If x + y + z = 0, then what is $\frac{xyz}{(x+y(y+z)(z+x)}$ equal to ? (where, x ≠ -y, y ≠ -z, z, ≠ -x)

Question

Accepted Answer

To solve the problem and justify why Option 1 is correct, we need to analyze the expression given the condition $x + y + z = 0$.

### Explanation:

Given:
$$ x + y + z = 0 $$

We need to find the value of:
$$ \frac{xyz}{(x+y)(y+z)(z+x)} $$

#### Step-by-step Solution:

1. **Substitute $z = -(x+y)$:**

Since $x + y + z = 0$, we can express $z$ as:
   $$ z = -(x + y) $$

2. **Substitute in the expression:**

Substitute $z = -(x+y)$ into the expression:
   $$ \frac{xyz}{(x+y)(y+z)(z+x)} = \frac{xy(-(x+y))}{(x+y)(y-(x+y))(z+x)} $$

3. **Simplify the denominator:**

- $y + z = y - (x+y) = -x$
   - $z + x = -(x+y) + x = -y$

So, the denominator becomes:
   $$ (x+y)(-x)(-y) = (x+y)xy $$

4. **Simplify the expression:**

The expression now becomes:
   $$ \frac{-xy(x+y)}{(x+y)xy} $$

Cancel out the common terms $(x+y)$ and $xy$ from the numerator and denominator:
   $$ = -1 $$

Thus, the value of the expression is $-1$, which corresponds to Option 1.

### Why Other Options are Incorrect:

- **Option 2 (1):** The expression simplifies to $-1$, not $1$. The negative sign arises from the substitution and simplification process.

- **Option 3 (xy + yz + zx):** This option suggests a polynomial expression, which does not match the simplified result of $-1$. The expression $(x+y)(y+z)(z+x)$ does not simplify to a form that results in $xy + yz + zx$.

- **Option 4 (None of the above):** Since we have shown that the expression simplifies to $-1$, this option is incorrect.

In conclusion, the correct answer is indeed Option 1, as the expression simplifies to $-1$ under the given condition $x + y + z = 0$.