SOLUTION: Find the roots of z^6=1 for z complex. Show that the sum of the roots you found is zero.

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Question 128835: Find the roots of z^6=1 for z complex. Show that the sum of the roots you found is zero.
Answer by Edwin McCravy(20060)   (Show Source): You can put this solution on YOUR website!
Find the roots of z^6=1 for z complex. Show that the sum of the roots you found is zero. 



Get 0 on the right:



Write as the difference of two squares:



Factor as the difference of squares:



Write the first factor as the difference of two cubes
and the second factor as the sum of two cubes:



Factor each by the rule for factoring the difference
(and sum) of two cubes



Set each factor = 0

 yields root 

 yields root 

, by the quadratic formula

yields roots  and 

, by the quadratic formula

yields roots  and .

------------------------------

So the 6 roots are:

, , , , , 

Add them up:

 =

 = 

Edwin
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