Math Is Fun Forum

Hixy

Member

Registered: 2011-09-24

Posts: 15

Finding the roots in a complex function

Hello everybody

What kind of help I'm looking for: A hint to where it goes wrong in my solution. A step by step solution that I can look at to verify that my work is correct after I've used the hint. Or any other kind of help you can give me.

What kind of level I'm at: Go ahead with the hard explanations.

I have the following problem due tomorrow. I know the steps, but doing the actual computation seems tremendously hairy to me and afterwards I can't seem to get a desired result.

The problem:

We are given the complex function

The problem statement gives 2 of the 4 roots as

and

My current thoughts:
The third root is obviously the complex conjugate:

Now my job is to find the last root.

Method: Factorize the entire polynomium with 1 unknown root

and isolate. However, what becomes my problem is that the a's from the complex numbers won't go away, and then I have 2 unknowns.

My try at a solution:

It was used that

and

Where do I go from here? Or should I have taken an entirely different road?

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Finding the roots in a complex function

Hi;

Forget about the complex root. They have only given you enough to cause problems with an otherwise easy question.

Deflate out the 2 after checking it is a root.

Graph the cubic or guess at roots they are very easy here. Then reduce the order by deflating or dividing out the next real root until you come to a quadratic and then use the quadratic formula.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Finding the roots in a complex function

hi hixy,

Your approach is OK although I agree with bobbym (I find that's generally best !    )

If you equate coefficients of each power of z you get some fairly easy equations and can solve for a as well as R.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Hixy

Member

Registered: 2011-09-24

Posts: 15

Re: Finding the roots in a complex function

Thanks a lot you guys! Totally did the trick. Didn't think about the deflate approach. I'll post my results later in case you want to see what I came up with.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Finding the roots in a complex function

Hi;

Welcome to the forum!

Yes, post your results.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Hixy

Member

Registered: 2011-09-24

Posts: 15

Re: Finding the roots in a complex function

Since

is known to be a solution, by deflating the polynomial we get

.
By inspecting the 3rd degree polynomial, we see that another root is 2. This can also be found by graphing it and find the intersection with the x-axis.
Thus the problem is already solved. 2 is a real root with multiplicity 2 and a(1+i) and a(1-i) are the complex roots.

However, deflating the 3rd degree polynomial gives

.
Taking the quadratic of the 2nd degree polynomial we get

.
So the two complex roots are -1+i and -1-i. Therefore a=1.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Finding the roots in a complex function

hi Hixy,

Well done!  You've done it!   

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Finding the roots in a complex function

Hi Hixy;

Well done!

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.