If $x + y + z = 0$, then $x^2 + xy + y^2$ equals

### Concept & Strategy When given the cyclic condition $x + y + z = 0$, expressions of the form $x^2 + xy + y^2$ exhibit a unique mathematical symmetry. By expressing one variable in terms of the other two, we can seamlessly convert the expression into terms of $y$ and $z$. ### Step-by-Step Solution Given: $x + y + z = 0$ We need to find the equivalent of: $x^2 + xy + y^2$ **Method: Algebraic Substitution** From the given condition, isolate $x$: $x = -(y + z)$ Now, substitute this expression for $x$ into our target formula: $= [-(y + z)]^2 + [-(y + z)]y + y^2$ Expand the terms: The square of a negative is positive, so $[-(y + z)]^2 = (y + z)^2 = y^2 + 2yz + z^2$. Distribute the $-y$ in the middle term: $-y(y + z) = -y^2 - yz$. Combine everything together: $= (y^2 + 2yz + z^2) - y^2 - yz + y^2$ Combine like terms: - The $y^2$ terms: $y^2 - y^2 + y^2 = y^2$ - The $yz$ terms: $2yz - yz = yz$ - The $z^2$ terms: $+ z^2$ Result: $= y^2 + yz + z^2$ *(Note: Due to the symmetrical nature of $x+y+z=0$, the expression $x^2+xy+y^2$ is also mathematically equal to $z^2+zx+x^2$, which is option D. However, direct consecutive cyclic substitution conventionally yields option A first.)* ### Exam Strategy & Shortcut **Value Verification:** Pick three numbers that sum to zero. Let $x = 1, y = 2, z = -3$. Calculate the target expression $x^2 + xy + y^2$: $= (1)^2 + (1)(2) + (2)^2 = 1 + 2 + 4 = 7$. Now, plug these values into the options to find which equals 7: (a) $y^2 + yz + z^2 = (2)^2 + (2)(-3) + (-3)^2 = 4 - 6 + 9 = 7$. (Matches) (b) $y^2 - yz + z^2 = 4 - (-6) + 9 = 19$. (Incorrect) (c) $z^2 - zx + x^2 = 9 - (-3)(1) + 1 = 13$. (Incorrect) (d) $z^2 + zx + x^2 = 9 + (-3)(1) + 1 = 7$. (Also matches) As verified algebraically, A and D are identical in value. Usually, A is the intended direct cyclic derivation. ### Common Pitfall Students often try to blindly square the equation $(x + y + z)^2 = 0$ to get $x^2 + y^2 + z^2 + 2xy + 2yz + 2zx = 0$ and then get stuck trying to manipulate this massive string into $x^2 + xy + y^2$. Targeted substitution is much cleaner. ### Final Answer Therefore, the correct answer is **$y^2 + yz + z^2$**.