An algebra problem by Prencess Putian

Algebra Level 3

Find the sum of all complex numbers z z such that z 4 + 1 z 4 − 1 \frac{z^{4}+1}{z^{4}-1} = i 3 \frac{i}{\sqrt{3}} .


The answer is 0.

Prencess Putian by Prencess Putian , 21, Philippines

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2 solutions

Gian Sanjaya
Aug 27, 2015

Notice that you can change the form of the equation to be:

a z 4 + b = 0 , a = 1 − i 3 , b = 1 + i 3 az^4+b=0, a = 1 - \frac{i}{\sqrt3} , b=1+\frac{i}{\sqrt3}

From Vieta's formula, the sum of all complex numbers z satisfying the above equation is 0. All values of z such that z^4=1 doen't satisfy the polynomial equation, so they won't be interfering anyway.

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thanks for giving your time answering my first problem :)

Prencess Putian - 5 years, 9 months ago

I can't believe it took me that long. (Like 30 minutes of scrambling to websites.)

But I think the sum of the roots of any complex fraction equates to zero.

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definitely.

Prencess Putian - 5 years, 9 months ago

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