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

Answered: Image /qna-images/answer/20994b3b-896c-4b80-a695-b645b57f3d1a.jpg

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) + 8 (x)) < max (deg f (x), deg g (x)). (ii) If f (x) g (x) #0, then deg (f (x) g (x)) < deg ƒ (x) + deg g (x). (iii) If R is an integral domain, then deg (f (x) g (x)) = deg f (x) + deg g (x). %3D

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) + 8 (x)) < max (deg f (x), deg g (x)). (ii) If f (x) g (x) #0, then deg (f (x) g (x)) < deg ƒ (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

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.4: Zeros Of A Polynomial

Problem 1E: 1. Find a monic polynomial of least degree over that has the given numbers as zeros, and a monic...

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