In algebra, two real numbers a and b satisfy a − b = 10 and a * b = −21. Using these conditions, find the exact value of the difference of cubes a^3 − b^3.

Introduction / Context:
This question tests knowledge of algebraic identities involving sums and differences of cubes, and how to use given information about a − b and a * b to find higher power expressions such as a^3 − b^3. Instead of solving for a and b directly, which would require solving a quadratic equation, the problem encourages the use of identities that express a^3 − b^3 in terms of simpler symmetric expressions involving a and b.

Given Data / Assumptions:

• Two real numbers a and b satisfy a − b = 10.
• Their product is a * b = −21.
• We need to compute a^3 − b^3 exactly.
• We can use standard algebraic identities for powers and symmetric sums.

Concept / Approach:
The central identity is a^3 − b^3 = (a − b)(a^2 + a b + b^2). We already know a − b, so we only need to find a^2 + a b + b^2. The expression a^2 + b^2 can be derived from (a − b)^2 and a b using the identity (a − b)^2 = a^2 + b^2 − 2 a b. Once we find a^2 + b^2, we add a b to get a^2 + a b + b^2, and then multiply by a − b to obtain a^3 − b^3.

Step-by-Step Solution:
1) Use the identity (a − b)^2 = a^2 + b^2 − 2 a b. 2) Substitute known values: (a − b)^2 = 10^2 = 100 and a b = −21. 3) So 100 = a^2 + b^2 − 2(−21) = a^2 + b^2 + 42. 4) Rearrange to find a^2 + b^2: a^2 + b^2 = 100 − 42 = 58. 5) Now compute a^2 + a b + b^2 = (a^2 + b^2) + a b = 58 + (−21) = 37. 6) Use the identity a^3 − b^3 = (a − b)(a^2 + a b + b^2). 7) Substitute a − b = 10 and a^2 + a b + b^2 = 37 to get a^3 − b^3 = 10 * 37. 8) Multiply: 10 * 37 = 370.

Verification / Alternative check:
As a check, you can find explicit values of a and b by solving the quadratic equation whose roots are a and b. Let t be a variable such that t^2 − (a + b)t + a b = 0. We know a − b and a b, so we can compute a + b from (a − b)^2 = (a + b)^2 − 4 a b. Using 100 = (a + b)^2 − 4(−21) = (a + b)^2 + 84 gives (a + b)^2 = 16, so a + b = 4 or a + b = −4. Solving the pair a − b = 10 and a + b = 4 gives one set of values, and similarly for a + b = −4. In either case, if you compute a^3 − b^3 numerically, you will obtain 370, which confirms the identity based result.

Why Other Options Are Wrong:
Options b (316), c (185), and d (158) arise from partial or incorrect use of identities, such as forgetting the a b term in a^2 + a b + b^2 or mis computing (a − b)^2. Option e (−370) has the correct magnitude but the wrong sign, which would correspond to swapping a and b or mis interpreting a − b. Only option a matches the correctly derived value of a^3 − b^3 using consistent algebra.

Common Pitfalls:
Common mistakes include trying to cube a − b directly or attempting to compute a^3 and b^3 separately without using identities, both of which lead to unnecessary complexity. Some learners also mis remember the identity for a^3 − b^3 or confuse it with (a − b)^3. Another pitfall is sign errors when substituting a b = −21 into formulas involving −2 a b. Writing each step clearly and checking intermediate values like a^2 + b^2 helps avoid these problems.

Final Answer:
Using the difference of cubes identity and the given values of a − b and a b, the exact value of a^3 − b^3 is 370.