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2

Part of 2017 Moldova Team Selection Test

Problems(1)

Complex root of a polynomial [Moldova TST 2017, D1, P2]

Source: Moldova TST 2017, Day 1, Problem 2

3/6/2017
Let f(X)=anXn+an−1Xn−1+⋯+a1X+a0f(X)=a_{n}X^{n}+a_{n-1}X^{n-1}+\cdots +a_{1}X+a_{0}be a polynomial with real coefficients which satisfies an≥an−1≥⋯≥a1≥a0>0.a_{n}\geq a_{n-1}\geq \cdots \geq a_{1}\geq a_{0}>0.Prove that for every complex root zz of this polynomial, we have ∣z∣≤1|z|\leq 1.
algebrapolynomial