomial and they are the zeroes of the function and the x-intercepts of its graph. If x = a is a root of a polynomial, then (x - a) is a factor of that polynomial.Some polynomials, such as f(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 distinct root; this follows from the fundamental theorem of algebra.
There is a difference between approximating roots and finding exact roots. Formulas for the roots of polynomials up to a degree of 2 have been known since ancient times (see quadratic equation) and up to a degree of 4 since the 16th century (see Gerolamo Cardano, Niccolo Fontana Tartaglia). But formulas for degree 5 eluded researchers. In 1824, Niels Henrik Abel proved the striking result that there can be no general formula (involving only the arithmetical operations and radicals) for 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.
Numerically solving a polynomial equation in one unknown is easily done on computer by the Durand-Kerner method or by some other root-finding algorithm. The reduction of equations in several unknowns to equations each in one unknown 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 characterizations of the roots of arbitrary polynomials are generalizations of the methods previously discovered to solve the quintic equation.