Show that all roots of the eq. (z + 1)^3 + z^3 = 0 lie on the line

x = -1/2

thank you.

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- Apr 14th 2006, 06:29 PMunicorna 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. - Apr 15th 2006, 12:24 AMCaptainBlackQuote:

Originally 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 - Apr 16th 2006, 01:51 PMunicorn
thanks :)