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```zn - w = 0
ï (*)
Let w = r (cos ï¦ + i sin ï¦) .
1
ï¦
ï¦ï¶
ï¦
+ i sin ï· .
Consider z' = r n ï§ cos
ï¨
n
nï¸
ï
(z' ) n = r ï¨cos ï¦ + i sin ï¦ ï© = w
â´
z' is a solution of the equation (*) .
Let z0 , z1 , ... , zn-1 be the n th roots of 1.
The roots of z n - w = 0 are z' z0 , z' z1 , ... , z' zn-1 .
ï¨
(ï z' z k
â´
n
= (z' ) n ( z k ) n = w ï 1 = w )
The n th roots of w are
2 kï°
1 ï¦
i
i
n
n
r e ïe n
for k = 0, 1, 2, ... , n-1
ï¦ ï« 2 kï°
1
i
n
= r ïe n
1
= rn
ï¦ 2kï° + ï¦ ï¶
ï¦ 2kï° + ï¦ ï¶ ï¹
ï· + i sin ï§
ï·
ïªcos ï§ï¨
ï¨ n ï¸ ïºï»
n ï¸
ï«
Equation reducible to the form z n - w = 0 :
(z + 1)11 - (z - 1)11 = 0
e.g.
Solution :
________________________________________________________________________
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ï
(z + 1)11 - (z - 1)11 = _______________________________________________________
= _______________________________________________________
= _______________________________________________________
= _______________________________________________________
Example :
Factorize z2n + z2n-1 + ... + z + 1 into real quadratic factors.
P.M/Complex/p.18
```
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