How Do You Find Roots of Complex Polynomial Equations?

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Homework Help Overview

The discussion revolves around finding the roots of a complex polynomial equation, specifically the equation z^4 + 7 - 24i = 0. Participants express varying levels of understanding and approaches to solving this problem.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss rewriting the equation in different forms, such as expressing it in exponential form or using polar coordinates. There are questions about the interpretation of complex numbers and how to handle the imaginary unit 'i' in the context of polynomial equations.

Discussion Status

Some participants have offered guidance on how to approach the problem, including suggestions to simplify the expression and consider the nature of complex numbers. Multiple interpretations of the problem are being explored, particularly regarding the use of different mathematical representations.

Contextual Notes

There is mention of potential confusion due to the presence of the imaginary unit 'i' and the complexity of handling roots in the context of complex numbers. Participants also reflect on their prior knowledge of similar polynomial equations.

meee
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find the four roots of the equation
z^4 + 7 -24i = 0

completely lost, some help please...
 
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ohhhh

z^4 = -7 + 24i...
= 25cis *** something
 
MANY WAYS TO DO THIS.you can write it as a exponent.
z^4 + 7 -24i = 0
z^4 = - 7 +24i

Z = Cos(2*n*pi+ theta) + i Sin(2*n*pi + theta) = exponent(i*(theta+2*n*pi))
n = 0, 1,2,3,
so
Z^4 = exponent(i*(theta+2*n*pi)*4)
= Cos((theta+2*n*pi)*4) + i Sin((theta+2*n*pi)*4) (De Moivre's theorem)now you can compare the coefficeints of real /imaginary parts to find theta.
YOU SHOULD GET A SET OF SOLUTIONS. 4 unique as you substitiute n=0,1,2,3. then they'll repeat for n>3).
 
Last edited:
I bet you know how to solve x^4 - 17 = 0. Why not do the same thing for your problem?
 
Ohhh yeah... thanksss! i think i mighta got it jolly... not sure coz i used cis not exponent

hurkyl... reali? because the 'i' made it confusing
 
Last edited:
hurkyl... reali? because the 'i' made it confusing
Yes, really! (7 - 24i) is just a number. Give it a name, like c, if it helps. and z^4 + (7-24i) = 0 (i.e. z^4 + c = 0) is just an ordinary polynomial equation.

The only difference (at least for this problem) is that it takes more work to simplify an expression involving a root.
 

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