History[ edit ] Lodovico Ferrari is credited with the discovery of the solution to the quartic inbut since this solution, like all algebraic solutions of the quartic, requires the solution of a cubic to be found, it could not be published immediately.

Polynomial The graph of a polynomial function of degree 3 In mathematicsa polynomial is an expression consisting of variables or indeterminates and coefficientsthat involves only the operations of additionsubtractionmultiplicationand non-negative integer exponents.

Polynomials appear in a wide variety of areas of mathematics and science. For example, they are used to form polynomial equationswhich 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 varietiescentral concepts in algebra and algebraic geometry.

Etymology The word polynomial joins two diverse roots: It was derived from the term binomial by replacing the Latin root bi- with the Greek poly. The word polynomial was first used in the 17th century. When the polynomial is considered as an expression, x is a fixed symbol which does not have any value its value is "indeterminate".

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. It is a common convention to use uppercase letters for the indeterminates and the corresponding lowercase letters for the variables arguments 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 x. Normally, the name of the polynomial is P, not P x. However, if a denotes a number, a variable, another polynomial, or, more generally any expression, then P a denotes, by convention, the result of substituting x by a in P.

Thus, the polynomial P defines the function which is the polynomial function associated to P. Frequently, when using this function, one supposes that a is a number. However one may use it over any domain where addition and multiplication are defined any ring.

In particular, when a is the indeterminate x, then the image of x by this function is the polynomial P itself substituting x to x does not change anything. In other words, This equality allows writing "let P x 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 s of the indeterminate s do not appear at each occurrence of the polynomial. Definition A polynomial is an expression that can be built from constants and symbols called indeterminates or variables by means of additionmultiplication and exponentiation to a non-negative power.

Two such expressions that may be transformed, one to the other, by applying the usual properties of commutativityassociativity and distributivity of addition and multiplication are considered as defining the same polynomial. A polynomial in a single indeterminate x can always be written or rewritten in the form where are constants and is the indeterminate.

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 functioncalled 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 [2] —and a finite number of indeterminates, raised to nonnegative 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.

A term and a polynomial with no indeterminates are called, respectively, a constant term and a constant polynomial.

The degree of the zero polynomial, 0, which has no terms at all is generally treated as not defined but see below. Forming a sum of several terms produces a polynomial. For example, the following is a polynomial: It consists of three terms: Polynomials of small degree have been given specific names.

A polynomial of degree zero is a constant polynomial or simply a constant.The degree of a polynomial is the highest degree of its monomials (individual terms) with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it, and thus is a non-negative integer.

So the terms would be 12x^4, 5x^3, -2x^2 and finally the constant term as its degree is zero. So all put together the descending order would be 12x^4+5x^3- 2x^2 – The powers are in the descending order and hence we can say that the polynomial is in the descending order.

Otherwise (if the remainder polynomial degree is lower than the divisor degree), the division is completed. The terms sum obtained on the step 2 is the quotient polynomial. Let's consider division example: 3x 4 +5x 3 +2x+4 / x 2 +2x+1.

Use this huge collection of polynomial worksheets to find the degree of monomial, binomial, trinomial and polynomial. Solve this set of worksheets that deals with writing the degree of binomials.

Find the degree of each term and then compare them. The highest power is the degree of the binomial. Identify the degree and leading coefficient of polynomial functions.

Figure (credit: Jason Bay, Flickr) Suppose a certain species of bird thrives on a small island. Write a polynomial function of least degree in standard form. First, let's change the zeros to factors. The rational zeros of -1, -2, and 5 mean that our factors are as follows.

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Monomials and polynomials (Algebra 1, Factoring and polynomials) – Mathplanet