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If re^{i theta} is a root of z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z + 1 = 0, where r > 0 and 0 <= theta < 2pi, then find the sum of all possible values of theta.

 Jul 23, 2022
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z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z^1 + 1 = 

(z^9-1)/(z-1) = 0. 

 

The roots will all have a magnitutude of 1 on the complex plane so r will always be 1. 

The 9th roots of unity(excluding 1) are:

 

e^{2pi/9i}

e^{4pi/9i}

....

e^{16pi/9i}

 

Adding up all of the values of theta gives us: 8pi. 

 Jul 23, 2022

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