Thread: Complex Numbers and Cubic Polynomial Roots

Hello everyone,

Let w be one of the three complex numbers having the property that
w^3 = -4 + 4*sqrt(3)i.

What is the |w|?

Also if a = 2Re(w) how can one show that a is a root of the polynomial:
p(z) = z^3 - 12z + 8.

Thanks for your time.

First note that . So

I do not follow the second part of the question.
Is it related to first part?

Originally Posted by JoAdams5000

Hello everyone,

Let w be one of the three complex numbers having the property that
w^3 = -4 + 4*sqrt(3)i.

What is the |w|?

Also if a = 2Re(w) how can one show that a is a root of the polynomial:
p(z) = z^3 - 12z + 8.

Thanks for your time.

Can you double-check your polynomial ?

Should it be ?

EDIT: no, it isn't, I used a 60 degree angle instead of a 30 degree one!

Thanks!

Part b is related to the first part, so

a= 2*Re(w) where w is the be one of the three complex numbers having the property that w^3 = -4 + 4*sqrt(3)i.
Then I must show that a is a root of the polynomial z^3 - 12*z + 8.

And the polynomial is p(z) = z^3 - 12z + 8

Thanks in advance.

Originally Posted by JoAdams5000

Thanks!

Part b is related to the first part, so

a= 2*Re(w) where w is the be one of the three complex numbers having the property that w^3 = -4 + 4*sqrt(3)i.
Then I must show that a is a root of the polynomial z^3 - 12*z + 8.

And the polynomial is p(z) = z^3 - 12z + 8

Thanks in advance.

Note that:

1. Using polar representation, .

2. Using standard trig identities, .