If x + y = 1, find the exact value of x^3 + y^3 + 3xy. Use the cube-of-sum identity to simplify.

Introduction / Context:Recognizing the pattern within the cube-of-sum identity allows you to simplify expressions that mix cubes and products. This is a classic manipulation in algebra-based aptitude questions.

Given Data / Assumptions:

• x + y = 1.
• Evaluate x^3 + y^3 + 3xy.

Concept / Approach:Recall the identity (x + y)^3 = x^3 + y^3 + 3xy(x + y). Rearranging isolates x^3 + y^3. After computing that part, add 3xy as requested, noticing cancellation.

Step-by-Step Solution:From identity: x^3 + y^3 = (x + y)^3 − 3xy(x + y).With x + y = 1, we have x^3 + y^3 = 1^3 − 3xy*1 = 1 − 3xy.Therefore x^3 + y^3 + 3xy = (1 − 3xy) + 3xy = 1.

Verification / Alternative check:Pick values summing to 1 (e.g., x = 0.4, y = 0.6). Compute x^3 + y^3 + 3xy numerically to see it equals 1, independent of the split.

Why Other Options Are Wrong:

• 3, 2, 0, −2: These ignore the cancellation of −3xy with +3xy or mistake the identity.

Common Pitfalls:Using x^3 + y^3 = (x + y)^3 directly without subtracting 3xy(x + y); forgetting that x + y = 1 simplifies the expression drastically.

Final Answer:1