Can any one help me with this:
Find all the REAL numbers m so that the equation
z^3 + (3+i)z^2 -3z - (m+i) = 0
has at least one REAL root.
NOTE: i = (-1)^1/2
The equation can be written as
$\displaystyle z^3+3z^2-3z-m+i(z^2-1)=0$
If z is a real root then
$\displaystyle \left\{\begin{array}{ll}z^3+3z^2-3z-m=0\\z^2-1=0\end{array}\right.$
From the second equation we have $\displaystyle z=\pm 1$.
Plugging z in the first equation we get $\displaystyle m=1, \ m=5$