Let a and b be real numbers such that a + b = 3. Simplify the expression a^3 + b^3 + 9ab − 27 and find its exact value.

Introduction / Context:
This algebra question tests your understanding of the cube expansion identity and how to work with symmetric expressions involving a and b. Instead of giving values of a and b directly, the question provides a relation a + b = 3 and asks you to simplify an expression involving a^3, b^3, and ab. Recognizing the structure of standard identities is the key to solving this quickly and without excessive calculation.

Given Data / Assumptions:

• a and b are real numbers.
• a + b = 3.
• We must evaluate a^3 + b^3 + 9ab − 27.
• No other conditions on a and b are specified.

Concept / Approach:
The main identity to use is the cube of a sum: (a + b)^3 = a^3 + b^3 + 3ab(a + b). From this identity, we can express a^3 + b^3 in terms of (a + b)^3 and ab. After substituting the known value a + b = 3, the expression a^3 + b^3 + 9ab − 27 will simplify dramatically. It turns out that the result does not depend on the particular values of a and b, only on their sum, which makes this expression a neat example of symmetric algebraic structure.

Step-by-Step Solution:
1) Start with the identity (a + b)^3 = a^3 + b^3 + 3ab(a + b). 2) Rearrange this to isolate a^3 + b^3: a^3 + b^3 = (a + b)^3 − 3ab(a + b). 3) Substitute a + b = 3. Then (a + b)^3 = 3^3 = 27. 4) So a^3 + b^3 = 27 − 3ab * 3 = 27 − 9ab. 5) Now consider the full expression: a^3 + b^3 + 9ab − 27. 6) Substitute a^3 + b^3 = 27 − 9ab into this expression: a^3 + b^3 + 9ab − 27 = (27 − 9ab) + 9ab − 27. 7) Combine like terms: 27 − 27 = 0 and −9ab + 9ab = 0. 8) The entire expression simplifies to 0.

Verification / Alternative check:
To confirm, choose specific numbers satisfying a + b = 3. For example, let a = 1 and b = 2. Then a^3 + b^3 + 9ab − 27 becomes 1^3 + 2^3 + 9 * 1 * 2 − 27 = 1 + 8 + 18 − 27 = 27 − 27 = 0. Try another pair, such as a = 0 and b = 3. Then the expression is 0^3 + 3^3 + 9 * 0 * 3 − 27 = 27 − 27 = 0 again. This consistency for different pairs with the same sum supports the identity based simplification.

Why Other Options Are Wrong:
Any non zero value would contradict the identity that we derived from (a + b)^3. Options b (24), c (25), d (27), and e (−27) suggest that the expression changes with specific choices of a and b, but direct substitution for different pairs shows that this is not the case. The expression always simplifies to zero when a + b = 3, so only option a is correct.

Common Pitfalls:
A common mistake is to expand (a + b)^3 incorrectly or to forget the 3ab(a + b) term. Some learners try to plug in arbitrary values for a and b without checking that they satisfy a + b = 3, which leads to inconsistent results. Others may try to simplify a^3 + b^3 + 9ab − 27 directly without recognizing the underlying identity. Remembering and using the cube expansion formula is the fastest and most reliable approach here.

Final Answer:
For any real numbers a and b with a + b = 3, the expression a^3 + b^3 + 9ab − 27 evaluates to 0.