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1.4. Equations of Low Degree 23 (b) Show that, if f, g, h are polynomials with f = gh and f and h are both reciprocal polynomial ...
24 1. Fundamentals 1.5 Polynomials of Several Variables If x and y are the roots of a quadratic equation at2 + bt + c = 0, then ...
1.5. Polynomials of Several Variables 25 Exercises What are the degrees of the following polynomials? Are they homo- geneous? s ...
26 1. Fundamentals Let p(t) = at3 +- bt2 + ct + d be a cubic polynomial whose zeros are x, Y, z* (a) Show that p(t) = p(t) - p ...
1.5. Polynomials of Several Variables 27 A polynomial of several variables tl, t2,... , t, is a finite sum of mono- mials of th ...
28 1. Fundamentals Explorations ES. Suppose that f(x, y) is a function of the two real variables x and y. For each llxed, value ...
1.5. Polynomials of Several Variables 29 solutions. These formulae may be in the form of polynomials with inte- ger coefficients ...
30 1. Fundamentals is satisfied by (A, B, C, D) = (6,23,32, w, (16,87,122,149), (39,70,91,108), (51,148,203,246), (59,228,317,38 ...
1.6. Basic Number Theory and Modular Arithmetic 31 an integer c for which b = UC. Thus, for example, 371111 since 111 = 37 x 3. ...
32 1. Fundamentals number into the larger to get 1606 = 3.418 + 352. Explain why the greatest common divisor of 418 and 1606 is ...
1.6. Basic Number Theory and Modular Arithmetic 33 Explain where the numbers in the top row come from. Show that the numbers in ...
34 1. Fundamentals (a) Let two positive integers m and n be written out as a product of prime powers. Express the greatest comm ...
1.6. Basic Number Theory and Modular Arithmetic 35 Explorations E.ll. Define the length of the Euclidean algorithm as the number ...
36 1. Fundamentals We can answer Fibonacci’s question by computing each F,, in turn: 1, 1,2,3,5,8,13,21,34,55,89,144,233,377,610 ...
1.7. Rings and Fields 37 real, complex-which we wish to consider, it is convenient to define abstract structures which embody th ...
38 1. Fundamentals Exercises Show that the following are fields with the usual definitions of addi- tion and multiplication: ( ...
1.8. Problems on Quadratics 39 (b) Show that Z, is a commutative ring with identity. (c) Characterize those values of m for whic ...
40 1. Fundamentals Prove that, if the roots of x2 +px + q = 0 are real, then the roots of x2 + px + q + (x + u)(2z + p) = 0 wil ...
1.8. Problems on Quadratics 41 (a) Find necessary and sufficient conditions on a, b, w so that the roots of z2 + 2u~ + b = 0 an ...
42 1. Fundamentals Find the equations of those conjugate diameters of the ellipse b2x2 + U2Y2 = a2b2 which are of equal length. ...
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