The roots of the equation $z^n = (z + 1)^n$ on the complex plane lie on the line
- $2x + 1 = 0$
- $2x - 1 = 0$
- $x + 1 = 0$
- $x - 1 = 0$
For z^n = (z + 1)^n, the magnitudes must be equal, so |z| = |z + 1|. This implies z is equidistant from 0 and -1 on the complex plane, which describes the perpendicular bisector of the segment connecting 0 and -1, specifically the line x = -1/2, or 2x + 1 = 0.
Divide both sides of the equation z to the power of n = (z + 1) to the power of n by (z + 1) to the power of n to get (z / (z + 1)) to the power of n = 1. This means the quantity z / (z + 1) must be an n-th root of unity. Let z / (z + 1) = e to the power of (i * 2 * pi * k / n), where k is an integer. Substituting z = x + iy and solving for z gives z = (z + 1) * e to the power of (i * theta), which rearranges to z (1 - e to the power of (i * theta)) = e to the power of (i * theta). Solving for z yields z = e to the power of (i * theta) / (1 - e to the power of (i * theta)). Using the half-angle identities for sine and cosine, this simplifies to z = -1/2 - (i/2) cot(theta/2). Since the real part of z is x, we have x = -1/2. This means all roots lie on the vertical line 2x + 1 = 0.