# Polynomials and complex numbers

1. Apr 8, 2014

### ivan_x3000

1. The problem statement, all variables and given/known data
Suppose that u and v are real numbers for which u + iv has modulus 3. Express the imaginary part of (u + iv)^−3 in terms of a polynomial in v.

2. Relevant equations

3. The attempt at a solution
|u+iv|=3 then sort(u^2+i^2) = 3 then
u = 3 and v=0 or u=0 and v=3

(0+3i)^-3

i swear i am missing something with these equations we've only being doing a topic on roots but wow it's different what i've been seeing in class

Last edited: Apr 8, 2014
2. Apr 8, 2014

### Staff: Mentor

If |u + iv| = 3, then u + iv is an arbitrary point on a circle of radius 3, centered at the origin in the complex plane. What effect does subtracting 3 from u + iv have on this circle?

3. Apr 8, 2014

### jbunniii

Those are two solutions, but there are infinitely many more.

By the way, is there a typo? The imaginary part of $(u+iv) - 3$ is simply $v$, which is already a polynomial in $v$.

4. Apr 8, 2014

### Staff: Mentor

The way I'm interpreting the problem, u + iv - 3 doesn't represent just a single point.

5. Apr 8, 2014

### ivan_x3000

Yes definitely a type it was meant to be (u+iv)^3

6. Apr 8, 2014

### Staff: Mentor

In the OP you changed it to (u + iv)-3 and above you have (u + iv)3. Which is it?

Really, you need to be more careful. If we don't know what the problem is, we can't help you.

7. Apr 8, 2014

### ivan_x3000

Sorry it is (u+iv)^-3 my bad

8. Apr 8, 2014

### jbunniii

OK, as hint I would recommend starting with
$$(u+iv)(u-iv) = |u+iv|^2$$
Rearrange this to get a formula for $(u+iv)^{-1}$ and use what you know about the modulus.

9. Apr 8, 2014

### ivan_x3000

Thank you so much that is a great hint haha

10. Mar 27, 2015

### krnysus

So (u + iv)^-1 = (u-iv)/9 then how do i solve it?

11. Mar 27, 2015

### Ray Vickson

Can you really not see how to get $(u + i v)^{-3}$ if you know $(u + iv)^{-1}$?

12. Mar 27, 2015

### krnysus

(U + iv)^-3 = (u - iv)/9(u + iv)^2 ? I seriously have no clue how to express the imaginary part in terms of polynomial in v...

13. Mar 27, 2015

### jbunniii

$$(u+iv)^{-3} = [(u+iv)^{-1}]^3 = \left(\frac{u-iv}{9}\right)^3 = ???$$