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Answers to Exercises; Chapter (^5 323) ai - c < ai + c < 2ai (for n 1 2), we have that u:+~ - c < (a: - c)/2, so that u ...
324 Answers to Exercises and Solutions to Problems 2.7. Let r1, r2 ,... , rk be all the positive zeros of p(t), each taken as of ...
Answers to Exercises; Chapter 5 325 2.15. (a) Let q(a) = q(b) = k. Th en a and b are zeros of the polynomial q(t) - k whose deri ...
326 Answers to Exercises and Solutions to Problems 2.21. (a) The real zeros are -a, 4, approx. 3.087. (b) Zeros are -7, 2/3, -5/ ...
Answers to Exercises; Chapter 5 327 ]w/u] > 1, so that ]w] > u. A similar argument using (b) and-the fact that anvn 1 an-l ...
328 Answers to Exercises and Solutions to Problems On the other hand, suppose that Ibd - cl < 1 - d2 and lb + dl < II+ cl. ...
Answers to Exercises; Chapter (^5 329) a few sketches, we can see that D2 U 03 (the image of the left semi-circle) must wind aro ...
330 Answers to Exercises and Solutions to Problems that the system has a solution u + vi with u > 0. Hence the given equation ...
Solutions to Problems; Chapter 5 331 We have that f(a) = f(b) = 0 and f’(t) = (n+2)(a-b)t”+‘-(a”+2-bn+2). If n is even, f’(t) h ...
332 Answers to Exercises and Solutions to Problems 4.10. Let p(x) = k(x - ar)(x - a2) ... (z - a,). Then p’(al) = k(a2 - al)(a3 ...
Solutions to Problems; Chapter 5 333 We solve by successive approximation, by evaluating the right side at x = With the aid of ...
334 Answers to Exercises and Solutions to Problems Remark. These polynomials occur as the denominators of a sequence of converge ...
Solutions to Problems; Chapter (^5 335) The distance between the outer zeros of x3+px+q is equal to the distance between the out ...
336 Answers to Exercises and Solutions to Problems 4.17. Let f(x) denote the left side. We make use of Rolle’s Theorem, f’(x) = ...
Solutions to Problems; Chapter 5 337 We know the real zero does not leave until a = 2. When will the nonreal zero leave?) Let a ...
338 Answers to Exercises and Solutions to Problems where g(z) is a sum of the form C f zk with /Z 2 2. By an estimate similar to ...
Solutions to Problems; Chapter 5 339 4.22. Denote the left side by p(t). S ince p(t) is positive for large It 1, p(t) assumes it ...
340 Answers to Exercises and Solutions to Problems following: if f(t) = Ca,t’ and g(t) = CZ,+,t’ satisfy a,g(t) = Eof(t), then a ...
Solutions to Problems; Chapter (^5 341) 4.31. Let Rez > a. Then Iz + a,-11 > Re(z + a,+l) > a + Re(a,-1) = la,-21aP1 + ...
342 Answers to Exercises and Solutions to Problems Let c = min{lg(u)l, Ig(v)l, Ig(ri)l} and let k be a large positive integer su ...
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