If a + b + c = 11 and ab + bc + ca = 17, find the value of a^3 + b^3 + c^3 − 3abc using standard algebraic identities.

Introduction / Context:

This problem is designed to test your familiarity with the symmetric identity for a^3 + b^3 + c^3 − 3abc. Instead of asking you to find the individual values of a, b, and c, it provides sums and pairwise products, and expects you to apply an identity that links them directly to the sum of cubes minus 3abc. This is a powerful method in algebra that saves time and reduces computation errors.

Given Data / Assumptions:

• a + b + c = 11.
• ab + bc + ca = 17.
• We must compute a^3 + b^3 + c^3 − 3abc.
• a, b, and c are real numbers for which these expressions are defined.

Concept / Approach:

The central identity is a^3 + b^3 + c^3 − 3abc = (a + b + c)(a^2 + b^2 + c^2 − ab − bc − ca). The problem directly gives a + b + c and ab + bc + ca, and we can obtain a^2 + b^2 + c^2 from the square of the sum (a + b + c)^2. Once we know both a^2 + b^2 + c^2 and ab + bc + ca, we can find a^2 + b^2 + c^2 − ab − bc − ca and plug everything into the cube identity.

Step-by-Step Solution:

Verification / Alternative check:

One way to verify is to notice that if a, b, and c were actual roots of some cubic equation, the given sums and products would correspond to coefficients. The identity used is a standard result in symmetric polynomials and is widely accepted. You could also attempt to construct explicit numbers satisfying the conditions, but that is unnecessary because the identity already uses only the provided symmetric data.

Why Other Options Are Wrong:

The values 121, 168, 300, and 231 may arise from forgetting the factor 2 in the square expansion, miscomputing 121 − 34, or using (a + b + c)^3 directly without subtracting 3(a + b + c)(ab + bc + ca). Only 770 is consistent with the correct step-by-step application of both identities.

Common Pitfalls:

A frequent mistake is to try to find a, b, and c individually, which is often difficult or impossible from the given information. Another common error is misremembering the cube identity as a^3 + b^3 + c^3 = (a + b + c)^3 − 3a^2b and similar incorrect forms. Memorising the correct form and practicing its use will help in many similar problems.

Final Answer:

The value of a^3 + b^3 + c^3 − 3abc is 770.