# Routh-Hurwitz Theorem

#### Zaryab Saeed

Joined Feb 10, 2016
3
Hi, I have a problem in which, given a polynomial, I have to calculate the number of roots with positive, zero and/or negative real parts. Now I know that every sign change in the first column of the Routh array indicates the number of roots with a positive real part, but how do I find out the roots with negative and/or zero real parts? I've tried reading up on it, but couldn't find the answer to this anywhere. Any help would be appreciated. Thanks!

#### Papabravo

Joined Feb 24, 2006
12,684
1. You can graph the polynomial.
2. You can use synthetic division to search for x-axis intersections, then use Newton's method to refine the result.
3. Roots with a zero real part are actually conjugate pairs.
4. Use the quadratic formula after finding one or more real roots.
5. Ask Matlab, Scilab, or Octave to give you the roots.

#### MrAl

Joined Jun 17, 2014
6,780
Hi,

Do you have to use R.H. or can you use any method you choose?

Also:

1. You can graph the polynomial.
2. You can use synthetic division to search for x-axis intersections, then use Newton's method to refine the result.
3. Roots with a zero real part are actually conjugate pairs.
4. Use the quadratic formula after finding one or more real roots.
5. Ask Matlab, Scilab, or Octave to give you the roots.
I think a better wording for #3 is: "Roots with an imaginary part appear as conjugate pairs".
They can have a non zero or zero real part.

#### Papabravo

Joined Feb 24, 2006
12,684
Hi,

Do you have to use R.H. or can you use any method you choose?

Also:

I think a better wording for #3 is: "Roots with an imaginary part appear as conjugate pairs".
They can have a non zero or zero real part.
Yes they can, but the answer was a response to the context of the original question.

#### MrAl

Joined Jun 17, 2014
6,780
Yes they can, but the answer was a response to the context of the original question.
Hi,

Yes, very good. I was being a bit more general with my reply.