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
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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all 3 proof.
![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)) < degf (x) + deg g (x).
(iii) If R is an integral domain, then
deg (f (x) g (x)) = deg f (x) + deg g (x).
%3D](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F86535937-7372-4fda-af49-5d593194d05a%2F20994b3b-896c-4b80-a695-b645b57f3d1a%2Fle3bkhd_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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)) < degf (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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