Question #41304

This question tests the 'Zero-Factor Cancellation Trap' and the omission of negative square root branches in complex equations.

When faced with $(z^5 - 1)^2 = z^4(z - 1)^2$, students typically take the square root to get $z^5 - 1 = \pm z^2(z - 1)$.

The primary trap occurs when they divide both sides by $(z - 1)$ to use the geometric series sum $z^4+z^3+z^2+z+1$, implicitly assuming $z \neq 1$ and thereby losing the valid root $z = 1$.

A secondary trap is forgetting the negative branch $-\sqrt{z^4} = -z^2$, which loses four additional roots.

The options are specifically calibrated to these omissions: 76 (missing $z=1$), 34 (missing the negative branch), and 30 (missing both).