help!!! complex number question! (1 Viewer)

moivre

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Can someone who is good at 4 unit please do this question for me?
I can't even start the question!!!~~~~~~~~~~
 

:: ck ::

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z^5 = -1
z<sub>k</sub>^5 = cis(pi + 2kpi)
z<sub>k</sub> = cis( (pi + 2kpi)/5 )

k = 0, z = cis (pi/5)
k = 1, z = cis 3pi/5
k = -1, z = cis (-pi/5)
k = 2, z = cis pi = -1
k = -2, z = cis(-3pi/5)

making a quadratic equation with roots cis pi/5, cis -pi/5

sum of roots = cis(pi/5) + cis(-pi/5) = 2cospi/5 [remember z + z conjugate =2Rez]
product of roots : cispi/5 * cis(-pi/5) = cis0 = 1
.'. quadratic equation roots cispi/5 , cis(-pi/5)
z^2 + (-sum of roots)z + (product of roots)
= z^2 -2zcospi/5 + 1

similarly for cis 3pi/5 and cis -3pi/5
sum of roots = 2cos3pi/5
product of roots = cis0 = 1
.'. quadratic equation roots cis 3pi/5 , cis(-3pi/5)
z^2 + (-sum of roots)z + (product of roots)
= z^2 -2zcos(3pi/5) + 1

.'. equation with roots cis(+- pi/5), cis(+- 3pi/5) and -1
= (z+1)(z^2-2zcos(3pi/5)+1)(z^2-2zcos(pi/5)+1), as required
 

SeDaTeD

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With a name like that I would have thought you would be an expert on complex numbers.
 

moivre

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haha...

fantastic!!! thanks very much!~~
 

:: ck ::

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no probs.. this should be in the other 4u thread tho
 

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