If `x^3 + y^3 + z^3 = 3(1 + xyz)`, P = y + z - x, Q = z + x - y and R = x + y - z, then what is the value of `P^3 + Q^3 + R^3- 3PQR`?

Options

A. 9
B. 8
C. 12
D. 6

Answer

To solve the problem, we need to find the value of \( P^3 + Q^3 + R^3 - 3PQR \) given the equation \( x^3 + y^3 + z^3 = 3(1 + xyz) \) and the definitions of \( P, Q, \) and \( R \).

### Step-by-Step Solution:

1. **Understanding the Given Equation**:
We start with the equation:
\[ x^3 + y^3 + z^3 = 3(1 + xyz) \]
This equation holds true if \( x, y, z \) are the roots of a cubic polynomial.

2. **Substituting Values**:
To simplify the calculations, we can set \( z = 0 \) and \( y = 0 \). This gives us:
\[ x^3 = 3(1 + 0) \implies x^3 = 3 \]
Thus, we have \( x = \sqrt[3]{3} \).

3. **Calculating \( P, Q, R \)**:
Now we can calculate \( P, Q, R \):
- \( P = y + z - x = 0 + 0 - \sqrt[3]{3} = -\sqrt[3]{3} \)
- \( Q = z + x - y = 0 + \sqrt[3]{3} - 0 = \sqrt[3]{3} \)
- \( R = x + y - z = \sqrt[3]{3} + 0 - 0 = \sqrt[3]{3} \)

4. **Calculating \( P^3, Q^3, R^3 \)**:
Now we compute \( P^3, Q^3, R^3 \):
- \( P^3 = (-\sqrt[3]{3})^3 = -3 \)
- \( Q^3 = (\sqrt[3]{3})^3 = 3 \)
- \( R^3 = (\sqrt[3]{3})^3 = 3 \)

5. **Calculating \( PQR \)**:
Next, we calculate \( PQR \):
\[ PQR = (-\sqrt[3]{3}) \cdot (\sqrt[3]{3}) \cdot (\sqrt[3]{3}) = -3 \]

6. **Putting it All Together**:
Now we substitute these values into the expression \( P^3 + Q^3 + R^3 - 3PQR \):
\[ P^3 + Q^3 + R^3 - 3PQR = -3 + 3 + 3 - 3(-3) \]
Simplifying this:
\[ = -3 + 3 + 3 + 9 = 12 \]

### Final Answer:
Thus, the value of \( P^3 + Q^3 + R^3 - 3PQR \) is \( 12 \).