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Complex Analysis -Markushevich's book1 Prob 9.1, 9.2: Division Algorithm for Polynomials
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Загружено: 2026-01-21
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Описание:
Theory of Functions of a Complex Variable (Vol 1) - A.I. Markushevich
I.9: Elementary Entire Functions
35: Polynomials
36: The mapping w = Pn(z)
37: The Mapping w = (z - a)^n
Prob 9.1: Given two polynomials f(z) and g(z), prove that there exist uniquely defined polynomials q(z) and r(z) such that
f(z) = q(z)g(z) + r(z),
where the degree of r(z) is less than the degree of g(z).
Prob 9.2: Use the result of the preceding problem to prove that a necessary and sufficient condition for a polynomial f(z) to be divisible by z - a without a remainder is that f(a) = 0, i.e., that z = a be a zero of f(z).
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