4.1%20Polynomial%20Arithmetic%20I

# 4.1%20Polynomial%20Arithmetic%20I - Math 302 Modern Algebra...

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Math 302 Modern Algebra Spring 2009 Homework Solutions 4.1 POLYNOMIAL ARITHMETIC AND THE DIVISION ALGORITHM 1. Perform the indicated operations and simplify your answer. (a) (3 x 4 + 2 x 3 4 x 2 + x + 4) + (4 x 3 + x 2 + 4 x + 3) in Z 5 [ x ] (3 x 4 + 2 x 3 4 x 2 + x + 4)+(4 x 3 + x 2 + 4 x + 3) = 3 x 4 + (2 + 4) x 3 + ( 4 + 1) x 2 + (1 + 4) x + (4 + 3) = 3 x 4 + x 3 + 2 x 2 + 2 (b) ( x + 1) 3 in Z 3 [ x ] ( x + 1) 3 = x 3 + 3 x 2 + 3 x + 1 = x 3 + 1 (c) ( x 1) 5 in Z 5 [ x ] ( x 1) 5 = x 5 5 x 4 + 10 x 3 10 x 2 + 5 x 1 = x 5 1 (d) ( x 2 3 x + 2)(2 x 3 4 x + 1) in Z 7 [ x ] ( x 2 3 x + 2)(2 x 3 4 x + 1) = 2 x 5 4 x 3 + x 2 6 x 4 + 12 x 2 3 x + 4 x 3 8 x + 2 = 2 x 5 + x 4 + 6 x 2 + 3 x + 2 3. (a) List all polynomials of degree 3 in Z 2 [ x ]. A polynomial of degree 3 must have the coeﬃcient of x 3 equal to 1. Since there are three other coeﬃcients, each of which is either 0 or 1, there are 2 3 = 8 such polynomials. x
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## This note was uploaded on 10/20/2009 for the course MATH 302 taught by Professor Edwards during the Spring '03 term at CSU Fullerton.

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