Find all complex solutions of $\displaystyle z^5=(z+1)^5.$ I guess it's possible to just substitute $\displaystyle a+bi,$ collect terms, and expand, but the solution intimated a quicker way. Anyone willing to crack it receives muchos kudos from me.
Find all complex solutions of $\displaystyle z^5=(z+1)^5.$ I guess it's possible to just substitute $\displaystyle a+bi,$ collect terms, and expand, but the solution intimated a quicker way. Anyone willing to crack it receives muchos kudos from me.
There may be a clever trick. I don't see it right off.
But this may save you some work.
$\displaystyle w^5-z^5=(w-z)(w^4+w^3z+w^2z^2+wz^3+z^4)$.
Letting $\displaystyle w=z+1$ the first factor is 1.
Then we get a polynomial in $\displaystyle z^4$.
There are four complex answers.
I'm not sure if you meant that you got this answer or you just know it's the answer and need to get to it.
If you want to know how to get to it, this may help:
$\displaystyle \displaystyle{z^5=(1+z)^5 \Leftrightarrow (1+z^{-1})^5=1}$
which means all the solutions are $\displaystyle 1+z^{-1}=w_k$ where the $\displaystyle w_k$
are the roots of unity of order 5. Equivalently, $\displaystyle z=\frac{1}{w_k - 1}$