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not all the roots are real

Source: 5-th Taiwanese Mathematical Olympiad 1996

January 12, 2007
algebrapolynomialVietaalgebra proposed

Problem Statement

Show that for any real numbers a3,a4,...,a85a_{3},a_{4},...,a_{85}, not all the roots of the equation a85x85+a84x84+...+a3x3+3x2+2x+1=0a_{85}x^{85}+a_{84}x^{84}+...+a_{3}x^{3}+3x^{2}+2x+1=0 are real.
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