Explanation Formula Used: ( 1 ) Division Algorithm: If f ( x ) and g ( x ) are polynomials in F [ x ] , with f ( x ) a nonzero polynomial. There exist unique elements q ( x ) and r ( x ) in F [ x ] such that g ( x ) = f ( x ) q ( x ) + r ( x ) with either r ( x ) = 0 or deg r ( x ) < deg f ( x ) . The polynomial q ( x ) is called the quotient and r ( x ) is called the remainder in the division of g ( x ) by f ( x ) . The Euclidean Algorithm consists of repeated application of division algorithm to yield the following sequence, where r n ( x ) is the last nonzero remainder. Euclidean Algorithm: g ( x ) = f ( x ) q 0 ( x ) + r 1 ( x ) , deg r 1 ( x ) < deg f ( x ) f ( x ) = r 1 ( x ) q 1 ( x ) + r 2 ( x ) , deg r 2 ( x ) < deg r 1 ( x ) r 1 ( x ) = r 2 ( x ) q 2 ( x ) + r 3 ( x ) deg r 3 ( x ) < deg r 2 ( x ) ⋮ ⋮ r n − 2 ( x ) = r n − 1 ( x ) q n − 1 ( x ) + r n ( x ) deg r n ( x ) < deg r n − 1 ( x ) r n − 2 ( x ) = r n ( x ) q n ( x ) Suppose that a is the leading coefficient of the last nonzero remainder, r n ( x ) .Then a − 1 r n ( x ) is the greatest common divisor of f ( x ) and g ( x ) . ( 2 ) If f ( x ) = a x j + … , g ( x ) = c x k + … with a ≠ 0 , c ≠ 0 , and j ≤ k , then a x j + … c a − 1 x k − j c x k + … c a − 1 x k − j f ( x ) _ g ( x ) − c a − 1 x k − j f ( x ) Explanation: Consider the functions provided. f ( x ) = 3 x 2 + 2 and g ( x ) = 4 x 4 + 2 x 3 + 6 x 2 + 4 x + 5 Here a = 3 , c = 4 , j = 2 and k = 4 . Divide g ( x ) by f ( x )

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Author: Gilbert, Linda, Jimmie

Publisher: Cengage Learning,

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Chapter 8.2, Problem 10E

To determine

The greatest common divisor d(x) of f(x) and g(x) in Z7[x].

where f(x)=3x2+2 and g(x)=4x4+2x3+6x2+4x+5 in Z7[x].

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Chapter 8.2, Problem 9E

Chapter 8.2, Problem 11E

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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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The greatest common divisor d ( x ) of f ( x ) and g ( x ) in Z 7 [ x ] . where f ( x ) = 3 x 2 + 2 and g ( x ) = 4 x 4 + 2 x 3 + 6 x 2 + 4 x + 5 in Z 7 [ x ] .

Solution Summary: The author explains the Euclidean Algorithm, where f(x) is a nonzero polynomial.

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The greatest common divisor d(x) of f(x) and g(x) in Z7[x].

where f(x)=3x2+2 and g(x)=4x4+2x3+6x2+4x+5 in Z7[x].

Want to see the full answer?

Chapter 8 Solutions

The greatest common divisor d ( x ) of f ( x ) and g ( x ) in Z 7 [ x ] . where f ( x ) = 3 x 2 + 2 and g ( x ) = 4 x 4 + 2 x 3 + 6 x 2 + 4 x + 5 in Z 7 [ x ] .

Solution Summary: The author explains the Euclidean Algorithm, where f(x) is a nonzero polynomial.

Videos

The greatest common divisor d(x) of f(x) and g(x) in Z7[x]. where f(x)=3x2+2 and g(x)=4x4+2x3+6x2+4x+5 in Z7[x].

Want to see the full answer?

Chapter 8 Solutions

The greatest common divisor d(x) of f(x) and g(x) in Z7[x].

where f(x)=3x2+2 and g(x)=4x4+2x3+6x2+4x+5 in Z7[x].