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Solutions to Problems; Chapter 1 263 for n 2 3. This is a straightforward manipulative exercise for the reader, 9.16. Let z be t ...
264 Answers to Exercises and Solutions to Problems 9.20. Since (Z + Y)~ = 8xy and (z - Y)~ = 4xy, the answer is a. Answers to Ex ...
Answers to Exercises; Chapter 2 265 3.5. (4 (~1~2.. -Pn)‘= ~;~~PlP2-P~-Pn. (e) @q)(t) = Wq(t)r. D’ff^1 erentiating yields (poq)‘ ...
266 4.8. (c) Answers to Exercises and Solutions to Problems ~=~4-33~2+ =(x2-2)(x2-1) J. X Cd) J \ i \ \ \ \ \ y=x4-5x-6 / x ‘\ \ ...
Solutions to Problems; Chapter 2 267 (b) Let q(x) = 6x5 - 15 x4 - 10x3 + 30x2 = x2(6x3 - 15x2 - 10x + 30). The number of zeros o ...
268 Answers to Exercises and Solutions to Problems (b) Setting u = x - 1 in the expression yields uv2{-nu - [l - (1+ u)“]} = 2 ( ...
Solutions to Problems; Chapter 2 269 5.6. (r + l)Sr + ( ’ i ’ ) Srel +.. - + (r + 1)Si = k k ( ’ f ’ ) k’+l-j k=l j=l = k[(l + k ...
270 Answers to Exercises and Solutions to Problems and p(t) = t”/9 - rt2/6 + c. Substitute t = 0 into (1) and (3) to obtain 3c = ...
Solutions to Problems; Chapter 2 271 x = 0 yields g(0) = 1, g’(0) = 3, g”(0) = 12. Since degg = 2, use Taylor’s Theorem to obtai ...
272 Answers to Exercises and Solutions to Problems positive. Then, since (x + u)(z + w) - (y + v)” - r(x + u)(y + v) = x-‘(x + u ...
Answers to Exercises; Chapter 3 273 1.8. Assume that a > 0 and note that at2 + bt + c = u[(t - b/2u)2 - (1/4e)(b2 - 4ac)]. Ap ...
274 Answers to Exercises and Solutions to Problems Since plcc and p2rt)cg, p must divide exactly one of a0 and bo, say plac, p$b ...
Answers to Exercises; Chapter 3 275 Since 1 ( ) is divisible by p for 2 5 k 5 p - 1, the irreducibility follows from the Eisenst ...
276 Answers to Exercises and Solutions to Problems (h) t2(t + 1)2 + (t + 1) = (t3 + t2 + l)(t + 1). (i) (3t + l)(t3 - t2 - 1). ( ...
Answers to Exercises; Chapter 3 277 2.13. No. 2t2+t-1 = (2t-l)(t+l) is reducible over Z, but irreducible over Z2. However, if th ...
278 Answers to Exercises and Solutions to Problems 3.4. The values of the polynomial at -2, -1, 1, 2 are, respectively, 40, 21, ...
Answers to Exercises; Chapter 3 279 (b) 83349 = 35.73. For a solution, we require t F 51, 193 or 242 (mod 243) and t - 41,48, 90 ...
280 Answers to Exercises and Solutions to Problems k be the smallest positive exponent for which (<a)k = 1. It can be seen th ...
Answers to Exercises; Chapter 3 281 On the other hand, if-c is a primitive Pkth root of unity, then (-0” = -1, from which < k ...
282 Answers to Exercises and Solutions to Problems Consider the difference p(t) -- A = PM - w(t) q(t) t - al q(t) where q(t) = ( ...
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