# Complex Number equations from roots

1. Feb 3, 2012

### DmytriE

1. The problem statement, all variables and given/known data
Determine the only real values a, b, c, and d such that the equation:
z4+az3+bz2+cz+d = 0​
has both z1 and z2 as roots.

z1 = 3 + j
z2 = -5 + 5j

2. Relevant equations
z = x + yj.
z = |z|ej$\theta$

3. The attempt at a solution
I am not sure where to begin. I can convert between Cartesian coordinates and Euler's formula but don't know where to go from there.

Any help would be greatly appreciated!

2. Feb 3, 2012

### tiny-tim

Hi DmytriE!

Hint: if 3 + j is a root (of a real-coefficient equation), can you spot another root?

3. Feb 3, 2012

### DmytriE

Indeed! Now, I do. Thanks for the hint!

Last edited: Feb 3, 2012
4. Feb 3, 2012

### DmytriE

I have calculated the different combination of roots but how do I know which variable (a,b,c,d) the values go to? When I multiplied z1, z1*, z2, and z2* I got a value of 500. Would that go into d?

It seems logical when I compare it to a second degree polynomial...

5. Feb 3, 2012

### tiny-tim

Yes.

But why not do it all in one go? …

Find (z - z1)(z - z1*), then (z - z2)(z - z2*), then multiply them.

6. Feb 3, 2012

### DmytriE

Great! Thank you again. I keep forgetting that z is a variable so I was strictly looking for the number rather than the equation with the coefficients.