Document Sample

Polynomials Polynomials A polynomial is an expression of finite length constructed from variables (also called indeterminates) and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. However, the division by a constant is allowed, because the multiplicative inverse of a non zero constant is also a constant. For example, x2 − x/4 + 7 is a polynomial, but x2 − 4/x + 7x3/2 is not, because its second term involves division by the variable x (4/x), and also because its third term contains an exponent that is not an integer (3/2). The term "polynomial" can also be used as an adjective, for quantities that can be expressed as a polynomial of some parameter, as in polynomial time, which is used in computational complexity theory. Polynomial comes from the Greek poly, "many" and medieval Latin binomium, "binomial".[1][2][3] The word was introduced in Latin by Franciscus Vieta. 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, a central concept in abstract algebra and algebraic geometry. Know More About :- Multiplication Identity Property Math.Edurite.com Page : 1/3 A polynomial is either zero or can be written as the sum of a finite number of non-zero terms. Each term consists of the product of a constant (called the coefficient of the term) and a finite number of variables (usually represented by letters), also called indeterminates, raised to whole number powers.[5] The exponent on a variable in a term is called the degree of that variable in that term; the degree of the term is the sum of the degrees of the variables in that term, and the degree of a polynomial is the largest degree of any one 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 (nonzero) constant term is 0. The coefficient of a term may be any number from a specified set. If that set is the set of real numbers, we speak of "polynomials over the reals". Other common kinds of polynomials are polynomials with integer coefficients, polynomials with complex coefficients, and polynomials with coefficients that are integers modulo of some prime number p. In most of the examples in this section, the coefficients are integers. In general any expression can be considered a polynomial if it is built from variables and constants using only addition, subtraction, multiplication, and raising expressions to whole number powers. Such an expression can always be rewritten as a sum of terms. For example, (x + 1)3 is a polynomial; its standard form is x3 + 3x2 + 3x + 1. However, in some situations, a more accurate terminology is needed in order to not be confusing. In such a case, one should say: (x + 1)3 is a polynomial expression which may be expanded or rewritten into the polynomial x3 + 3x2 + 3x + 1. Although different as expressions, these two expressions are equal in the ring of the polynomials in the indeterminate x with integer coefficients. As for the integers, two kinds of divisions are considered for the polynomials. The Euclidean division that generalizes the Euclidean division of the integers. It results in two polynomials, a quotient and a remainder that are characterized by the following property of the polynomials: given two polynomials a and b such that b ≠ 0, there exists a unique pair of polynomials, q, the quotient, and r, the remainder, such that a = b q + r and degree(r) < degree(b) (here the polynomial zero is supposed to have a negative degree). By hand as well as with a computer, this division can be computed by the polynomial long division algorithm. A formal quotient of polynomials, that is, an algebraic fraction where the numerator and denominator are polynomials, is called a "rational expression" or "rational fraction" and is not, in general, a polynomial. Division of a polynomial by a number, however, does yield another polynomial. Read More About :- Addition Column Math.Edurite.com Page : 2/3 Thank You Math.Edurite.Com

DOCUMENT INFO

Shared By:

Categories:

Tags:

Stats:

views: | 4 |

posted: | 8/23/2012 |

language: | |

pages: | 3 |

OTHER DOCS BY mathedutireteam

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.