Explanation Given that: f ( x ) , g ( x ) , h ( x ) ∈ F [ x ] , where f ( x ) and g ( x ) are relatively prime and f ( x ) and h ( x ) are relatively prime. Formula used: Relatively prime polynomial: Two non-zero polynomials f ( x ) and g ( x ) in F [ x ] are called relatively prime over field F if their greatest common divisor in F [ x ] is 1. Proof: Let f ( x ) , g ( x ) , h ( x ) ∈ F [ x ] , such that f ( x ) and g ( x ) are relatively prime. Let d ( x ) is a greatest common divisor of f ( x ) and g ( x ) h ( x ) To prove that f ( x ) and g ( x ) h ( x ) are relatively prime, we have to prove that d = 1 . Since, d ( x ) is a greatest common divisor of f ( x ) and g ( x ) h ( x ) , we have d ( x ) | f ( x ) and d ( x ) | g ( x ) h ( x ) Let p ( x ) is the greatest common divisor of d ( x ) and g ( x ) Then, p ( x ) | d ( x ) and p ( x ) | g ( x

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ISBN: 9781285463230

Author: Gilbert, Linda, Jimmie

Publisher: Cengage Learning,

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Chapter 8.3, Problem 25E

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To prove: f( x ) and g( x )h( x ) are relatively prime

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Chapter 8 Solutions

Elements Of Modern Algebra

Ch. 8.1 - True or False Label each of the following...Ch. 8.1 - Prob. 2TFE Ch. 8.1 - Prob. 3TFE Ch. 8.1 - Prob. 4TFE Ch. 8.1 - Prob. 5TFE Ch. 8.1 - Prob. 6TFE Ch. 8.1 - Prob. 7TFE Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E

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To prove: f ( x ) and g ( x ) h ( x ) are relatively prime

Solution Summary: The author explains that two non-zero polynomials f(x ), g, and h are relatively prime over field F.

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