If x + y = 3, then what is the value of the algebraic expression x^3 + y^3 + 9xy?

Introduction / Context:
This aptitude question tests your understanding of algebraic identities and how to use them to evaluate expressions efficiently. Instead of trying random values for x and y, you are expected to recognise a standard cube identity and convert the given expression into a simpler form in terms of the known value x + y.

Given Data / Assumptions:

• x + y = 3
• Required expression: x^3 + y^3 + 9xy
• x and y are real numbers

Concept / Approach:
The key idea is to use the identity for the cube of a binomial. For any real numbers x and y, the following identity holds:
(x + y)^3 = x^3 + y^3 + 3xy(x + y)
By comparing this identity with the expression in the question, we can try to rewrite x^3 + y^3 + 9xy in a similar pattern. Since x + y is already given, the goal is to express the entire expression only in terms of x + y, which will then allow direct substitution.

Step-by-Step Solution:
Step 1: Start from the identity (x + y)^3 = x^3 + y^3 + 3xy(x + y). Step 2: Compare 3xy(x + y) with the term 9xy given in the question. Step 3: Since x + y = 3, we get 3xy(x + y) = 3xy × 3 = 9xy. Step 4: Therefore x^3 + y^3 + 9xy is exactly equal to x^3 + y^3 + 3xy(x + y). Step 5: Using the identity, x^3 + y^3 + 3xy(x + y) = (x + y)^3. Step 6: Substitute x + y = 3 to get (x + y)^3 = 3^3 = 27. Step 7: Hence x^3 + y^3 + 9xy = 27.

Verification / Alternative check:
You can verify the result by choosing specific values of x and y that satisfy x + y = 3, such as x = 1 and y = 2. Compute x^3 + y^3 + 9xy for these values to confirm that the expression indeed equals 27, which provides a useful numerical check for your algebraic manipulation.

Why Other Options Are Wrong:
15: This might result from incorrect substitution or using a wrong identity without the 3xy(x + y) term.
81: This would correspond to evaluating (3^4) instead of (3^3) and does not relate to the correct cube identity.
9: This value could appear if someone mistakenly uses x + y instead of (x + y)^3 or forgets the contribution of the 9xy term.
3: This is simply the value of x + y, not of the required cubic expression.

Common Pitfalls:
Students often forget the middle term in the identity for (x + y)^3, writing only x^3 + y^3 instead of x^3 + y^3 + 3xy(x + y). Another common mistake is to try to find individual values of x and y unnecessarily. In aptitude exams, efficient use of identities saves time and reduces calculation errors.

Final Answer:
Therefore, the value of x^3 + y^3 + 9xy is 27.