The number of solutions of `z^3+ z =0` is (a) 2 (b) 3 (c) 4 (d) 5

Options

A. 4
B. 3
C. 5
D. 2

Answer

To solve the equation \( z^3 + z = 0 \), we can follow these steps:

### Step 1: Factor the equation
We start with the equation:
\[ z^3 + z = 0 \]
We can factor out \( z \):
\[ z(z^2 + 1) = 0 \]

### Step 2: Set each factor to zero
This gives us two factors to consider:
1. \( z = 0 \)
2. \( z^2 + 1 = 0 \)

### Step 3: Solve the first factor
From the first factor, we have:
\[ z = 0 \]
This gives us one solution.

### Step 4: Solve the second factor
Now, we solve the second factor:
\[ z^2 + 1 = 0 \]
Rearranging gives:
\[ z^2 = -1 \]
Taking the square root of both sides, we find:
\[ z = \pm i \]
This gives us two additional solutions: \( z = i \) and \( z = -i \).

### Step 5: Count the total number of solutions
Now, we can count the total number of solutions:
1. \( z = 0 \)
2. \( z = i \)
3. \( z = -i \)

Thus, we have a total of **3 solutions**.

### Conclusion
The number of solutions of the equation \( z^3 + z = 0 \) is **3**.

### Answer
The correct option is (b) 3.

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