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1. Let f(x) and g (x) be two non-zero polynomials in R[x], R being any ring. (i) If f (x) +g (x)# 0, then deg (f (x) + g (x)) < max (deg f (x), deg g (x)). (ii) If f (x) g (x) #0, then deg (f (x) g (x)) < deg f (x) + deg g (x). (iii) If R is an integral domain, then deg (f (x) g (x)) = deg f (x) + deg g (x). %3D

1. Let f(x) and g (x) be two non-zero polynomials in R[x], R being any ring. (i) If f (x) +g (x)# 0, then deg (f (x) + g (x)) < max (deg f (x), deg g (x)). (ii) If f (x) g (x) #0, then deg (f (x) g (x)) < deg f (x) + deg g (x). (iii) If R is an integral domain, then deg (f (x) g (x)) = deg f (x) + deg g (x). %3D

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.6: Algebraic Extensions Of A Field
Problem 7E
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1. Let f(x) and g (x) be two non-zero polynomials in R[x], R being any ring. (i) If f (x) +g (x) #0, then deg (f (x) + g (x)) < max (degf(x), deg g (x)). (ii) If f (x) g (x) # 0, then deg (f (x) g (x)) < deg f (x) + deg g (x). (iii) If R is an integral domain, then deg (f (x) g (x)) = deg f (x) + deg g (x). %3D
Transcribed Image Text:1. Let f(x) and g (x) be two non-zero polynomials in R[x], R being any ring. (i) If f (x) +g (x) #0, then deg (f (x) + g (x)) < max (degf(x), deg g (x)). (ii) If f (x) g (x) # 0, then deg (f (x) g (x)) < deg f (x) + deg g (x). (iii) If R is an integral domain, then deg (f (x) g (x)) = deg f (x) + deg g (x). %3D
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