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
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 28E
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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).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8ff93d80-f747-4f15-803d-56788ff3d116%2F42c0d904-49cd-40bb-9b7b-8adf6c20fc79%2Fmlffpup_processed.jpeg&w=3840&q=75)
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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