# Adv Alegbra HW Solutions 142 - . (ii) Find the roots of f (...

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142 4.76 (i) Write x 15 1 as a product of irreducible polynomials in F 2 [ x ] . Solution. Absent. (ii) Find an irreducible quartic polynomial g ( x ) F 2 [ x ] , and use it to de f ne a primitive 15th root of unity ζ F 16 . Solution. Absent. (iii) Find a BCH code C over F 2 of length 15 having minimum distance d ( C ) 3. Solution. Absent. 4.77 Let C be the 2-error correcting Reed-Solomon [ 7 , 3 ] code over F 8 in Ex- ample 4.131. Decode the word y =[ ζ 3 ,ζ, 1 3 + ζ, 0 3 3 ] assuming that it has an error vector of weight 2. Solution. Use Table 4.1 in Example 4.127. The answer is c =[ ζ 3 ,ζ, 1 + ζ 3 3 + ζ, 0 3 , 1 ] . Exercises for Chapter 5 5.1 (i) Find the roots of f ( x ) = x 3 3 x + 1 . Solution. Here, q =− 3, r = 1, and D = 27 r 2 + 4 q 3 =− 81 < 0. Using Vi`ete s formula, we have t = 2 and 3 θ = 120 , so that the roots are 2 cos 40 , 2 cos 160 , 2 cos 280
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Unformatted text preview: . (ii) Find the roots of f ( x ) = x 3 9 x + 28 . Solution. 4 , 2 i 3. (iii) Find the roots of f ( x ) = x 3 24 x 2 24 x 25 . Solution. 17 , 1 2 i 3 2 . 5.2 (i) Find the roots of f ( x ) = x 3 15 x 4 using the cubic formula. Solution. g = 3 p 2 + 121 and h = 3 p 2 121 . (ii) Find the roots of f ( x ) using the trigonometric formula. Solution. 4 , 2 3. 5.3 Find the roots of f ( x ) = x 3 6 x + 4 . Solution. 2 , 1 3 . 5.4 Find the roots of x 4 15 x 2 20 x 6 . Solution. 3 , 1 , 2 6 ....
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## This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.

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