If a + b + c = 0 for real numbers a, b and c, then what is the value of the sum of cubes a^3 + b^3 + c^3?

Introduction / Context:
This question explores a well known identity involving three variables and their cubes. When the sum a + b + c equals zero, the sum of cubes a^3 + b^3 + c^3 has a particularly simple expression in terms of the product abc.

Given Data / Assumptions:

• a, b, and c are real numbers.
• a + b + c = 0.
• We must find a^3 + b^3 + c^3.

Concept / Approach:
The identity for three cubes is: a^3 + b^3 + c^3 − 3abc = (a + b + c)(a^2 + b^2 + c^2 − ab − bc − ca). If a + b + c = 0, the right side becomes zero and we obtain a direct relation between a^3 + b^3 + c^3 and abc. We use this identity directly to get the result.

Step-by-Step Solution:
Step 1: Recall the identity a^3 + b^3 + c^3 − 3abc = (a + b + c)(a^2 + b^2 + c^2 − ab − bc − ca). Step 2: Substitute a + b + c = 0 into the right hand side. Step 3: Then (a + b + c)(a^2 + b^2 + c^2 − ab − bc − ca) = 0 × (a^2 + b^2 + c^2 − ab − bc − ca) = 0. Step 4: Hence the identity simplifies to a^3 + b^3 + c^3 − 3abc = 0. Step 5: Rearranging, we get a^3 + b^3 + c^3 = 3abc. Step 6: This relation holds for any real a, b, and c satisfying a + b + c = 0.

Verification / Alternative check:
We can check with an example. Let a = 1, b = 2, and c = −3, so a + b + c = 0. Then a^3 + b^3 + c^3 = 1 + 8 − 27 = −18. Also abc = 1 × 2 × (−3) = −6 and 3abc = 3 × (−6) = −18, which matches the sum of cubes. This confirms the identity in a concrete case.

Why Other Options Are Wrong:
Option A (abc) and Option B (2abc) underestimate the sum of cubes by a fixed factor. Option D (0) is only true if abc itself is zero, which is not guaranteed by a + b + c = 0. Option E (abc/3) is an arbitrary fraction with no support from the standard identity.

Common Pitfalls:
Some learners misremember the identity and write a^3 + b^3 + c^3 = (a + b + c)^3, which is not correct. Others forget the −3abc term entirely. It is important to learn the full three variable cube identity, especially because it simplifies beautifully in the special case a + b + c = 0.

Final Answer:

The correct value of a^3 + b^3 + c^3 is 3abc.