Download END

Survey
yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts
no text concepts found
Transcript
zn - w = 0
 (*)
Let w = r (cos  + i sin ) .
1



+ i sin  .
Consider z' = r n  cos

n
n

(z' ) n = r cos  + i sin   = w
∴
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 .

( z' z k
∴

n
= (z' ) n ( z k ) n = w  1 = w )
The n th roots of w are
2 k
1 
i
i
n
n
r e e n
for k = 0, 1, 2, ... , n-1
  2 k
1
i
n
= r e n
1
= rn

 2k +  
 2k +   
 + i sin 

cos 
 n  
n 

Equation reducible to the form z n - w = 0 :
(z + 1)11 - (z - 1)11 = 0
e.g.
Solution :
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________

(z + 1)11 - (z - 1)11 = _______________________________________________________
= _______________________________________________________
= _______________________________________________________
= _______________________________________________________
Example :
Factorize z2n + z2n-1 + ... + z + 1 into real quadratic factors.
P.M/Complex/p.18
Related documents