Prove that all the roots of the equation $(1+z)^6+z^6=0$ are collinear.

Question

Prove that all the roots of the equation $(1+z)^6+z^6=0$ are collinear.

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$(1+z)^6+z^6=0$

$\frac{1+z}{z}=(-1)^{1/6}=e^{i\pi/6},e^{i3\pi/6},e^{i5\pi/6},e^{i7\pi/6},e^{i9\pi/6},e^{11i\pi/6}$,but i do not know how to solve further.

Answer 1

Suppose $z$ satisfies $(1+z)^6 + z^6 = 0$. Then we have $$|1+z|^6 = |z|^6 \implies |z| = |z+1|$$ It follows that $$z\overline{z} = |z|^2 = |z+1|^2 = (z+1)(\overline{z} + 1) = z\overline{z} + 2 \text{Re}(z) + 1 \implies \text{Re}(z) = \frac{-1}{2}$$ Thus, all roots of the equation $(1+z)^6 + z^6$ lie on the line given by $\text{Re}(z) = -1/2$.