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Help with Complex Roots

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The resulting expression looks suspiciously similar to a binomial expansion with n = 5; try multiplying by z and then subtracting 1 from both sides, then go from there. Just remember to check your answers afterwards, since this multiplication will likely introduce one or more false solutions.
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z^(4) \- 5(z-1)\[z^(2) \- z + 1\] = z^(4) \- 5(z-1)\[z^(2) \- (z - 1)\] = z^(4)  \- 5z^(2)(z - 1) + 5(z - 1)^(2) = \[z^(2) \- 5(z-1)/2\]^(2) \- 5(z-1)^(2)/4,

and you can factor the last expression as a difference of squares.

Edit: Alternatively, if you'll accept roots in trigonometric form, you can rewrite the expression as \[(z-1)^(5) \+ 1\]/z.
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