Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
2 Let m € R[x] be a polynomial with degm≥ 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 S 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).

2 Let m € R[x] be a polynomial with degm≥ 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 S 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).

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 28E
See similar textbooks
icon
Related questions
Question
100%
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 {f €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 {f €R[x] : ƒ(2) = 3} is an equivalence class of Sm. Give a brief justification (one or two sentences).
Expert Solution
steps

Step by step

Solved in 4 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Similar questions
Recommended textbooks for you
Elements Of Modern Algebra
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Algebra for College Students
Algebra for College Students
Algebra
ISBN:
9781285195780
Author:
Jerome E. Kaufmann, Karen L. Schwitters
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS