complex analysis: Find real m so z^3+(3+i)z^2-3z-(m+i) has

[N1CKY]

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

 

[pka]

Suppose that a is a real root of the polynomial, then
a³+(3+i)a²-3a-(m+i)=0.

But Im(a³+(3+i)a²-3a-(m+i))=0 or a²-1=0. That means that a=±1.

Using those two possible roots, you find the values of m.

 

[N1CKY]

1)

1^3 + (3+i)1 - 3(1) -m - i =0
m=1


2)

-1^3 + (3+i)1 - 3(-1) -m -i =0
m=5