If x + y + z = 0 , then `( x + y - z)^(3) + ( y + z - x)^(3) + (z + x - y) ^(3) = k (xyz)` , where k is equal to :

Options

A. `-3`
B. `3`
C. `-24`
D. `9`

Answer

To solve the problem, we start with the given equation:

**Step 1:** We know that \( x + y + z = 0 \).

**Step 2:** We need to simplify the expression \( (x + y - z)^3 + (y + z - x)^3 + (z + x - y)^3 \).

**Step 3:** We can rewrite each term using the condition \( x + y + z = 0 \):
- \( x + y - z = (x + y + z) - 2z = -2z \)
- \( y + z - x = (x + y + z) - 2x = -2x \)
- \( z + x - y = (x + y + z) - 2y = -2y \)

So, we can rewrite the expression as:
\[ (-2z)^3 + (-2x)^3 + (-2y)^3 \]

**Step 4:** Now, we calculate each cube:
\[ (-2z)^3 = -8z^3, \quad (-2x)^3 = -8x^3, \quad (-2y)^3 = -8y^3 \]

**Step 5:** Combining these results:
\[ -8z^3 - 8x^3 - 8y^3 = -8(x^3 + y^3 + z^3) \]

**Step 6:** We use the identity for the sum of cubes when \( x + y + z = 0 \):
\[ x^3 + y^3 + z^3 = 3xyz \]

**Step 7:** Substitute this back into our expression:
\[ -8(x^3 + y^3 + z^3) = -8(3xyz) = -24xyz \]

**Step 8:** We compare this with the form \( k(xyz) \):
\[ -24xyz = k(xyz) \]

Thus, we find that \( k = -24 \).

**Final Answer:** Therefore, the value of \( k \) is \( -24 \).

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