If `a + b + c =0,` then what is the value of `(a + b -c) ^(3) + (c + a -b) ^(3) + (b + c - a) ^(3)`?

Options

A. `-8 (a ^(3) + b ^(3) + c ^(3))`
B. `a^(3) + b ^(3) + c ^(3)`
C. `24 abc`
D. `-24 abc`

Answer

To solve the problem, we need to find the value of \((a + b - c)^3 + (c + a - b)^3 + (b + c - a)^3\) given that \(a + b + c = 0\).

### Step-by-Step Solution:

1. **Substitute Values**:
Since \(a + b + c = 0\), we can express \(c\) in terms of \(a\) and \(b\):
\[ c = - (a + b) \]
Now, we can rewrite the expressions \(a + b - c\), \(c + a - b\), and \(b + c - a\).

2. **Calculate Each Expression**:
- For \(a + b - c\):
\[ a + b - c = a + b - (- (a + b)) = a + b + a + b = 2(a + b) \]
- For \(c + a - b\):
\[ c + a - b = - (a + b) + a - b = -a - b + a - b = -2b \]
- For \(b + c - a\):
\[ b + c - a = b - (a + b) - a = b - a - b - a = -2a \]

3. **Substitute Back**:
Now we substitute these values back into the original expression:
\[ (2(a + b))^3 + (-2b)^3 + (-2a)^3 \]

4. **Calculate Each Cube**:
- \( (2(a + b))^3 = 8(a + b)^3 \)
- \( (-2b)^3 = -8b^3 \)
- \( (-2a)^3 = -8a^3 \)

5. **Combine the Terms**:
Now we can combine all these terms:
\[ 8(a + b)^3 - 8b^3 - 8a^3 \]
Factor out the common term:
\[ 8 \left( (a + b)^3 - b^3 - a^3 \right) \]

6. **Use the Identity**:
We can use the identity for the expansion of cubes:
\[ (x + y)^3 = x^3 + y^3 + 3xy(x + y) \]
Here, let \(x = a\) and \(y = b\):
\[ (a + b)^3 = a^3 + b^3 + 3ab(a + b) \]
Therefore, substituting this back gives:
\[ 8 \left( a^3 + b^3 + 3ab(a + b) - b^3 - a^3 \right) = 8 \cdot 3ab(a + b) \]

7. **Final Result**:
Since \(a + b + c = 0\) implies \(a + b = -c\):
\[ 8 \cdot 3ab(-c) = -24abc \]

Thus, the final value of \((a + b - c)^3 + (c + a - b)^3 + (b + c - a)^3\) is:
\[ \boxed{-24abc} \]