If ` x + y + z = 6` then the value of `( x - 1) ^(3) + ( y - 2) ^(3) + ( z - 3) ^(3)` is

Options

A. ` 3 ( x - 1) ( y + 2) ( z - 3)`
B. `3 ( x + 1) ( y - 2) ( z - 3)`
C. ` 3 ( x - 1) ( y - 2) ( z + 3)`
D. ` 3 ( x - 1) ( y - 2) ( z - 3)`

Answer

To solve the problem, we need to find the value of \( (x - 1)^3 + (y - 2)^3 + (z - 3)^3 \) given that \( x + y + z = 6 \).

### Step 1: Set Up the Variables
Let:
- \( a = x - 1 \)
- \( b = y - 2 \)
- \( c = z - 3 \)

Then we can express \( x, y, z \) in terms of \( a, b, c \):
- \( x = a + 1 \)
- \( y = b + 2 \)
- \( z = c + 3 \)

### Step 2: Substitute into the Equation
Substituting these into the equation \( x + y + z = 6 \):
\[ (a + 1) + (b + 2) + (c + 3) = 6 \]
This simplifies to:
\[ a + b + c + 6 = 6 \]
Thus, we find:
\[ a + b + c = 0 \]

### Step 3: Use the Identity for Cubes
Now we can use the identity for the sum of cubes:
\[ a^3 + b^3 + c^3 - 3abc = (a + b + c) \left( a^2 + b^2 + c^2 - ab - ac - bc \right) \]
Since we found \( a + b + c = 0 \), we can simplify the equation:
\[ a^3 + b^3 + c^3 - 3abc = 0 \]
Thus:
\[ a^3 + b^3 + c^3 = 3abc \]

### Step 4: Calculate \( abc \)
We need to express \( abc \) in terms of \( x, y, z \):
\[ abc = (x - 1)(y - 2)(z - 3) \]
To find \( abc \), we can substitute \( x = 1, y = 2, z = 3 \) (which satisfies \( x + y + z = 6 \)):
\[ abc = (1 - 1)(2 - 2)(3 - 3) = 0 \]

### Step 5: Final Calculation
Now substituting back into our equation:
\[ (x - 1)^3 + (y - 2)^3 + (z - 3)^3 = 3abc = 3 \cdot 0 = 0 \]

### Conclusion
The value of \( (x - 1)^3 + (y - 2)^3 + (z - 3)^3 \) is \( \boxed{0} \).