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2 Let m = R[x] be a polynomial with deg m≥ 1. Define a relation Sm on R[x] by the rule that (f,g) € S if and only if m is a factor of g - f. (a) Prove that Sm is an equivalence relation on R[x]. (b) The division rule for polynomials implies that every equivalence class of Sm con- tains one polynomial with a special property. What is this property? (c) Write down a polynomial m € R[x] such that the set {ƒ ≤ R[x] : ƒ(2) = 3} is an equivalence class of Sm. Give a brief justification (one or two sentences).

2 Let m = R[x] be a polynomial with deg m≥ 1. Define a relation Sm on R[x] by the rule that (f,g) € S if and only if m is a factor of g - f. (a) Prove that Sm is an equivalence relation on R[x]. (b) The division rule for polynomials implies that every equivalence class of Sm con- tains one polynomial with a special property. What is this property? (c) Write down a polynomial m € R[x] such that the set {ƒ ≤ R[x] : ƒ(2) = 3} is an equivalence class of Sm. Give a brief justification (one or two sentences).

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 22E: A relation R on a nonempty set A is called asymmetric if, for x and y in A, xRy implies yRx. Which...
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2 Let m = R[x] be a polynomial with deg m > 1. Define a relation Sm on R[x] by the rule that (f, g) € S if and only if m is a factor of g – f. - (a) Prove that Sm is an equivalence relation on R[x]. (b) The division rule for polynomials implies that every equivalence class of Sm con- tains one polynomial with a special property. What is this property? (c) Write down a polynomial m € R[x] such that the set {ƒ € R[x] : ƒ(2) = 3} is an equivalence class of Sm. Give a brief justification (one or two sentences).
Transcribed Image Text:2 Let m = R[x] be a polynomial with deg m > 1. Define a relation Sm on R[x] by the rule that (f, g) € S if and only if m is a factor of g – f. - (a) Prove that Sm is an equivalence relation on R[x]. (b) The division rule for polynomials implies that every equivalence class of Sm con- tains one polynomial with a special property. What is this property? (c) Write down a polynomial m € R[x] such that the set {ƒ € R[x] : ƒ(2) = 3} is an equivalence class of Sm. Give a brief justification (one or two sentences).
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