Factorzing polynomials with complex coefficients

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Discussion Overview

The discussion revolves around the factorization of a polynomial with complex coefficients, specifically the polynomial z^2-(2i+4)z + 8i. Participants explore methods for solving the polynomial and express uncertainty regarding the application of the quadratic formula and the simplification of complex square roots.

Discussion Character

  • Homework-related
  • Mathematical reasoning

Main Points Raised

  • One participant expresses difficulty in factoring the polynomial and seeks guidance on techniques for doing so.
  • Another participant requests to see the application of the quadratic formula to understand the issue better.
  • A participant provides the quadratic formula application and simplifies it to a form involving the square root of a complex number, indicating it does not yield the expected solutions.
  • There is a question regarding the square root of the complex number 3-4i, which remains unresolved.
  • A participant acknowledges their lack of knowledge about taking the square root of a complex number but indicates they have since learned how to do so.

Areas of Agreement / Disagreement

The discussion remains unresolved, with participants expressing differing levels of understanding and knowledge about complex square roots and their implications for solving the polynomial.

Contextual Notes

Participants do not clarify the method for taking the square root of complex numbers, which may affect their ability to reach the expected solutions.

FelixHelix
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Not sure if this is the right place to post (but its related to a complex analysis questions)

I'm doing a past paper for my revision and am stuck at the first hurdle. I simply cannot factor this polynomial in z for the life of me. I've tried completing the square and the usual quadratic formula but do not get the answer as given.

I know what the answers are supposed to be so if anyone could help walk me through how you get there and if there is a technique of generally doing so I'd be most grateful.

z^2-(2i+4)z + 8i = (z-4)(z+2i)
 
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Why don't you show us how you tried doing the quadratic formula? Because it should work
 
I can:

z1,z2= \frac{(2i+4) \pm \sqrt{(2i+4)^2-(4)(1)(8i)}}{(2)(1)}
this simplifies to:
z1,z2 = (i+2) \pm \sqrt{3-4i}

Which isn't what I need...

Do you get the solutions z = 4 and z = 2i?
 
Last edited:
You did get those numbers. Do you know what the square root of 3-4i is?
 
Hi Shredder - No I didn't know how to take the sqrt of a complex number... but I do now. Thanks for pointing this out - I looked it up and am happy with dealing with these now. Your help is much appreciated!

Felix
 

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