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3.5. Roots of Unity 103 The nth cyclotomic polynomials Q,,(t) is the product of all the linear polynomials (t - (‘) where C ran ...
104 3. Factors and Zeros (c) Let p be a prime and let C be a primitive pth root of unity. Define u = C([” : a is a quadratic non ...
3.5. Roots of Unity 105 (d) Explain how to construct Sicherman dice, two nonstandard cu- bical dice whose faces are assigned num ...
106 3. Factors and Zeros (d) Suppose n is the product of powers of distinct primes p. Show that 4(n)=nn(l-i). Pb (e) n = Cdl” 4( ...
3.6. Rational Functions 107 (v) ala2a3 .--ak^3 nk(ula2. ..ak) (mod m). Deduce that n4(*) 3 1 (mod m), and obtain the Little Ferm ...
108 3. Factors and Zeros in the form -- A B t-1 +t+4 for suitable constants A and B. (4 Let N/q(t) b e a rational function for ...
3.6. Rational Functions 109 For 2 < n 5 5, write out the partial fraction decomposition of l/(P - 1) over Q and C. What is t ...
110 3. Factors and Zeros Show, by dividing w(t) by v(t), that there are polynomials ai (1 < i 5 Ic), for which degai < deg ...
3.7. Problems on Factorization 111 Find a polynomial p(x) such that [P(x)]~ - x is divisible by (x - 1) (x - 2)(x - 3). Find va ...
112 3. Factors and Zeros (a) Determine a necessary and sufficient condition on the integers b and c such that both the polynomi ...
3.8. Other Problems 113 3.8 Other Problems Suppose that x2 = yx - 1 and y2 = 1 - y. Show that x5 = 1 but that 2 # 1. Is correc ...
114 3. Factors and Zeros Determine the range of w(w + x)(w + y)(w + 2) where z, ‘y, z, w are real numbers which satisfy If 2 + ...
3.8. Other Problems 115 Let f, g, h be polynomials over R or C. Suppose that f = g”‘h, where g does not divide h. (a) Show tha ...
116 3. Factors and Zeros A regular polygon of seven sides is inscribed in a circle of unit radius. Prove that the length of a s ...
Hints 117 2.8. Let g = u + v and h be two polynomials with 0 5 degv < deg u. Show that gh cannot be homogeneous. Recall Exerc ...
118 3. Factors and Zeros 6.8. For example 1 1 -=t-1+ Bt3+Ct2+Dt+E t5 - 1 @+P+P+t+l for suitable A, B, C, D, E. 7.1. (c) Rearrang ...
Hints 119 7.10. Let u = r + s and v = rs. Express the quadratic factors in terms of u and v. 7.11. Let x = gn+l and try to facto ...
120 3. Factors and Zeros 8.12. Use Exercise 2.12 to factor 0 = -w7 - x7 - y7 - z7. 8.13. Use Exercise 2.12. Note that 2(~~+y~+2~ ...
Equations 4.1 Simultaneous Equations in Two or Three Unknowns A man standing on a railway bridge 32 meters long observes a train ...
122 4. Equations ~22 + bzy + c2z = 0. Suppose that neither equation is a constant multiple of the other. (a) Manipulate the firs ...
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