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Beautiful polynomial

Source: Serbia and Montenegro TST 2004 Problem 3

January 30, 2006
algebrapolynomialalgebra unsolved

Problem Statement

Let P(x)P(x) be a polynomial of degree nn whose roots are i−1,i−2,⋅⋅⋅,i−ni-1, i-2,\cdot\cdot\cdot, i-n (where i2=−1i^2=-1), and let R(x)R(x) and S(x)S(x) be the polynomials with real coefficients such that P(x)=R(x)+iS(x)P(x)=R(x)+iS(x). Show that the polynomial RR has nn real roots. (R. Stanojevic)
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