# Complex numbers and roots

1. Jan 30, 2013

### Saitama

1. The problem statement, all variables and given/known data
Let z be a complex number satisfying the equation $z^3-(3+i)z+m+2i=0$, where mεR. Suppose the equation has a real root, then find the value of m.

2. Relevant equations

3. The attempt at a solution
The equation has one real root which means that the other two roots are complex and are conjugates of each other.
Let $\alpha, \beta, \gamma$ be the three roots. The coefficient of z^2 is zero.
Hence $\alpha+\beta+\gamma=0$. Let $\gamma$ be the real root and the other two are complex. The sum of the complex roots is zero. From here, i get $\gamma=0$.
Product of roots, $\alpha \beta \gamma=m+2i=0$. This gives me $m=-2i$ which is incorrect as m is a real number.

Any help is appreciated. Thanks!

2. Jan 30, 2013

### Staff: Mentor

Why? Their imaginary parts cancel, but the real parts do not have to.

Your approach is way too complicated. If the equation has a real root, there is a real z which satisfies the equation. What is the imaginary part of the left side? It has to be zero...
This allows to calculate z and m.

3. Jan 30, 2013

### Saitama

Do you mean I have to substitute z=x+iy and compare the real and imaginary parts?

4. Jan 30, 2013

### Staff: Mentor

What is the imaginary part of z^3 if z is real? What is the imaginary part of (3+i)z?
m is real, and the imaginary part of 2i is obvious.

5. Jan 30, 2013

### Saitama

Thanks a lot mfb!