For each f (x), g(x), p(x) ∈ F [x], Determine whether f (x) ≡ g(x) (mod p(x)). (a) f (x) = x5 − 2x4 + 4x3 + x + 1 F = Q g(x) = 3x4 + 2x3 − 5x2 − 9 p(x) = x2 + 1 (b) f (x) = x4 + x2 + x + 1 F = Z2 g(x) = x4 + x3 + x2 + 1 p(x) = x2 + x

For each f (x), g(x), p(x) ∈ F [x], Determine whether f (x) ≡ g(x) (mod p(x)). (a) f (x) = x5 − 2x4 + 4x3 + x + 1 F = Q g(x) = 3x4 + 2x3 − 5x2 − 9 p(x) = x2 + 1 (b) f (x) = x4 + x2 + x + 1 F = Z2 g(x) = x4 + x3 + x2 + 1 p(x) = x2 + x

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter4: Polynomial And Rational Functions

Section: Chapter Questions

Problem 14T

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Question

For each f (x), g(x), p(x) ∈ F [x], Determine whether f (x) ≡ g(x) (mod p(x)).
(a) f (x) = x⁵ − 2x⁴ + 4x³ + x + 1 F = Q
g(x) = 3x⁴ + 2x³ − 5x² − 9 p(x) = x² + 1

(b) f (x) = x⁴ + x² + x + 1 F = Z₂
g(x) = x⁴ + x³ + x² + 1 p(x) = x² + x

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