How Do You Find Roots of Complex Polynomial Equations?

[meee]

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

completely lost, some help please...

 

[meee]

ohhhh

z^4 = -7 + 24i...
= 25cis *** something

 

[jollygood]

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).

 

[Hurkyl]

I bet you know how to solve x^4 - 17 = 0. Why not do the same thing for your problem?

 

[meee]

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

 

[Hurkyl]

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.