4.1%20Polynomial%20Arithmetic%20I

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

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.

Ask a homework question - tutors are online