If `x+y+z=0`, then`[(y-z-x)//2]^(3)+[(z-x-y)//2]^(3)+[(x-y-z)//2]^(3)` equals

Options

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

Answer

To solve the problem, we need to evaluate the expression:

\[ \left(\frac{y - z - x}{2}\right)^3 + \left(\frac{z - x - y}{2}\right)^3 + \left(\frac{x - y - z}{2}\right)^3 \]

given that \(x + y + z = 0\).

### Step 1: Rewrite the expression using the given condition

Since \(x + y + z = 0\), we can express \(z\) in terms of \(x\) and \(y\):

\[ z = - (x + y) \]

Now, substitute \(z\) into the expression:

1. For the first term:
\[ \frac{y - z - x}{2} = \frac{y - (- (x + y)) - x}{2} = \frac{y + x + y - x}{2} = \frac{2y}{2} = y \]

2. For the second term:
\[ \frac{z - x - y}{2} = \frac{- (x + y) - x - y}{2} = \frac{-2x - 2y}{2} = - (x + y) \]

3. For the third term:
\[ \frac{x - y - z}{2} = \frac{x - y - (- (x + y))}{2} = \frac{x - y + x + y}{2} = \frac{2x}{2} = x \]

Now, substituting these back into the expression gives us:

\[ y^3 + (- (x + y))^3 + x^3 \]

### Step 2: Simplify the expression

Now we can simplify the expression:

\[ y^3 + (- (x + y))^3 + x^3 = y^3 + -(x + y)^3 + x^3 \]

Using the identity \((a + b)^3 = a^3 + b^3 + 3ab(a + b)\), we can expand \(-(x + y)^3\):

\[ -(x + y)^3 = - (x^3 + y^3 + 3xy(x + y)) \]

Thus, the expression becomes:

\[ y^3 - (x^3 + y^3 + 3xy(x + y)) + x^3 \]

### Step 3: Combine like terms

Combining the terms gives us:

\[ y^3 - x^3 - y^3 - 3xy(x + y) + x^3 = -3xy(x + y) \]

### Step 4: Substitute back \(x + y + z = 0\)

Since \(x + y + z = 0\), we can replace \(x + y\) with \(-z\):

\[ -3xy(-z) = 3xyz \]

### Final Result

Thus, we have:

\[ \left(\frac{y - z - x}{2}\right)^3 + \left(\frac{z - x - y}{2}\right)^3 + \left(\frac{x - y - z}{2}\right)^3 = 3xyz \]