4. Express the greatest common divisor of the following pairs of polynomials as a combination of the given polynomials: i. f(x) = x³ + x² - 5x - 2 and g(x) = x4 - 2x3 -x +2 in Q[x] ii. f(x) = x³ + 2x +2 and g(x) = x4 + 3x³ + 4x + 2 in Z5 [x] iii. f(x) = x³ + x² + x + 1 and g(x) = x² + x² + 1 in Z₂[x]

Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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(3.2) #4i
4. Express the greatest common divisor of the following pairs of polynomials
PROP as a combination of the given polynomials:
i. f(x) = x³ +x² - 5x – 2 and g(x) = x² − 2x³ -x+2 in Q[x]
ii. f(x) = x³ + 2x + 2 and g(x) = x4 + 3x³ + 4x + 2 in Z5[x]
iii. f(x) = x³ + x² + x + 1 and g(x) = x² + x² + 1 in Z₂[x]
5. Factor each of the following polynomials completely into irreducible poly-
Transcribed Image Text:4. Express the greatest common divisor of the following pairs of polynomials PROP as a combination of the given polynomials: i. f(x) = x³ +x² - 5x – 2 and g(x) = x² − 2x³ -x+2 in Q[x] ii. f(x) = x³ + 2x + 2 and g(x) = x4 + 3x³ + 4x + 2 in Z5[x] iii. f(x) = x³ + x² + x + 1 and g(x) = x² + x² + 1 in Z₂[x] 5. Factor each of the following polynomials completely into irreducible poly-
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