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1.1. The Anatomy of a Polynomial of a Single Variable 3 Exercises State the degree, and the constant, linear and leading coeffi ...
4 1. Fundamentals (4) (1+ t2)-’ ( r 1 3+t-2t= t+7. Let p(t) = 3t - 4 and q(t) = 2t2 - 5t + 8. Verify that (a) (p + q)(t) = 2t2 ...
1.1. The Anatomy of a Polynomial of a Single Variable 5 (4 degtp + d 6 max(deg p, deg q). ( max(a, 6) is the larger of the two n ...
6 1. Fundamentals Show that, for any positive integer k, (1+ t)(1 + t2)(1 + t4>... (1+ t2y = 1 + t + t2 + t3 +. * * + t? C ...
1.1. The Anatomy of a Polynomial of a Single Variable 7 which the sum of the kth powers of the numbers of the two subsets are eq ...
8 1. Fundamentals can enlist the aid of rook polynomials to help us avoid omissions or repeats in our counting. Let an m x n (m ...
1.2. Quadratic Polynomials 9 (c) Let n be a positive integer and let al,... , a,, be n symbols. What is the coefficient oft’ in ...
10 1. Fundamentals (b) Show that if the discriminant of the polynomial does not vanish, then it has two zeros. (c) Let m and n d ...
1.2. Quadratic Polynomials 11 (b) Find a counterexample to (a) if the word “nonrational” is re- placed by “rational.” Show that ...
12 1. Fundamentals (Let the equation of a typical chord be y = mx + k, where k is a parameter. The midpoint of the chord is give ...
1.3. Complex Numbers 13 E.5. Polynomials, some of whose values are squares. The square integers 1 = 12, 25 = 52 and 49 = 72 are ...
14 1. Fundamentals The polar decomposition. Let r = 1z1,6’ = arg z, then z = r(cos B+isin 0). z=xtyi Through the introduction of ...
1.3. Complex Numbers 15 (c) %+w=F+?Tj (d) zUr=;iv (e) I = z (f) Re z = rcosf3 = f(z+F) < [zl (g) Im z = rsin0 = $(z - Z?) 5 1 ...
16 1. Fundamentals (a) z3 = 1 (b) z4 = 1 (c) 26 = 1 (d) zd= 1. Indicate the solutions of each equation on an Argand diagram. (a ...
1.4. Equations of Low Degree^17 Exploration E.6. Commuting Polynomials. Two polynomials are said to commute under composition if ...
18 1. Fundamentals (b) Determine by inspection a root of the polynomial equation t3 - 4t + 3 = 0, and use this information to fi ...
1.4. Equations of Low Degree 19 (b) Show that u3 and v3 are roots of the quadratic equation x2 + qx - p3/27 = 0. (c) Let D = 27q ...
20 1. Fundamentals On August 7, 1877 (?), Arthur Cayley (1821-1895) wrote to Rudolf Lipschitz (1832-1903) a letter containing t ...
1.4. Equations of Low Degree^21 (b) Show that the quartic polynomial in (a) can be written as the product of two factors (t” + u ...
22 1. Fundamentals (a) Verify that each of the following polynomials is a reciprocal poly- nomial: x3 + 4x2 + 4x + 1 3x6 - 7x5 + ...
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