z is a complex number satisfying `z^(4)+z^(3)+2z^(2)+z+1=0`, then `|z|` is equal to

Options

A. `1/2`
B. `3/4`
C. 1
D. none of these

Explanation

To solve the equation \( z^4 + z^3 + 2z^2 + z + 1 = 0 \) and find the modulus \( |z| \), we can follow these steps:

### Step 1: Factor the polynomial
We start with the polynomial:
\[ z^4 + z^3 + 2z^2 + z + 1 = 0 \]

We can rearrange it as:
\[ z^4 + z^3 + z^2 + z^2 + z + 1 = 0 \]
Now, we can group the terms:
\[ (z^4 + z^3 + z^2) + (z^2 + z + 1) = 0 \]
We can factor out \( z^2 \) from the first group:
\[ z^2(z^2 + z + 1) + (z^2 + z + 1) = 0 \]
Now, we can factor out \( z^2 + z + 1 \):
\[ (z^2 + z + 1)(z^2 + 1) = 0 \]

### Step 2: Solve each factor
Now we have two factors to solve:
1. \( z^2 + z + 1 = 0 \)
2. \( z^2 + 1 = 0 \)

#### For \( z^2 + 1 = 0 \):
\[ z^2 = -1 \implies z = i \quad \text{or} \quad z = -i \]

#### For \( z^2 + z + 1 = 0 \):
Using the quadratic formula:
\[ z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = 1, c = 1 \):
\[ z = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{-1 \pm \sqrt{1 - 4}}{2} = \frac{-1 \pm \sqrt{-3}}{2} \]
This simplifies to:
\[ z = \frac{-1 \pm i\sqrt{3}}{2} \]

### Step 3: Calculate the modulus for each solution
#### For \( z = i \) and \( z = -i \):
The modulus is:
\[ |z| = |i| = 1 \quad \text{and} \quad |-i| = 1 \]

#### For \( z = \frac{-1 \pm i\sqrt{3}}{2} \):
The modulus is calculated as follows:
\[ |z| = \sqrt{\left(\frac{-1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2} = \sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1 \]

### Conclusion
In all cases, we find that:
\[ |z| = 1 \]

Thus, the final answer is:
\[ \boxed{1} \]