Question

The number of all possible solutions of the equation \( z^3 + z = 0 \) is:

• A. \( 4 \)
• B. \( 5 \)
• C. \( 3 \)
• D. \( 6 \)

Hint:

When solving polynomial equations, first try to factor the equation and solve each factor individually.

Answer 1

The Correct Option is C

We are given the equation \( z^3 + z = 0 \).
Step 1: Factor the equation. \[ z^3 + z = 0 \implies z(z^2 + 1) = 0 \]
Step 2: Solve for \( z \).
The solutions to this equation are found by setting each factor equal to zero: \[ z = 0 \quad \text{or} \quad z^2 + 1 = 0 \] For \( z^2 + 1 = 0 \), we get: \[ z^2 = -1 \implies z = i \quad \text{or} \quad z = -i \] Thus, the solutions are \( z = 0, i, -i \).
Step 3: Conclusion.
Therefore, the number of possible solutions is 3.