Finding Real and Complex Roots of a Polynomial Equation

**ft_fan** · May 6th 2009, 12:41 PM

Hi there,

I have to answer to following question, and am having issues:

Given that $2-i$ is a root of the polynomial equation

$z^6-5z^5+8z^4-2z^3-3z^2+3z-10=0$ ,

find all other roots.

I can see that $i$ is a root, and that because of conjugate pairs, $-2-i$ must be, and I understand that there must be 6 roots due to the degree of the polynomial, but I don't know how to find the other 3. Any advice or pointers would be hugely appreciated!

Thanks in advance!

**TheAbstractionist** · May 6th 2009, 01:01 PM

Originally Posted by ft_fan

I can see that $i$ is a root, and that because of conjugate pairs, $-2-i$ must be, and I understand that there must be 6 roots due to the degree of the polynomial, but I don't know how to find the other 3. Any advice or pointers would be hugely appreciated!

Hi ft_fan.

If $i$ is a root, then $-i$ must also be a root. So you know four of the six roots of the equation already. You should be able to find the remaining two roots by solving a quadratic equation.

**Plato** · May 6th 2009, 01:03 PM

Originally Posted by ft_fan

Hi there,

I have to answer to following question, and am having issues:

Given that $2-i$ is a root of the polynomial equation

$z^6-5z^5+8z^4-2z^3-3z^2+3z-10=0$ ,

find all other roots.

I can see that $i$ is a root, and that because of conjugate pairs, $\color{red}-2-i$ must be, and I understand that there must be 6 roots due to the degree of the polynomial, but I don't know how to find the other 3.

The conjugate of $2+i$ is $2\color{blue}-i$ not $\color{red}-2-i$ .

You know four roots $2+i,~2-i,~i,~-i$ .
So here are four factors $[z-(2+i)][z-(2-i)][z-i][z+i]$
Can you finish?

**ft_fan** · May 6th 2009, 04:58 PM

Ok, getting there, and thanks for picking up my mistake in typing above. I am still having a slight issue though;
I have taken the 4 roots,

$2-i$ , $2+i$ , $i$ and $-i$ ,

and multiplied them together as follows,

$(z-(2-i))(z-(2+i))(z-(i))(z-(-i))=0$ ,

giving the following polynomial,

$z^4-4z^3+6z^2-4z+5=0$ .

I have divided this by the original polynomial,

$z^6-5z^5+8z^4-2z^3-3z^2+3z-10=0$ ,

but each time I get the same thing,
$z^2+z+1$ remainder $-10z^2-z-5$ ,

which clearly isn't correct.

Am I doing something wrong in thr division of polynomials, or have I perhaps multiplied the roots together incorrectly? I've reached the point where I have retried the problem on so many occasions I cant work out what's going wrong.

Any help much appreciated!