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5.3. Location of Complex Roots 183 (6cosB+ 1)12(cosf?+isin0) - 39. Sketch the Nyquist diagram for this polynomial (taking M = 6) ...
184 5. Approximation and Location of Zeros Verify that there are two possible values of r: r1 = (l/2)(1 + &), r2 = (l/2)(1 - ...
5.3. Location of Complex Roots 185 In general, for fixed k, a recursion of the type U,+l = UOV, + al&-1 + “. + akv,-k can be ...
186 5. Approximation and Location of Zeros 5.4 Problems Prove that, for complex k the polynomial t3 - 3t + k never has more tha ...
5.4. Problems 187 Prove that the roots of the cubic equation t3 - (a + b + c)t2 + (ab + bc + co - d2 - e2 - f2)t + (ad2 + be2 ...
188 5. Approximation and Location of Zeros Let 731 < 782 < ... < nk be a set of positive integers. Prove that the poly ...
Hints 189 Show that every zero z of the complex polynomial f(%) = %” + an-l%“-’ +. * * + aI% + 00 satisfies -b 5 Re z < a, ...
190 5. Approximation and Location of Zeros 3.6. (a) A standard way to isolate the difference of the coefficients is to multiply ...
Hints 191 4.13. What is the sign of the value of the polynomial at 0 and l? What about the discriminant? What about the product ...
192 5. Approximation and Location of Zeros 4.29. Multiply up the denominators and evaluate the two sides at x = ei. 4.31. If Re ...
6 Symmetric Functions of the Zeros 6.1 Interpreting the Coefficients of a Polynomial Does the polynomial t6+2t5+3t4 -4t3+5t2+6t+ ...
194 6. Symmetric Functions of the Zeros Let ZL, v, w be the zeros of the cubic polynomial 4t3 - 7t2 - 3t + 2. Determine a cubic ...
6.1. Interpreting the Coefficients of a Polynomial 195 (b) Prove that, if x, y, z are real, then a2 >3b and that z (and, by s ...
196 6. Symmetric Functions of the Zeros (b) Let (Y = a - b - c,P=b-c-u,y=c-a-b,S=a+b+c. Verify that a+P+y+6=0 (YP+CYY+a6+py+Pb+y ...
6.2. The Discriminant 197 (b) Let D(tl,h ,... , tn) be the square of the expression in (a). This is called the discriminant of t ...
198 6. Symmetric Functions of the Zeros 6.3 Sums of the Powers of the Roots While the determination of an arbitrary symmetric fu ...
6.3. Sums of the Powers of the Roots 199 (c) Assuming for the moment that (b) is valid, what must the coef- ficients ki be? Try ...
200 6. Symmetric Functions of the Zeros Another approach to obtaining the recursion relation for the pk is to make use of infini ...
6.4. Problems 201 n b(n,O) b(n, 1) b(n,2) b(n,3) b(n,J) b(n,5) b(n,6) 0 1 1 1 -1 2 1 -3 2 3 1 -6 11 -6 4 1 -10^35 -50^24 5 1 -15 ...
202 6. Symmetric Functions of the Zeros The product of two of the four roots of the quartic equation x4 - 18x3 + kx2 + 200x - ...
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