Forest brown

        4th period

        Mr.Lathan

     

             In mathematics, a polynomial is an expression consisting of variables  and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. An example of a polynomial of a single indeterminate,, is, which is a quadratic polynomial. An example in three variables is .

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 and algebraic varieties, central concepts in algebra and algebraic geometry.

Etymology  

According to the Oxford English Dictionary, polynomial succeeded the term binomial, and was made simply by replacing the Latin root bi- with the Greek poly-, which comes from the Greek word for many. The word polynomial was first used in the 17th century.Notation and terminology,  

the x occurring in a polynomial is commonly called either a variable or an indeterminate. When the polynomial is considered for itself, x is a fixed symbol which does not have any value . It is thus more correct to call it an "indeterminate". However, when one considers the function 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, but this may be sometimes confusing and is not done in this article.

It is a common convention to use uppercase letters for the indeterminates and the corresponding lowercase letters for the variables  of the associated function.

It may be confusing that a polynomial P in the indeterminate X may appear in the formulas either as P or as P.

Normally, the name of the polynomial is P, not P. However, if a denotes a number, a variable, another polynomial, or, more generally any expression, then P denotes, by convention, the result of substituting X by a in P. For example, the polynomial P defines the function.This equality allows writing "let P be a polynomial" as 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 of the indeterminate do not appear at each occurrence of the polynomial.

Definition  

A polynomial in a single indeterminate can be written in the form

your y and x numbers, or more generally elements of a ring, and x is a symbol which is called an indeterminate or, for historical reasons, a variable. The symbol x does not represent any value, although the usual  laws valid for arithmetic operations also apply to it.

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—and a finite number of indeterminates, raised to nonegative integer powers. 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 one term with nonzero coefficient. Because, the degree of an indeterminate without a written exponent is one. A term and a polynomial with no indeterminates are called respectively a constant term and a constant polynomial; the degree of a constant term and of a nonzero constant polynomial is 0. The degree of the zero polynomial  is generally treated as not defined .

For example:

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:

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, quadratic polynomials and cubic polynomials. For higher degrees the specific names are not commonly used, although quartic polynomial  and quintic polynomial  are sometimes used. The names for the degrees may be applied to the polynomial or to its terms. For example, in the term 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 . These conventions are useful 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. In the case of polynomials in more than one indeterminate, a polynomial is called homogeneous of if all its non-zero terms have . The zero polynomial is homogeneous, and, as homogeneous polynomial, its degree is undefined. For example, is homogeneous of degree 5. For more details, see homogeneous polynomial.The commutative law 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 in the example above 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 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. a two-term polynomial is called a binomial, and a three-term polynomial is called a trinomial. The term "quadrinomial" is occasionally used for a four-term polynomial.

    In conclusion we can say that polynomials aren’t the most used thing in math but they are very useful and have help develop many great things.