Solve the equation $x^{5} - 5 x^{4} + 9 x^{3} - 9 x^{2} + 5 x - 1 = 0$ .

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Solution

The above equation can be re-arranged as
$(x^{5} - 1) + 9 x^{2} (x - 1) + 5 x (1 - x^{3}) = 0$
$(x - 1) (x^{4} + x^{3} + x^{2} + x + 1) + 9 x^{2} (x - 1) - 5 x (x - 1) (x^{2} + x + 1) = 0$
$(x - 1) (x^{4} + x^{3} + x^{2} + x + 1 + 9 x^{2} - 5 x^{3} - 5 x^{2} - 5 x) = 0$
$(x - 1) (x^{4} - 4 x^{3} + 5 x^{2} - 4 x + 1) = 0$
Hence $x = 1$ is a solution.

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