Explanation Given information: f ( x ) and g ( x ) be two polynomials over the field F , both of degree n or less, m > n , There exist m distinct elements c 1 , c 2 , c 3 , . . . , c m of F , such that f ( c i ) = g ( c i ) for i = 1 , 2 , . . . , m . Formula used: Zero, Root, Solution: Let f ( x ) = a 0 + a 1 x + . . . + a n x n ∈ F [ x ] . If c ∈ F is an element of F , such that f ( c ) = 0 , then c is called a zero of f ( x ) , then we say that c is root of the equation f ( x ) = 0 . Explanation: Let h ( x ) = f ( x ) − g ( x ) be a nonzero polynomial with deg h ( x ) ≤ n , m > n . There exist m distinct elements c 1 , c 2 , c 3 , . . . , c m of F , such that f ( c i ) = g ( c i ) for i = 1 , 2 , . . . , m . h ( x ) = f ( x ) − g ( x ) ⇒ h ( c i ) = f ( c i ) − g ( c i ) , i = 1 , 2 ,

8th Edition

ISBN: 9781285965918

Author: Gilbert

Publisher: Cengage

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Chapter 8.3, Problem 10E

To determine

To prove: If m > n and if there exist m distinct elements c1, c2, c3, . . ., cm of F, such that f(ci) = g(ci) for i = 1, 2, . . ., m, then f(x) = g(x) where f(x) and g(x) be two polynomials over the field F, both of degree n or less.

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Chapter 8.3, Problem 9E

Chapter 8.3, 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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To prove: If m > n and if there exist m distinct elements c 1 , c 2 , c 3 , . . . , c m of F , such that f ( c i ) = g ( c i ) for i = 1 , 2 , . . . , m , then f ( x ) = g ( x ) where f ( x ) and g ( x ) be two polynomials over the field F , both of degree n or less.

Solution Summary: The author demonstrates that m distinct elements exist over the field F.

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To prove: If m > n and if there exist m distinct elements c1, c2, c3, . . ., cm of F, such that f(ci) = g(ci) for i = 1, 2, . . ., m, then f(x) = g(x) where f(x) and g(x) be two polynomials over the field F, both of degree n or less.

Want to see the full answer?

Chapter 8 Solutions

To prove: If m &gt; n and if there exist m distinct elements c 1 , c 2 , c 3 , . . . , c m of F , such that f ( c i ) = g ( c i ) for i = 1 , 2 , . . . , m , then f ( x ) = g ( x ) where f ( x ) and g ( x ) be two polynomials over the field F , both of degree n or less.

Solution Summary: The author demonstrates that m distinct elements exist over the field F.

Videos

To prove: If m > n and if there exist m distinct elements c1, c2, c3, . . ., cm of F, such that f(ci) = g(ci) for i = 1, 2, . . ., m, then f(x) = g(x) where f(x) and g(x) be two polynomials over the field F, both of degree n or less.

Want to see the full answer?

Chapter 8 Solutions

To prove: If m > n and if there exist m distinct elements c 1 , c 2 , c 3 , . . . , c m of F , such that f ( c i ) = g ( c i ) for i = 1 , 2 , . . . , m , then f ( x ) = g ( x ) where f ( x ) and g ( x ) be two polynomials over the field F , both of degree n or less.