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P has nonreal root: x^100+20x^99+198x^98+...+1

Source: VTRMC 2011 P7

May 14, 2021
algebrapolynomialPolynomialsroots

Problem Statement

Let P(x)=x100+20x99+198x98+a97x97+…+a1x+1P(x)=x^{100}+20x^{99}+198x^{98}+a_{97}x^{97}+\ldots+a_1x+1 be a polynomial where the ai (1≤i≤97)a_i~(1\le i\le97) are real numbers. Prove that the equation P(x)=0P(x)=0 has at least one nonreal root.
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