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week11 - Fundamental Theorem of Algebra Original Notes...

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Fundamental Theorem of Algebra Original Notes adopted from November 20, 2001 (Week 11) © P. Rosenthal , MAT246Y1, University of Toronto, Department of Mathematics typed by A. Ku Ong a + bi + c + di = ( a + c) + (b + d) i Triangle Inequality For z 1 , z 2 C, |z 1 + z 2 | |z 1 | + |z 2 | ( Or | z 1 + z 2 +... z n | | z 1 | + | z 1 | + ... + |z n | Fundamental Theorem of Algebra : Every polynomial with complex coeffients other than constant polynomials has a complex proof. p(z) = a n z n + a n-1 z n-1 + a 1 z + a 0 , a j C, n N. a n 0. n degree of polynomial. Definition : A closed curve in the plane is a continuous function from [0, 2 π ] into C such its values at 0 and 2 π are the same. Eg. A Function φ (t)= f(t) + i g(t), f,g into functions mapping [0, 2 π ] into R & f(0) = f(2 π ), g(0) = g (2 π ). Eg. φ (t) = cost + sint , t (0, 2 π ) Winding number is 1. Eg. φ (t) = cos3t + isin3t , t [0, 2 π ] PICTURE Winding number is 3 Definition: If φ is a closed curve in C that doesn't go through (0,0), its winding number about (0,0) is the

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