- #1
jisbon
- 476
- 30
- Homework Statement
- NIL
- Relevant Equations
- NIL
--Continued--
7)
Let
##\sum_{k=0}^9 x^k = 0##
Find smallest positive argument. Same thing as previous question, but I guess I can expand to
##z+z_{2}+z_{3}+...+z_{9}=0##
##z=re^{iθ}##
##re^{iθ}+re^{2iθ}+re^{3iθ}+...##
What do I do to proceed on?
Cheers
4)
Take ##3+7i## is a solution of ##3x^2+Ax+B=0##
Since ##3+7i## is a solution, I can only gather :
##(z−(3+7i))(...)=3x2+Ax+B##
Not sure on how to go from here.
EDIT: I got A =18 and B=174, is this correct?
I recognized that since there's a 3, this means the other root must be a conjugate, hence
##(z-(3+7i))(z-(3-7i))##
##(z-3)^2-(7i)^2 =0##
##z^2+6z+58=0##
##3z^2+18z+174=0##
6)
Suppose ##z=2e^{ikπ}##and
##z^{n}=2^5 e^{iπ/8}##
Find k such that z has smallest positive argument?
I don't understand this question :/ For z to have smallest positive principal argument, what does it entail/mean?
EDIT: Tried again. Got the following:
##z^{n}=2^n e^{inkπ} = 2^5 e^{iπ/8}##
## nk = 1/8##
##5k =1/8##
##k = 1/40##?
Take ##3+7i## is a solution of ##3x^2+Ax+B=0##
Since ##3+7i## is a solution, I can only gather :
##(z−(3+7i))(...)=3x2+Ax+B##
Not sure on how to go from here.
EDIT: I got A =18 and B=174, is this correct?
I recognized that since there's a 3, this means the other root must be a conjugate, hence
##(z-(3+7i))(z-(3-7i))##
##(z-3)^2-(7i)^2 =0##
##z^2+6z+58=0##
##3z^2+18z+174=0##
6)
Suppose ##z=2e^{ikπ}##and
##z^{n}=2^5 e^{iπ/8}##
Find k such that z has smallest positive argument?
I don't understand this question :/ For z to have smallest positive principal argument, what does it entail/mean?
EDIT: Tried again. Got the following:
##z^{n}=2^n e^{inkπ} = 2^5 e^{iπ/8}##
## nk = 1/8##
##5k =1/8##
##k = 1/40##?
7)
Let
##\sum_{k=0}^9 x^k = 0##
Find smallest positive argument. Same thing as previous question, but I guess I can expand to
##z+z_{2}+z_{3}+...+z_{9}=0##
##z=re^{iθ}##
##re^{iθ}+re^{2iθ}+re^{3iθ}+...##
What do I do to proceed on?
Cheers
Last edited: