(Degree Rule) Let D be an integral domain and f(x), g(x) E D[x]. Prove that deg (f(x) · g(x)) = deg f(x) + deg g(x). Show, by ex- ample, that for commutative ring R it is possible that deg f(x)g(x) < deg f(x) + deg g(x), where f(x) and g(x) are nonzero elements in R[x].

(Degree Rule) Let D be an integral domain and f(x), g(x) E D[x]. Prove that deg (f(x) · g(x)) = deg f(x) + deg g(x). Show, by ex- ample, that for commutative ring R it is possible that deg f(x)g(x) < deg f(x) + deg g(x), where f(x) and g(x) are nonzero elements in R[x].

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

(Degree Rule) Let D be an integral domain and f(x), g(x) E D[x].
Prove that deg (f(x) · g(x)) = deg f(x) + deg g(x). Show, by ex-
ample, that for commutative ring R it is possible that deg f(x)g(x) <
deg f(x) + deg g(x), where f(x) and g(x) are nonzero elements in
R[x].

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