I know how to show that, z^(n+1) = (z-1)[z^n + z^(n-1) + ... + 1). But how do I use this to find the complex roots of z^4 + z^3 + z^2 + z + 1 = 0 ? Thanks for any help in advance.
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Originally Posted by PvtBillPilgrim I know how to show that, z^(n+1) - 1 = (z-1)[z^n + z^(n-1) + ... + 1). Mr F says: The correction is in red. But how do I use this to find the complex roots of z^4 + z^3 + z^2 + z + 1 = 0 ? Thanks for any help in advance. z^5 - 1 = 0 => z^5 = 1 => z = the fifth roots of 1. But z^5 - 1 = (z-1)(z^4 + z^3 + ... + 1). Therefore the fifth roots of 1 are solutions to (z-1)(z^4 + z^3 + ... + 1) = 0. Therefore the solutions to z^4 + z^3 + ... + 1 = 0 are the non-real fifth roots of 1.
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