The equation zⁿ + z + 1 = 0 doesn't have all roots on the unit circle in general. For different values of n, the roots have different moduli. For example, when n = 2: z² + z + 1 = 0 gives z = ω, ω² (cube roots of unity with |z| = 1). When n = 3: z³ + z + 1 = 0 has roots with varying moduli. The sum |z₁| + |z₂| + ... + |zₙ| depends on specific n and root positions. Answer cannot be determined without knowing n.