2 Let mE R[x] be a polynomial with deg m > 1. Define a relation Sm on R[x] by the rule that (f,g) ES 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 mE R[x] such that the set {fe R[x]: f(2)= 3} is an equivalence class of Sm. Give a brief justification (one or two sentences).
2 Let mE R[x] be a polynomial with deg m > 1. Define a relation Sm on R[x] by the rule that (f,g) ES 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 mE R[x] such that the set {fe R[x]: f(2)= 3} is an equivalence class of Sm. Give a brief justification (one or two sentences).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![2 Let me R[x] be a polynomial with deg m≥ 1. Define a relation Sm on R[x] by the rule
that (f,g) ES 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 {fe R[x]: f(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%2Fa14fd66c-f167-4783-8c85-a5372189d527%2F3a27d6ce-c2a2-41ae-bad3-9815fb1927fc%2F34sdhaz_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2 Let me R[x] be a polynomial with deg m≥ 1. Define a relation Sm on R[x] by the rule
that (f,g) ES 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 {fe R[x]: f(2)= 3} is an
equivalence class of Sm. Give a brief justification (one or two sentences).
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