Some times the simplest way is a division

Algebra Level 3

\[ \large { x }^{ 4 }-{ 6x }^{ 3 }+{ 7x }^{ 2 }-{ 6x }+1=0 \]

Note that \( \left(x + \frac1x\right)^2 = x^2 + \frac1{x^2} + 2 \) and \( \left(x + \frac1x\right)^3 = x^3 + \frac1{x^3} + 3\left(x+ \frac1x\right) \). With that in mind, find the sum of real values of \(x\) that satisfy the equation above.

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