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2.2. Division of Polynomials 63 (b) Show that, if G is a graph with exactly n vertices and no connecting edges, then Cc(t) = tn. ...
64 2. Evaluation, Division, and Expansion (b) 2t3 + 9t2 + 8t - 4 and t2 + 5t + 6. E.21. Investigate formulae for the remainder w ...
2.3. The Derivative 65 Show that the first two terms in the expansion of a polynomial p(t) = a&” + a,-#-l + * * * + a1t + ...
66 2. Evaluation, Division, and Expansion (a) Verify that if u(t) = t 5-4t3+2t2-7, then u’(t) = 5t4-12t2+4t, u” t) = 20t3 - 24t ...
2.3. The Derivative 67 (c) Differentiate both sides of the equation in (b) repeatedly. Set t = c to obtain P(C) = co P’(C) = Cl ...
68 2. Evaluation, Division, and Expansion (a) Show that, for any number c and any nonzero polynomial p(t), we can find a nonnega ...
2.3. The Derivative 69 (ii) as a ring of polynomials in the variable y over F[z]. Corresponding to these, we can consider two ty ...
70 2. Evaluation, Division, and Expansion quantity xfz + yfY. What does this equal when deg f = l? deg f = 2? Make a conjecture ...
2.4. Graphing Polynomials 71 realized through the medium of limits). A second order differential equation satisfied by the real ...
72 2. Evaluation, Division, and Expansion Let f(z) be a polynomial defined on R. We say that f(x) is increasing on an interval [ ...
2.4. Graphing Polynomials 73 (e) Show that, if a quadratic polynomial has two real roots, then its derivative must have its root ...
74 2. Evaluation, Division, and Expansion Discuss the possible graphs of the general quartic ax4 + bx3 + cx2 + dx + e. You may ...
2.5. Problems 75 f’(x) has k real zeros, what can be said about the number of real zeros of f (x)3 Show that a real polynomial o ...
76 2. Evaluation, Division, and Expansion PI, P2, P3 are quadratic polynomials with positive leading coefficients and real zero ...
2.5. Problems 77 Let zcn) = z(z - 1)... (z - n + 1) for n a positive integer and let t(O) = 1. Prove that (x + y)(n) = 2 ( ; ) ...
78 2. Evaluation, Division, and Expansion Find all odd manic quintic polynomials over Z which have at least two integer zeros a ...
Hints 79 5.3. Put over a common denominator and check the numerator for a dou- ble zero at x = 1. (b) Set u = x - 1 and expand b ...
3 Factors and Zeros 3.1 Irreducible Polynomials 30 = 6.5; t3-6t+4 = (t-2)(t2+2t-2). Both equationsexpress an element as a produc ...
3.1. Irreducible Polynomials 81 factor indefinitely, and after a finite number of factorizations must arrive at factors which ar ...
(^82) 3. Factors and Zeros (a) Show that, if c is positive, t2 +c is irreducible over Z, Q and R, but reducible over C. (b) Supp ...
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