If `x+y+z=2s`, then `(s-x)^(3) +(s-y)^(3) +3(s-x) (s-y)z` equals

Options

A. `z^(3)`
B. `-z^(3)`
C. `x^(3)`
D. `y^(3)`

Answer

To solve the problem, we need to simplify the expression \((s-x)^{3} + (s-y)^{3} + 3(s-x)(s-y)z\) given that \(x + y + z = 2s\).

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ x + y + z = 2s \]

2. **Rearrange the equation to express \(z\)**:
\[ z = 2s - x - y \]

3. **Substitute \(z\) into the expression**:
We need to evaluate:
\[ (s-x)^{3} + (s-y)^{3} + 3(s-x)(s-y)z \]
Substitute \(z\):
\[ (s-x)^{3} + (s-y)^{3} + 3(s-x)(s-y)(2s - x - y) \]

4. **Use the identity for the sum of cubes**:
Recall that:
\[ a^{3} + b^{3} = (a+b)(a^{2} - ab + b^{2}) \]
Let \(a = s-x\) and \(b = s-y\). Then:
\[ (s-x)^{3} + (s-y)^{3} = (s-x + s-y)((s-x)^{2} - (s-x)(s-y) + (s-y)^{2}) \]
Simplifying \(s-x + s-y\):
\[ = 2s - (x+y) \]
Since \(x + y = 2s - z\), we have:
\[ = z \]

5. **Combine the terms**:
Now we can rewrite the expression:
\[ z((s-x)^{2} - (s-x)(s-y) + (s-y)^{2}) + 3(s-x)(s-y)(2s - x - y) \]

6. **Simplify the expression**:
Notice that:
\[ (s-x)(s-y) = s^{2} - s(x+y) + xy \]
Using \(x + y = 2s - z\):
\[ = s^{2} - s(2s - z) + xy = s^{2} - 2s^{2} + sz + xy = sz - s^{2} + xy \]

7. **Final simplification**:
The entire expression simplifies to:
\[ z^{3} \]
Thus:
\[ (s-x)^{3} + (s-y)^{3} + 3(s-x)(s-y)z = z^{3} \]

### Conclusion:
The final result is:
\[ \boxed{z^{3}} \]