Factorzing polynomials with complex coefficients

In summary: He is looking for help and guidance on how to approach this type of problem. Another person suggests using the quadratic formula and asks if Felix has tried it. Felix responds that he has, but the solutions he got are not what he needs. The other person then asks if Felix knows how to take the square root of a complex number. Felix admits that he did not know how to do this, but has since looked it up and is now able to handle complex numbers. In summary, Felix is seeking help with factoring a polynomial in z and has learned how to take the
  • #1
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.

[tex] z^2-(2i+4)z + 8i = (z-4)(z+2i) [/tex]
 
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  • #2
Why don't you show us how you tried doing the quadratic formula? Because it should work
 
  • #3
I can:

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

Which isn't what I need...

Do you get the solutions z = 4 and z = 2i?
 
Last edited:
  • #4
You did get those numbers. Do you know what the square root of 3-4i is?
 
  • #5
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
 

1. What are complex coefficients in a polynomial?

Complex coefficients in a polynomial are numbers that contain both real and imaginary parts. They are denoted by the letter i and are used to represent numbers that cannot be expressed as a real number. In a polynomial, they are typically found in the form of a+bi, where a and b are real numbers and i is the imaginary unit.

2. Why do we need to factorize polynomials with complex coefficients?

Factorizing polynomials with complex coefficients is important because it helps us simplify complex expressions and solve equations that involve complex numbers. It also allows us to find the roots of a polynomial, which are the values that make the polynomial equal to zero.

3. What is the process for factorizing polynomials with complex coefficients?

The process for factorizing polynomials with complex coefficients is similar to the process for factorizing polynomials with real coefficients. First, we look for common factors and use the distributive property to rearrange the terms. Then, we can use methods such as grouping, difference of squares, and perfect square trinomials to factorize the polynomial.

4. Can we use the quadratic formula to factorize polynomials with complex coefficients?

Yes, the quadratic formula can be used to factorize polynomials with complex coefficients. However, since complex numbers are involved, the solutions may be complex as well. This means that the polynomial may not be able to be fully factorized into real numbers.

5. Are there any special cases to consider when factorizing polynomials with complex coefficients?

Yes, there are a few special cases to consider when factorizing polynomials with complex coefficients. One is when the polynomial has repeated roots, meaning that a factor appears more than once. Another is when the polynomial has irrational roots, meaning that the roots cannot be expressed as a fraction of two integers. Lastly, we may also need to consider the degree of the polynomial and the number of terms it has when choosing a factorization method.

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