The sum of the moduli of the roots of the equation $x^{4}-5x^{3}+6x^{2}-5x+1=0$ is:

Divide the equation x^4 - 5x^3 + 6x^2 - 5x + 1 = 0 by x^2 to get x^2 - 5x + 6 - 5(1/x) + 1/x^2 = 0. Grouping terms gives (x^2 + 1/x^2) - 5(x + 1/x) + 6 = 0. Using the substitution y = x + 1/x we have y^2 - 2 - 5y + 6 = 0 which simplifies to y^2 - 5y + 4 = 0. Factoring gives (y - 1)(y - 4) = 0 so y = 1 or y = 4. For y = 4 the roots are 2 +/- sqrt(3) and for y = 1 the equation x^2 - x + 1 = 0 gives complex roots (1 +/- i sqrt(3))/2 which have a modulus of 1 each. The sum of the moduli of all four roots is 2 + sqrt(3) + 2 - sqrt(3) + 1 + 1 = 6. The sum is 6.

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