# Thread: complex conjugate roots

1. ## complex conjugate roots

I have the following problem which i hope someone can point me in the right direction of solving:

The equation X^4+40x+39 has 4 roots, if two of the roots are the complex conjugate roots 2+J3 and 2-J3 by a process of long divison and slving a quadratic equation find the other two roots.

The exaple I am given in the book i have gives you the real roots to start with but gives no example of an equation that gives you the imaginary roots. I am looking for somehelp with how to get started on this one.

Any help is apprecciated

2. Hello,

$\displaystyle 2 \pm 3j$ is a root.
Therefore the polynomial can be factored by $\displaystyle [x-(2+3j)][x-(2-3j)]=[(x-2)-3j][(x-2)+3j]$

We know that $\displaystyle (a-b)(a+b)=a^2-b^2$.

So the previous line equals to :

$\displaystyle =(x-2)^2-(3j)^2=x^2-4x+4-9\underbrace{j^2}_{-1}=x^2-4x+13$

Now, you can try the division process...

3. Originally Posted by ally79
I have the following problem which i hope someone can point me in the right direction of solving:

The equation X^4+40x+39 has 4 roots, if two of the roots are the complex conjugate roots 2+J3 and 2-J3 by a process of long divison and slving a quadratic equation find the other two roots.

The exaple I am given in the book i have gives you the real roots to start with but gives no example of an equation that gives you the imaginary roots. I am looking for somehelp with how to get started on this one.

Any help is apprecciated
Note: Only one of the roots needed to be given since all coefficients of the quartic are real and so the conjugate root theorem could be used to get the other.