- 257

- 23

- Homework Statement
- NIL

- Homework Equations
- NIL

--Continued--

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##?

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

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: