Evaluate a cubic identity under a linear condition: If a + b + 1 = 0, compute the value of a^3 + b^3 + 1 − 3ab.

Question

Solve this multiple-choice question and choose the correct option.

Accepted Answer

Introduction / Context:This is a direct application of the identity a^3 + b^3 + c^3 − 3abc = (a + b + c)(a^2 + b^2 + c^2 − ab − bc − ca). Recognizing c = 1 and that a + b + 1 is given simplifies the expression to zero immediately. Given Data / Assumptions: a + b + 1 = 0 Target: a^3 + b^3 + 1 − 3ab Concept / Approach:Set c = 1 in the well-known identity. Because the factor (a + b + c) appears, if a + b + 1 equals zero, then the whole product must vanish, provided everything is finite. This makes the computation simple and robust. Step-by-Step Solution: Use: a^3 + b^3 + 1 − 3ab = (a + b + 1)(a^2 + b^2 + 1 − ab − a − b).Given a + b + 1 = 0 ⇒ the product equals 0.Therefore, a^3 + b^3 + 1 − 3ab = 0. Verification / Alternative check: Pick a, b satisfying a + b + 1 = 0, e.g., a = −1, b = 0. Then LHS = (−1)^3 + 0 + 1 − 0 = 0, confirming the identity. Why Other Options Are Wrong: 1, 3, −1: These contradict the factorization result which forces the value to zero under the given condition. Common Pitfalls: Applying a^3 + b^3 formula incorrectly instead of the three-term identity. Forgetting to set c = 1 explicitly. Final Answer: 0

Evaluate a cubic identity under a linear condition: If a + b + 1 = 0, compute the value of a^3 + b^3 + 1 − 3ab.

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