Consider f(x) = x³ + 2x² + 3, g(x) = x³ + 2x² + 4 in the polynomial ring F5[2]. Denote F125 = F5[x]/(f(x)) and R125 = F5[x]/(g(x)). 1. Since f(x) has degree 3, it is irreducible if and only if it has no roots. Verify this by computing the values of f on the 5 elements of F5. This shows that F125 is a field. f(0) = 3 . f(1) = 1 f(2)= 1 f(3) = 4 f(4) = 0 2. Find all the roots of g(x). Give your answer as a list of numbers separated by commas. Roots of g(x)= 2 Since there is at least one, g(z) is reducible and so R125 is not a field. 3. In F125, compute (2x² + x +1)(3x² + 2x + 2) = x^4+2x^3+4x^2+4x+2 with coefficients from F5. 4. 3. In R125, compute (2x² + x + 1)(3x² + 2x + 2) = less with coefficients from F5. 5. In F125, compute (2x² + x + 1)¬¹ ; coefficients from F5. giving your answer as polynomial of degree 2 or less giving your answer as polynomial of degree 2 or giving your answer as polynomial of degree 2 or less with

Consider f(x) = x³ + 2x² + 3, g(x) = x³ + 2x² + 4 in the polynomial ring F5[2]. Denote F125 = F5[x]/(f(x)) and R125 = F5[x]/(g(x)). 1. Since f(x) has degree 3, it is irreducible if and only if it has no roots. Verify this by computing the values of f on the 5 elements of F5. This shows that F125 is a field. f(0) = 3 . f(1) = 1 f(2)= 1 f(3) = 4 f(4) = 0 2. Find all the roots of g(x). Give your answer as a list of numbers separated by commas. Roots of g(x)= 2 Since there is at least one, g(z) is reducible and so R125 is not a field. 3. In F125, compute (2x² + x +1)(3x² + 2x + 2) = x^4+2x^3+4x^2+4x+2 with coefficients from F5. 4. 3. In R125, compute (2x² + x + 1)(3x² + 2x + 2) = less with coefficients from F5. 5. In F125, compute (2x² + x + 1)¬¹ ; coefficients from F5. giving your answer as polynomial of degree 2 or less giving your answer as polynomial of degree 2 or giving your answer as polynomial of degree 2 or less with

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter4: Polynomial And Rational Functions

Section4.3: Zeros Of Polynomials

Problem 3E

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