Question

If $x + y + z = 0$, then what is the value of $(x + y)(y + z)(z + x)$?

• A. \(x^2 + y^2 + z^2 \)
• B. \((x + y + z)^2 \)
• C. \(-xyz \)
• D. \(xyz \)
• E. \((x + y + z)^3 \)

Hint:

If $x + y + z = 0$, many symmetric expressions simplify drastically. Always try substituting directly into identities.

Answer 1

The Correct Option is C

Concept: Use the identity: \[ (x + y)(y + z)(z + x) = (x + y + z)(xy + yz + zx) - xyz \]
Step 1: Substitute given condition.
Given: \[ x + y + z = 0 \] Substitute into identity: \[ (x + y)(y + z)(z + x) = 0 \cdot (xy + yz + zx) - xyz \]

Step 2: Simplify.
\[ = -xyz \]