The solutions of the equation `z^4 + 4i z^3 - 6z^2 - 4iz -i =0 ` represent vertices of a convex polygon in the complex plane. The area of the polygon is :

Options

A. `2^(1//2)`
B. `2^(3//2)`
C. `2^(5//2)`
D. `2^(5//4)`

Answer

To solve the equation \( z^4 + 4iz^3 - 6z^2 - 4iz - i = 0 \) and find the area of the polygon represented by its roots in the complex plane, we can follow these steps:

### Step 1: Rewrite the equation
We start with the given polynomial:
\[ z^4 + 4iz^3 - 6z^2 - 4iz - i = 0 \]

### Step 2: Identify the transformation
We notice that the polynomial can be transformed by substituting \( z = w + i \) to simplify the expression. This substitution will help us eliminate the imaginary unit \( i \) from the constant term.

### Step 3: Substitute and simplify
Substituting \( z = w + i \):
\[ (w + i)^4 + 4i(w + i)^3 - 6(w + i)^2 - 4i(w + i) - i = 0 \]
Expanding this expression will yield a polynomial in \( w \).

### Step 4: Expand the polynomial
Using the binomial theorem, we expand each term:
1. \( (w + i)^4 = w^4 + 4w^3i + 6w^2(-1) + 4w(-i) + 1 \)
2. \( 4i(w + i)^3 = 4i(w^3 + 3w^2i - 3w - i) \)
3. \( -6(w + i)^2 = -6(w^2 + 2wi - 1) \)
4. \( -4i(w + i) = -4iw - 4i^2 = -4iw + 4 \)

Combining all these expansions will lead to a polynomial in \( w \).

### Step 5: Solve for roots
Once we have the polynomial in \( w \), we can solve for its roots. The roots \( w_1, w_2, w_3, w_4 \) can be found using numerical methods or factoring if possible.

### Step 6: Convert back to \( z \)
Convert the roots back to \( z \) using \( z = w + i \).

### Step 7: Identify the vertices of the polygon
The roots \( z_1, z_2, z_3, z_4 \) will represent the vertices of a convex polygon in the complex plane.

### Step 8: Calculate the area of the polygon
To find the area of the polygon formed by the vertices \( z_1, z_2, z_3, z_4 \), we can use the formula for the area of a polygon given its vertices:
\[ \text{Area} = \frac{1}{2} \left| \sum_{j=1}^{n} (x_j y_{j+1} - x_{j+1} y_j) \right| \]
where \( (x_{n+1}, y_{n+1}) = (x_1, y_1) \).

### Step 9: Final calculation
After calculating the area using the coordinates of the vertices, we will arrive at the final answer.