Complex Numbers: Roots of an EQN

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June 2nd 2011, 06:54 PM
Blizzardy

Complex Numbers: Roots of an EQN

Hi guys, need help with this question:

Show that the roots of the EQN z^5 - (z - i)^5 = 0, z is not equal to i, are
(1/2)[cot(k(pi)/5) + i] where k = 1,2,3,4.

Thanks in advance!
June 2nd 2011, 07:42 PM
TKHunny

Since z = i is not a solution, I'm puzzled by the question.
June 2nd 2011, 08:55 PM
Prove It

Quote:

Originally Posted by Blizzardy

Hi guys, need help with this question:

Show that the roots of the EQN z^5 - (z - i)^5 = 0, z is not equal to i, are
(1/2)[cot(k(pi)/5) + i] where k = 1,2,3,4.

Thanks in advance!

Why not start by expanding $\displaystyle (z- i)^5$ and simplifying before trying to solve the equation?
June 3rd 2011, 02:06 AM
Ackbeet

Try this line of reasoning:

$z^{5}-(z-i)^{5}=0$

$z^{5}=(z-i)^{5}$

$\left(\frac{z-i}{z}\right)^{\!\!5}=1.$

Let

$w=\frac{z-i}{z}.$

Then $w^{5}=1,$ the fifth roots of unity (exactly). So I would try writing down the $w$ 's, and then solving $zw=z-i$ for $z$ once you've done that.

Make sense?

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