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P_n(x) = x^{n+2}-2x+1, one root c_n in (0,1), lim_{n \to \infty}c_n

Source: Spanish Mathematical Olympiad 1987 P6

August 2, 2018
polynomialpolynomial equationreal rootlimitalgebra

Problem Statement

For all natural numbers nn, consider the polynomial Pn(x)=xn+2−2x+1P_n(x) = x^{n+2}-2x+1. (a) Show that the equation Pn(x)=0P_n(x)=0 has exactly one root cnc_n in the open interval (0,1)(0,1). (b) Find limn→∞cnlim_{n \to \infty}c_n.
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