What is the value of ` (2 - 2 z + z^(2)) ( 2 + 2 z - z^(2)) ` when ` z + (2) /(z) = 2` ?

Options

A. 0
B. 2
C. 4
D. 8

Answer

To solve the problem, we need to find the value of the expression \( (2 - 2z + z^2)(2 + 2z - z^2) \) given that \( z + \frac{2}{z} = 2 \).

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ z + \frac{2}{z} = 2 \]
To eliminate the fraction, multiply both sides by \( z \) (assuming \( z \neq 0 \)):
\[ z^2 + 2 = 2z \]

2. **Rearrange the equation**:
\[ z^2 - 2z + 2 = 0 \]
This is a quadratic equation in \( z \).

3. **Substitute the quadratic expression into the original expression**:
We need to evaluate:
\[ (2 - 2z + z^2)(2 + 2z - z^2) \]
Notice that \( z^2 - 2z + 2 = 0 \) implies:
\[ z^2 = 2z - 2 \]
Therefore, we can replace \( z^2 \) in the expression.

4. **Substituting \( z^2 \)**:
Replace \( z^2 \) in the first part:
\[ 2 - 2z + z^2 = 2 - 2z + (2z - 2) = 0 \]

5. **Now substitute back into the expression**:
The expression now simplifies to:
\[ 0 \cdot (2 + 2z - z^2) \]
Since anything multiplied by 0 is 0, we have:
\[ 0 \]

6. **Final result**:
Therefore, the value of the expression \( (2 - 2z + z^2)(2 + 2z - z^2) \) is:
\[ \boxed{0} \]