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 rootsOriginally Posted by unicorn
lie on $\displaystyle x=-1/2$ then $\displaystyle z=-1/2$ should be a real root.
This checks out, so we know that $\displaystyle z+1/2$ is a factor of the cubic,
and so we use long division to find the remaining quadratic factor:
$\displaystyle
(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 $\displaystyle z^2 + z + 1$) have real parts $\displaystyle -1/2$
(which they do).
RonL