Explanation Formula used: Degree: Let R be a commutative ring with unity, and let f ( x ) = a 0 x 0 + a 1 x 1 + a 2 x 2 + . . . + a n x n be a non-zero element of R [ x ] . Then, the degree of f ( x ) is the largest integer k , such that the coefficient of x k is not zero, and this coefficient a k is called the leading coefficient of f ( x ) . Explanation: Consider the statement, “The degree of the sum of any two polynomials f ( x ) and g ( x ) over R is always the sum of the degree of f ( x ) and g ( x ) ”. Let, f ( x ) = 3 + 5 x 2 and g ( x ) = 1 + x + x 3 . In f ( x ) , 2 is the largest integer, such that the coefficient of x 2 is not zero. ∴ deg ( f ( x ) ) = 2 . In g ( x ) , 3 is the largest integer, such that the coefficient of x 3 is not zero. ∴ deg ( g ( x ) ) = 3

8th Edition

ISBN: 9781285463230

Author: Gilbert, Linda, Jimmie

Publisher: Cengage Learning,

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Chapter 8.1, Problem 4TFE

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Whether the statement, “The degree of the sum of any two polynomials f(x) and g(x) over R is always the sum of the degree of f(x) and g(x)” is true or false.

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Chapter 8.1, Problem 3TFE

Chapter 8.1, Problem 5TFE

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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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Whether the statement, “The degree of the sum of any two polynomials f ( x ) and g ( x ) over R is always the sum of the degree of f ( x ) and g ( x ) ” is true or false.

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Whether the statement, “The degree of the sum of any two polynomials f(x) and g(x) over R is always the sum of the degree of f(x) and g(x)” is true or false.

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Chapter 8 Solutions