If $x + y + z = 0$, then $x^3 + y^3 + z^3 + 3xyz$ is equal to

### Concept & Formula This problem is built upon a well-known algebraic identity regarding cubic sums: $$x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)$$ If the condition $x + y + z = 0$ is fulfilled, the entire right side becomes zero, leading to: $$x^3 + y^3 + z^3 = 3xyz$$ ### Step-by-Step Solution * **Given:** $x + y + z = 0$ * **Step 1:** From the standard cubic identity, we know that if $x+y+z=0$, then: $$x^3 + y^3 + z^3 = 3xyz$$ * **Step 2:** The question asks for the value of the expression $x^3 + y^3 + z^3 + 3xyz$. Substitute $3xyz$ in place of $x^3 + y^3 + z^3$: $$(3xyz) + 3xyz$$ * **Step 3:** Combine the like terms: $$6xyz$$ ### Exam Strategy & Shortcut **Value Substitution Method:** Choose simple values for $x, y, z$ that satisfy the condition $x + y + z = 0$. Let $x = 1, y = -1, z = 0$. Substitute these test values into the required expression: $$1^3 + (-1)^3 + 0^3 + 3(1)(-1)(0) = 1 - 1 + 0 + 0 = 0$$ Now check the options using our selected values ($6xyz = 6(1)(-1)(0) = 0$). While option (a) is also $0$, trying a second set like $1, 1, -2$ quickly eliminates option (a) because $1 + 1 - 8 + 3(1)(1)(-2) = -6 - 6 = -12$, matching $6xyz = 6(1)(1)(-2) = -12$. ### Common Pitfall * **Misreading the Expression:** Students frequently mix up $x^3 + y^3 + z^3 - 3xyz = 0$ with the required expression. They assume the answer is $0$ automatically without noticing that the question contains a **plus** sign ($+ 3xyz$) instead of a minus sign. ### Final Answer **Therefore, the correct answer is 6 xyz.**