4. (i) Let f(x) = x² + 3x³ + 2x² +2 and g(x) = x² + 2x + 1 € Q[x]. Find the unique polynomials q(x), r(x) E Q[x] such that f(x) = g(x)g(x) +r(x), where either r(x) = 0 or 0≤ deg r(x) < deg g(x). (ii) Let f(x) = x¹ + [3] x³ + [2]x² + [2] and g(x) = x² + [2] + [1] € Zs [x]. Find g(x), r(x) = Z₁ [x] such that f(x)= q(x)g(x) +r(x), where either r(x) = 0 or 0 ≤ deg r(x) < deg g(x).
4. (i) Let f(x) = x² + 3x³ + 2x² +2 and g(x) = x² + 2x + 1 € Q[x]. Find the unique polynomials q(x), r(x) E Q[x] such that f(x) = g(x)g(x) +r(x), where either r(x) = 0 or 0≤ deg r(x) < deg g(x). (ii) Let f(x) = x¹ + [3] x³ + [2]x² + [2] and g(x) = x² + [2] + [1] € Zs [x]. Find g(x), r(x) = Z₁ [x] such that f(x)= q(x)g(x) +r(x), where either r(x) = 0 or 0 ≤ deg r(x) < deg g(x).
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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Solve all parts of this question ( 2 parts). Write step by step clearly !
![4. (i) Let f(x) = x² + 3x³ + 2x² + 2 and g(x) = x² + 2x + 1 € Q[x]. Find the unique polynomials q(x),
r(x) E Q[x] such that f(x) = q (x)g(x) +r(x), where either r(x) = 0 or 0 ≤ deg r(x) < deg g(x).
(ii) Let f(x) = x¹ + [3]x³+ [2]x² + [2] and g(x) = x² + [2]x+ [1] € Z5 [x]. Find q(x), r(x) € Z5 [2] such that
f(x) = g(x)g(x) +r(x), where either r(x) = 0 or 0≤ deg r(x) < deg g(x).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc218f049-0f8e-48eb-a22b-2f6089ff642f%2F96fe731b-cce5-47bc-962f-de8bb5f1b9c5%2F3n534b4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4. (i) Let f(x) = x² + 3x³ + 2x² + 2 and g(x) = x² + 2x + 1 € Q[x]. Find the unique polynomials q(x),
r(x) E Q[x] such that f(x) = q (x)g(x) +r(x), where either r(x) = 0 or 0 ≤ deg r(x) < deg g(x).
(ii) Let f(x) = x¹ + [3]x³+ [2]x² + [2] and g(x) = x² + [2]x+ [1] € Z5 [x]. Find q(x), r(x) € Z5 [2] such that
f(x) = g(x)g(x) +r(x), where either r(x) = 0 or 0≤ deg r(x) < deg g(x).
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