Question

If \( xy + yz + zx = 0 \), then \( (x + y + z)^2 \) equals

• A. \( (x + y)^2 + xz \)
• B. \( (x + z)^2 + xy \)
• C. \( x^2 + y^2 + z^2 \)
• D. \( 2(xy + yz + xz) \)

Hint:

Remember to expand squares and simplify based on given conditions. If an expression equals zero, it can often simplify your calculations significantly.

Answer 1

The Correct Option is C

Starting with the given equation \( xy + yz + zx = 0 \), we need to simplify \( (x + y + z)^2 \).
Expanding \( (x + y + z)^2 \) gives us: \[ (x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + zx) \] Since we are given that \( xy + yz + zx = 0 \), we substitute that into the equation, resulting in: \[ (x + y + z)^2 = x^2 + y^2 + z^2 + 2(0) = x^2 + y^2 + z^2 \] Thus, the correct answer is \( x^2 + y^2 + z^2 \), corresponding to option (C).