The value of `x+ y + z` is 2 and `xy + yz + zx ` is 11 , then find the value of `x^3 + y^3 + z^3 - 3xyz`.

Options

A. 72
B. 76
C. 74
D. -58

Answer

To solve the problem, we need to find the value of \( x^3 + y^3 + z^3 - 3xyz \) given that \( x + y + z = 2 \) and \( xy + yz + zx = 11 \).

### Step-by-step Solution:

1. **Identify the given values**:
- \( x + y + z = 2 \)
- \( xy + yz + zx = 11 \)

2. **Use the formula**:
We will use the identity:
\[ x^3 + y^3 + z^3 - 3xyz = (x + y + z) \left( (x + y + z)^2 - 3(xy + yz + zx) \right) \]

3. **Calculate \( (x + y + z)^2 \)**:
\[ (x + y + z)^2 = 2^2 = 4 \]

4. **Substitute into the formula**:
Now substitute \( x + y + z \) and \( xy + yz + zx \) into the formula:
\[ x^3 + y^3 + z^3 - 3xyz = (2) \left( 4 - 3 \times 11 \right) \]

5. **Calculate \( 3 \times 11 \)**:
\[ 3 \times 11 = 33 \]

6. **Substitute and simplify**:
\[ x^3 + y^3 + z^3 - 3xyz = 2 \left( 4 - 33 \right) = 2 \left( -29 \right) \]

7. **Final calculation**:
\[ 2 \times -29 = -58 \]

Thus, the value of \( x^3 + y^3 + z^3 - 3xyz \) is \(-58\).

### Final Answer:
\[ \boxed{-58} \]