# Thread: Complex Numbers and Cubic Polynomial Roots

1. ## 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.

2. First note that $\left| {z^3 } \right| = \sqrt {16 + 48} = 8$. So $|z|=?$

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

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.

Can you double-check your polynomial ?

Should it be $p(z)=z^3-12z+8\sqrt{3}$ ?

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

4. 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!

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

1. Using polar representation, $a = 4 \cos \left( \frac{2 \pi}{9} \right)$.
2. Using standard trig identities, $\cos^3 (\theta) = \frac{3}{4} \cos (\theta) + \frac{1}{4} \cos (3 \theta)$.