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College Algebra Exam Review 35

College Algebra Exam Review 35 - this out Example

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1.8. POLYNOMIALS 45 algebraic systems that generalize the examples Q , R , and C ; we will give a careful definition of fields later.) Polynomials with coefficients in K are expressions of the form a n x n C a n 1 x n 1 C C a 0 , where the a i are elements in K . The set of all poly- nomials with coefficients in K is denoted by KOExŁ . Addition and multipli- cation of polynomials are defined according to the familiar rules: . X j a j x j / C . X j b j x j / D X j .a j C b j /x j ; and . X i a i x i /. X j b j x j / D X i X j .a i b j /x i C j D X k . X i;j W i C j D k a i b j /x k D X k . X i a i b k i /x k : I trust that few readers will be disturbed by the informality of defining polynomials as “an expression of the form ...” . However, it is possible to formalize the concept by defining a polynomial to be an infinite sequence of elements of K , with all but finitely many entries equal to zero (namely, the sequence of coefficients of the polynomial). Thus 7x 2 C 2x C 3 would be interpreted as the sequence .3; 2; 7; 0; 0; : : : / . The operations of addi- tion and multiplication of polynomials can be recast as operations on such functions. It is straightforward (but not especially enlightening) to carry
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Unformatted text preview: this out. Example 1.8.1. .2x 3 C 4x C 5/ C .5x 7 C 9x 3 C x 2 C 2x C 8/ D 5x 7 C 11x 3 C x 2 C 6x C 13; and .2x 3 C 4x C 5/.5x 7 C 9x 3 C x 2 C 2x C 8/ D 10x 10 C 20x 8 C 25x 7 C 18x 6 C 2x 5 C 40x 4 C 65x 3 C 13x 2 C 42x C 40: K can be regarded as subset of KŒxŁ , and the addition and multiplica-tion operations on KŒxŁ extend those on K ; that is, for any two elements in K , their sum and product as elements of K agree with their sum and product as elements of KŒxŁ . The operations of addition and multiplication of polynomials satisfy properties exactly analogous to those listed for the integers in Proposition 1.6.1 ; see Proposition 1.8.2 on the next page. All of these properties can be verified by straightforward computa-tions, using the definitions of the operations and the corresponding prop-erties of the operations in K ....
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