If z is a complex number and `iz^(3) + z^(2) - z + I = 0`, the value of |z| is

Options

A. 0
B. `sqrt(2)`
C. 1
D. `1 + sqrt(2)`

Answer

To solve the equation \( iz^3 + z^2 - z + i = 0 \) and find the value of \( |z| \), we can follow these steps:

### Step 1: Rearranging the equation
We start with the equation:
\[ iz^3 + z^2 - z + i = 0 \]
We can rearrange this as:
\[ iz^3 + z^2 - z = -i \]

### Step 2: Grouping terms
Next, we can group the terms involving \( z \):
\[ iz^3 + z^2 - z = -i \]

### Step 3: Factor out common terms
We can factor out \( z \) from the first two terms:
\[ z^2(iz + 1) - z = -i \]
Rearranging gives us:
\[ z^2(iz + 1) = z - i \]

### Step 4: Isolate \( z \)
We can rewrite the equation as:
\[ z^2(iz + 1) - z + i = 0 \]
This suggests that we can set \( z^2 = \frac{1}{i} \) or \( z^2 = -i \).

### Step 5: Solve for \( z \)
To find \( z \), we can express \( \frac{1}{i} \) as:
\[ \frac{1}{i} = -i \]
Thus, we have:
\[ z^2 = -i \]
Taking the square root gives us:
\[ z = \sqrt{-i} \]

### Step 6: Finding the modulus
To find \( |z| \), we can use the property that \( |z^2| = |z|^2 \). Since \( z^2 = -i \), we find:
\[ |z^2| = |-i| = 1 \]
Thus,
\[ |z|^2 = 1 \implies |z| = 1 \]

### Conclusion
The value of \( |z| \) is:
\[ \boxed{1} \]