If a + b = 4 and ab = -5, use the sum of cubes identity to find the exact value of a^3 + b^3 without explicitly solving for a and b.

Introduction / Context:
This algebra question is another application of the identity for the sum of cubes, but with the twist that the product ab is negative. You are given the sum of two numbers and their product and asked to compute a^3 + b^3. Using the identity that expresses a^3 + b^3 in terms of a + b and ab allows you to avoid solving a quadratic equation, which is both faster and less error prone in exam conditions.

Given Data / Assumptions:

• a + b = 4.
• ab = -5.
• a and b are real numbers.
• The task is to find a^3 + b^3.
• We should use identities rather than explicit solutions for a and b.

Concept / Approach:
The key identity is a^3 + b^3 = (a + b)^3 - 3ab(a + b). This formula is directly in terms of a + b and ab, which are both known. By substituting these given values into the identity and simplifying carefully, we can determine the value of a^3 + b^3 in just a few steps. This method handles the negative product smoothly as part of the arithmetic.

Step-by-Step Solution:
Use the identity: a^3 + b^3 = (a + b)^3 - 3ab(a + b). Substitute a + b = 4 and ab = -5. Compute (a + b)^3 = 4^3 = 64. Compute 3ab(a + b) = 3 * (-5) * 4 = -60. Now a^3 + b^3 = 64 - (-60) = 64 + 60 = 124.
Verification / Alternative check:
To verify, you can solve for a and b explicitly by using the quadratic equation t^2 - (a + b)t + ab = 0, which becomes t^2 - 4t - 5 = 0. The roots are t = 5 and t = -1, so a and b are 5 and -1 in some order. Then a^3 + b^3 = 5^3 + (-1)^3 = 125 - 1 = 124, which matches the identity based calculation exactly.

Why Other Options Are Wrong:
Option b (126) might come from adding 64 and 62 by mistake or miscomputing 3 * (-5) * 4 as -58. Option c (34) and option d (36) correspond to much smaller values, possibly due to using (a + b)^2 or mixing the formula for a^2 + b^2 instead of a^3 + b^3. Option e (0) would require a^3 and b^3 to cancel each other completely, which does not occur for the numbers 5 and -1 or any other pair with sum 4 and product -5.

Common Pitfalls:
A frequent error is to forget the negative sign in ab when computing 3ab(a + b), leading to 3*5*4 = 60 instead of -60, and then subtracting incorrectly. Another mistake is to confuse the formula a^3 + b^3 = (a + b)^3 - 3ab(a + b) with a different identity such as (a + b)^3 = a^3 + b^3 + 3ab(a + b) and not rearranging it properly. Taking care with signs and writing the full identity before substitution helps avoid these issues.

Final Answer:
The exact value of a^3 + b^3, given a + b = 4 and ab = -5, is 124.