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

Options

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

Explanation

To find the number of solutions of 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
Taking the modulus on both sides, we have:
\[ |z^3| = |-\bar{z}| \]
Since the modulus of a complex number is always non-negative, we can simplify this to:
\[ |z|^3 = |\bar{z}| \]
We know that \( |\bar{z}| = |z| \), so we can write:
\[ |z|^3 = |z| \]

### Step 3: Solve for modulus
Now, we can factor out \( |z| \) (assuming \( |z| \neq 0 \)):
\[ |z|^3 - |z| = 0 \]
This gives us:
\[ |z|(|z|^2 - 1) = 0 \]
From this equation, we have two cases:
1. \( |z| = 0 \)
2. \( |z|^2 - 1 = 0 \) which gives \( |z| = 1 \)

### Step 4: Analyze the cases
**Case 1**: \( |z| = 0 \) implies \( z = 0 \). This is one solution.

**Case 2**: \( |z| = 1 \) means \( z \) lies on the unit circle. We can express \( z \) as:
\[ z = e^{i\theta} \]
for some angle \( \theta \).

### Step 5: Substitute back into the original equation
Substituting \( z = e^{i\theta} \) 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:
\[ 4\theta = (2n + 1)\pi \quad \text{for } n \in \mathbb{Z} \]
So:
\[ \theta = \frac{(2n + 1)\pi}{4} \]

### Step 6: Find the distinct solutions
The values of \( n \) will give us distinct angles for \( \theta \):
- For \( n = 0 \): \( \theta = \frac{\pi}{4} \)
- For \( n = 1 \): \( \theta = \frac{3\pi}{4} \)
- For \( n = 2 \): \( \theta = \frac{5\pi}{4} \)
- For \( n = 3 \): \( \theta = \frac{7\pi}{4} \)

Thus, we have four distinct solutions on the unit circle corresponding to these angles.

### Step 7: Total solutions
Including the solution \( z = 0 \), we have a total of:
\[ 4 \text{ (from the unit circle)} + 1 \text{ (the zero solution)} = 5 \text{ solutions} \]

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

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