Math Help - a complex numbers problem

1. a complex numbers problem

Show that all roots of the eq. (z + 1)^3 + z^3 = 0 lie on the line
x = -1/2
thank you.

2. Originally Posted by unicorn
Show that all roots of the eq. (z + 1)^3 + z^3 = 0 lie on the line
x = -1/2
thank you.
First observe that a cubic has at least one real root. So if all the roots
lie on $x=-1/2$ then $z=-1/2$ should be a real root.

This checks out, so we know that $z+1/2$ is a factor of the cubic,
and so we use long division to find the remaining quadratic factor:

$
(z + 1)^3 + z^3=2 (z+1/2)(z^2 + z + 1)=0
$

Now we can use that quadratic formula to verify that the two remaining
roots (which are the roots of $z^2 + z + 1$) have real parts $-1/2$
(which they do).

RonL

3. thanks