Unit 1 Section 4 - 1.4. Ring of Polynomials We have seen...

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1.4. Ring of Polynomials We have seen from the past sections that the basic objects of Algebra are numbers. We have several types of numbers ranging from counting or natural, to integers, rational, irrational, real and complex numbers. A set of numbers may have an underlying algebraic structure depending on the operations defined on them. The world of mathematics will be small if only numbers are to be considered as its objects. At times, quantities have no definite values while there are occasions when a particular value is often used so that a symbol for it is found very useful. The first type of quantities is called variables , or those which are represented only by letters and whose values may be arbitrarily chosen depending on the situation. The second type is called a constant , or a quantity whose value is fixed and may not be changed during a particular discussion. Illustration 1. In the formula, F = ma, relating force to the product of mass and acceleration, the variables are m (mass), a (acceleration) and F (force). In the expression, ½ gt 2 , t is the variable while g is a constant referring to the value of gravity. Of course, ½ is also a constant. The use of symbols in representing quantities (or numbers) led to the notion that algebra is generalized arithmetic. Any combination of numbers and symbols related by the operations we have described in the earlier sections will be called an algebraic expression . Illustration 2 . 2x + 3y, (x 2 y + 2xy 2 ) – x/y + x , and π x 3 + 2 x are algebraic expressions. Any algebraic expression consisting of distinct parts separated by plus or minus signs is called an algebraic sum. Each distinct part, together with its sign, is called a term of the algebraic expression. An algebraic expression consisting of just one term is called a monomial ; a binomial , if it is composed of two terms; a trinomial if it has three terms; and in general, a multinomial if it has any number of terms. A particular term of an algebraic expression is composed of one or more factors. Each of the factors may be called the coefficient of the others. For example, in 3u 4 v , 3 is the coefficient of u 4 v , 3u 4 is the coefficient of v and 3v is the coefficient of u 4 . At times, we need to distinguish between numerical and literal (letter-symbol) coefficients. In the given example, 3 is the numerical coefficient while u 4 v is the literal coefficient. Terms that have the same literal coefficients are called similar terms . Time to think ! 1) Draw a Venn diagram describing the relationship among the sets of numbers enumerated above. 2) Define the operations that can be defined on each set above and the underlying algebraic structures.
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Unit 1. Algebra as the Study of Structures Section 4 page 2 In this section, we shall focus on a special type of algebraic expressions, namely the polynomials . A polynomial in a variable t is an algebraic expression involving only non- negative integral powers of t . For example, 2xt 2 + 3 y t
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This note was uploaded on 05/13/2011 for the course MATH 17 taught by Professor Dikopaalam during the Spring '11 term at University of the Philippines Los Baños.

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Unit 1 Section 4 - 1.4. Ring of Polynomials We have seen...

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