The number of solutions of the equation `z^(3)+barz=0`, is

Options

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

Explanation

To solve the equation \( z^3 + \bar{z} = 0 \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ z^3 + \bar{z} = 0 \]
This can be rewritten as:
\[ z^3 = -\bar{z} \]

### Step 2: Take the modulus of both sides
Taking the modulus of both sides, we have:
\[ |z^3| = |-\bar{z}| \]
Since the modulus of a complex number is always non-negative, we can simplify this to:
\[ |z^3| = |\bar{z}| \]
Using the property that \( |\bar{z}| = |z| \), we can write:
\[ |z|^3 = |z| \]

### Step 3: Analyze the modulus equation
From the equation \( |z|^3 = |z| \), we can factor it as:
\[ |z|^3 - |z| = 0 \]
Factoring gives:
\[ |z|(|z|^2 - 1) = 0 \]
This leads to two cases:
1. \( |z| = 0 \)
2. \( |z|^2 - 1 = 0 \) which gives \( |z| = 1 \)

### Step 4: Solve for \( |z| = 0 \)
If \( |z| = 0 \), then:
\[ z = 0 \]
This is one solution.

### Step 5: Solve for \( |z| = 1 \)
If \( |z| = 1 \), we can express \( z \) in polar form as:
\[ z = e^{i\theta} \]
Substituting this into the original equation:
\[ (e^{i\theta})^3 + \bar{e^{i\theta}} = 0 \]
This simplifies to:
\[ e^{3i\theta} + e^{-i\theta} = 0 \]
Multiplying through by \( e^{i\theta} \) gives:
\[ e^{4i\theta} + 1 = 0 \]
Thus:
\[ e^{4i\theta} = -1 \]
This implies:
\[ 4\theta = (2n + 1)\pi \quad \text{for } n \in \mathbb{Z} \]
Solving for \( \theta \):
\[ \theta = \frac{(2n + 1)\pi}{4} \]
This gives us four distinct angles for \( n = 0, 1, 2, 3 \):
- \( \theta = \frac{\pi}{4} \)
- \( \theta = \frac{3\pi}{4} \)
- \( \theta = \frac{5\pi}{4} \)
- \( \theta = \frac{7\pi}{4} \)

### Step 6: Count the total solutions
From \( |z| = 0 \), we have 1 solution: \( z = 0 \).

From \( |z| = 1 \), we have 4 solutions corresponding to the angles above.

Thus, the total number of solutions is:
\[ 1 + 4 = 5 \]

### Final Answer
The number of solutions of the equation \( z^3 + \bar{z} = 0 \) is \( \boxed{5} \).

---