If `x=(beta -gamma)(alpha-delta)" "y=(gamma-alpha)(beta-delta), z=(alpha-beta)(gamma-delta)`, then the value of `x^(3)+y^(3)+z^(3)-3xyz` is

Options

A. 0
B. `alpha^(6) +beta^(6)+gamma^(6)+delta^(6)`
C. `alpha^(6)beta^(6)gamma^(6) delta^(6)`
D. none of these

Answer

To solve the problem, we need to find the value of \( x^3 + y^3 + z^3 - 3xyz \) given the definitions of \( x \), \( y \), and \( z \):

1. **Define the Variables**:
\[ x = (\beta - \gamma)(\alpha - \delta) \]
\[ y = (\gamma - \alpha)(\beta - \delta) \]
\[ z = (\alpha - \beta)(\gamma - \delta) \]

2. **Use the Identity**:
We can use the identity:
\[ x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx) \]
Thus, we need to calculate \( x + y + z \) and \( x^2 + y^2 + z^2 - xy - yz - zx \).

3. **Calculate \( x + y + z \)**:
\[ x + y + z = (\beta - \gamma)(\alpha - \delta) + (\gamma - \alpha)(\beta - \delta) + (\alpha - \beta)(\gamma - \delta) \]
Let's simplify this expression step by step.

- Expanding each term:
- \( x = \beta\alpha - \beta\delta - \gamma\alpha + \gamma\delta \)
- \( y = \gamma\beta - \gamma\delta - \alpha\beta + \alpha\delta \)
- \( z = \alpha\gamma - \alpha\delta - \beta\gamma + \beta\delta \)

- Combine all terms:
\[ x + y + z = (\beta\alpha - \gamma\alpha + \gamma\beta - \alpha\beta + \alpha\gamma - \beta\gamma) + (-\beta\delta + \gamma\delta - \alpha\delta + \beta\delta) \]
The terms involving \( \delta \) cancel out, and we are left with:
\[ x + y + z = 0 \]

4. **Substituting into the Identity**:
Since \( x + y + z = 0 \), we can substitute this into our identity:
\[ x^3 + y^3 + z^3 - 3xyz = (0)(x^2 + y^2 + z^2 - xy - yz - zx) = 0 \]

5. **Final Result**:
Therefore, the value of \( x^3 + y^3 + z^3 - 3xyz \) is:
\[ \boxed{0} \]