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Number of real roots of an equation

Source: Finnish Mathematics Competition 2007, Final Round, Problem 2

November 14, 2011
algebrapolynomialalgebra unsolved

Problem Statement

Determine the number of real roots of the equation x8−x7+2x6−2x5+3x4−3x3+4x2−4x+52=0.x^8 - x^7 + 2x^6 - 2x^5 + 3x^4 - 3x^3 + 4x^2 - 4x +\frac{5}{2}= 0.
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