# Complex polynomial- HELP

1. Aug 23, 2009

### Clef

1. The problem statement, all variables and given/known data

(b) Using your answer to part (a), write down all complex roots of the polynomial
z4 + iz3 - z2 - iz + 1:

2. Relevant equations

(a) Determine all of the complex solutions of
z5 - i = 0:
Write your answers in polar form with -pi < x < pi.

3. The attempt at a solution

I've worked out all of the complex solutions of part a to be :

0= 1e^(ipi/10)

1= 1e^(ipi/2)

2= 1e^(9pi/10)

3= 1e^ (-7pi/10)

4= 1e^(-3pi/10)

But I have no idea how to go about the second part, having only solved complex polynomials with Z not iZ. :S
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Aug 23, 2009

### HallsofIvy

Use the fact that $z^5- a^5= (z- a)(z^4+ az^3+ a^2z^2+ a^3z+ a^4)$
With a= i, using the facts that $i^2= -1$, $i^3= -i$, $i^4= 1$, and $i^5= i$, that becomes $z^5- i= (z- i)(z^4+ iz^3- z^2- iz^3+ 1$.

Do you see the point?

3. Aug 23, 2009

### Clef

okay, I understand how that is the expanded form.
So would the solutions for part b) just be the polar forms of part a) divided by (z-i)?

Which would in turn make it:

(z5-a5)/(z1-a3)

=

z4-a2

=

z4-1

??

4. Aug 23, 2009

### kuruman

The answer to part (b) would be easier to see if you convert the roots of the polynomial from complex exponentials to complex numbers in the form z = a + i b. Once you've done this, is there another way you can express the equation z5 - i = 0 ?

5. Aug 23, 2009

### terminator88

Are you by any chance taking calculus 2 at melbourne university? LoL