If a − b = 4 and the product ab = −3 in algebra, then using standard identities what is the value of the difference of cubes a^3 − b^3?

Introduction / Context:
This algebra problem again focuses on the identity for the difference of cubes together with the square of a difference. Instead of solving for a and b directly, we use a − b and ab to compute a^3 − b^3. This style of question is common in aptitude tests because it rewards fluency with algebraic identities and efficient reasoning.

Given Data / Assumptions:

• a − b = 4
• ab = −3
• We want to find a^3 − b^3.
• a and b are algebraic numbers and can be treated symbolically.

Concept / Approach:
The core identity is a^3 − b^3 = (a − b)(a^2 + ab + b^2). We already know a − b and ab, so the key objective is to find a^2 + b^2. We use the expansion (a − b)^2 = a^2 + b^2 − 2ab. Once a^2 + b^2 is obtained, we can add ab to form a^2 + ab + b^2 and then multiply by a − b for the final result.

Step-by-Step Solution:
Start from the identity (a − b)^2 = a^2 + b^2 − 2ab. Given a − b = 4, so (a − b)^2 = 4^2 = 16. Substitute into the identity: 16 = a^2 + b^2 − 2ab. Given ab = −3, so −2ab = −2 * (−3) = 6. Thus 16 = a^2 + b^2 + 6, so a^2 + b^2 = 10. Now compute a^2 + ab + b^2 = (a^2 + b^2) + ab = 10 + (−3) = 7. Use the identity a^3 − b^3 = (a − b)(a^2 + ab + b^2). Substitute a − b = 4 and a^2 + ab + b^2 = 7. Hence a^3 − b^3 = 4 * 7 = 28.

Verification / Alternative check:
We can cross check by constructing a quadratic whose roots are a and b. For example, one such quadratic is t^2 − (a + b)t + ab = 0. From a − b and ab we could find (a + b)^2 = a^2 + b^2 + 2ab = 10 + 2 * (−3) = 4, giving a + b = ±2. Using any consistent pair of a and b that satisfy both a − b = 4 and a + b = 2, we obtain specific values, then compute a^3 − b^3 explicitly and verify that it equals 28. This confirms the identity based computation.

Why Other Options Are Wrong:
The values 21 and 23 result from arithmetic errors, such as miscalculating a^2 + b^2 or ab. The option −20 appears if one misplaces a sign in the identity or uses a + b in place of a − b. The value 12 is too small and would correspond to incorrect use of 2ab instead of ab in the final step.

Common Pitfalls:
Misremembering the identity for (a − b)^2 or misusing a^3 − b^3 = (a − b)(a^2 − ab + b^2) instead of the correct form for a^3 − b^3 can lead to wrong results. Careful attention to signs, especially when ab is negative, is very important. Writing out each identity fully and substituting values slowly helps avoid these mistakes.

Final Answer:
The value of the difference of cubes is 28.