Difficulty: Easy
Correct Answer: 0
Explanation:
Introduction / Context:
This algebra question asks you to evaluate a^3 + b^3 + c^3 − 3abc for specific values of a, b, and c. Instead of expanding each cube separately, you can recognise this as a well known symmetric expression. The identity for a^3 + b^3 + c^3 − 3abc often appears in factorisation problems and is a useful tool in many competitive examinations.
Given Data / Assumptions:
Concept / Approach:
We could directly compute each cube and the product abc, but recognising the identity a^3 + b^3 + c^3 − 3abc = (a + b + c)(a^2 + b^2 + c^2 − ab − bc − ca) provides insight into when the expression becomes zero. Here, it is quicker to check whether a + b + c is zero. If so, the entire expression must be zero, which saves time and reduces arithmetic.
Step-by-Step Solution:
Step 1: Compute the sum a + b + c = 4 + 2 + (−6). Step 2: Simplify the sum: 4 + 2 − 6 = 0. Step 3: Recall the identity a^3 + b^3 + c^3 − 3abc = (a + b + c)(a^2 + b^2 + c^2 − ab − bc − ca). Step 4: Since a + b + c = 0, the expression becomes 0 * (a^2 + b^2 + c^2 − ab − bc − ca). Step 5: Any number multiplied by 0 is 0, so a^3 + b^3 + c^3 − 3abc = 0. Step 6: If desired, verify numerically: a^3 = 64, b^3 = 8, c^3 = −216, so their sum is 64 + 8 − 216 = −144. Step 7: Compute abc = 4 * 2 * (−6) = −48, and −3abc = −3 * (−48) = 144. Step 8: Add cubes sum and −3abc: −144 + 144 = 0, confirming the identity based reasoning.Verification / Alternative check:
The direct numerical method confirms the identity. Calculating the cubes and the product abc separately gives exactly the same result as the shortcut using the condition a + b + c = 0. This double check assures you that the answer is accurate and the identity is applied correctly.
Why Other Options Are Wrong:
The values 2, 4, 6, and 8 arise only if arithmetic is mishandled when computing cubes or the product −3abc. Since both the identity method and direct computation yield 0, any nonzero option contradicts correct algebraic manipulation.
Common Pitfalls:
Students may try to expand everything without noticing the simple condition that a + b + c = 0, which guarantees the expression is zero. Others might miscalculate the sign of c^3 or abc, leading to incorrect sums. Recognising patterns like a + b + c = 0 is an important skill that simplifies many algebraic problems.
Final Answer:
The value of a^3 + b^3 + c^3 − 3abc for a = 4, b = 2, c = −6 is 0.
Discussion & Comments