Polynomial with complex numbers

P

Physicsfreak-2000

New member
Joined
Sep 6, 2019
Messages
4
Hey guys,

I just started my physics major at college. I am taking a course in Calculus and I should hand in my second homework problems next week. The homework is on complex numbers and I finished almost everything but I can't figure out how to solve the last problem.

We are given a certian polynomial p(z) = z^16 + z^15 + z^14 +z^13.... and so on until z + 1. We should get 16 answers. I figured out I should multiply this whole polynomail with z - 1. I thought it would be an easier equation than: z^17 - z - 1. (I hope I haven't made a mistake there)

I can't figure out what the next steps are in order to solve this problem and I would appreciate some help :)
 
Physicsfreak-2000 said:
Hey guys,

I just started my physics major at college. I am taking a course in Calculus and I should hand in my second homework problems next week. The homework is on complex numbers and I finished almost everything but I can't figure out how to solve the last problem.

We are given a certian polynomial p(z) = z^16 + z^15 + z^14 +z^13.... and so on until z + 1. We should get 16 answers. I figured out I should multiply this whole polynomail with z - 1. I thought it would be an easier equation than: z^17 - z - 1. (I hope I haven't made a mistake there)

I can't figure out what the next steps are in order to solve this problem and I would appreciate some help :)
Click to expand...
You say ...... We should get 16 answers.

Answers for what? What did you want to "find"?

Do you see that p(z) is a geometric series?

What would be the "sum" of the series with 17 terms?
 
Physicsfreak-2000 said:
I just started my physics major at college. I am taking a course in Calculus and I should hand in my second homework problems next week. The homework is on complex numbers and I finished almost everything but I can't figure out how to solve the last problem. We are given a certian polynomial p(z) = z^16 + z^15 + z^14 +z^13.... and so on until z + 1. We should get 16 answers. I figured out I should multiply this whole polynomail with z - 1. I thought it would be an easier equation than: z^17 - z - 1. (I hope I haven't made a mistake there)
Click to expand...
It seems that your question evolves \(\displaystyle P(z) = \sum\limits_{k = 0}^{16} {{z^k}} = 0\).
If that is correct see this LINK
 
So you're trying to solve the equation z^16 + z^15 + z^14 +z^13 ... + z + 1 = 0. You multiplied the function p(z) by (z - 1), and you seem to be saying you got z^17 - z - 1. That isn't quite right. It's actually z^17 - 1. That is, the equation

(z - 1)(z^16 + z^15 + z^14 +z^13 ... + z + 1) = 0​

(which is true whenever the given equation is true) becomes

z^17 - 1 = 0​

So you are solving for the roots of z^17 = 1 other than the trivial real root, 1. Have you learned how to find roots using polar form of a complex number?
 
We haven't learned how to find the roots of a complex number using polar form, but I did some research on it and now I think I get it. Thanks for the help guys!
 
Physicsfreak-2000 said:
We haven't learned how to find the roots of a complex number using polar form, but I did some research on it and now I think I get it. Thanks for the help guys!
Click to expand...
Great! Please post your solution here to help other students with similar problems.
 
You must log in or register to reply here.
Top