If a + b = 4 and ab = 3, use the sum of cubes identity to compute the exact value of a^3 + b^3 without solving separately for a and b.

Introduction / Context:
This algebra problem focuses on the identity for the sum of cubes. Rather than solving explicitly for a and b, you are expected to recognise that a^3 + b^3 can be expressed in terms of a + b and ab. This technique allows rapid simplification and is very useful in aptitude exams where direct computation of roots would be time consuming.

Given Data / Assumptions:

• a + b = 4.
• ab = 3.
• a and b are real numbers.
• The goal is to find a^3 + b^3.
• We want to avoid solving for a and b individually.

Concept / Approach:
The identity for the sum of cubes is a^3 + b^3 = (a + b)^3 - 3ab(a + b). This identity expresses a^3 + b^3 in terms of the sum a + b and the product ab, both of which are given. By substituting these values and simplifying, we can directly obtain the value of a^3 + b^3 in a few steps. This method prevents unnecessary quadratic equation solving and reduces algebraic errors.

Step-by-Step Solution:
Use the identity: a^3 + b^3 = (a + b)^3 - 3ab(a + b). Substitute a + b = 4 and ab = 3 into the identity. Compute (a + b)^3 = 4^3 = 64. Compute 3ab(a + b) = 3 * 3 * 4 = 36. Therefore a^3 + b^3 = 64 - 36 = 28.
Verification / Alternative check:
As a check, you can solve for a and b explicitly. From a + b = 4 and ab = 3, the pair (a, b) satisfies t^2 - 4t + 3 = 0, whose roots are t = 1 and t = 3. So a and b are 1 and 3 in some order. Then a^3 + b^3 = 1^3 + 3^3 = 1 + 27 = 28, which matches the identity based result. This confirms that the formula has been applied correctly.

Why Other Options Are Wrong:
Option b (21), option c (17), option d (31), and option e (26) result from common errors such as using 3ab instead of 3ab(a + b), miscomputing 4^3 as 16, or getting 3*3*4 as 34. None of these values are consistent with both the identity and the explicit evaluation using actual values of a and b.

Common Pitfalls:
Learners sometimes confuse the identities for a^3 - b^3 and a^3 + b^3 or mistakenly use (a + b)^3 = a^3 + b^3 + 3ab(a + b) without rearranging. Another pitfall is to forget to multiply by a + b in the term 3ab(a + b), leading to partial expressions. Carefully writing down the entire identity and substituting the given values step by step helps avoid these mistakes.

Final Answer:
The exact value of a^3 + b^3, given a + b = 4 and ab = 3, is 28.