If $(x + y) = 3$, $xy = 2$, then what is the value of $x^3 + y^3$?

### Concept & Formula This requires directly calculating the sum of cubes using the sum and product of the variables, circumventing the need to find individual variable values. $$x^3 + y^3 = (x + y)^3 - 3xy(x + y)$$ ### Step-by-Step Solution * **Given:** The sum $(x + y) = 3$ and the product $xy = 2$. * Substitute these known values directly into the derived identity: $$x^3 + y^3 = (3)^3 - 3(2)(3)$$ * Calculate the cube: $$3^3 = 27$$ * Calculate the product term: $$3 \times 2 \times 3 = 18$$ * Subtract: $$27 - 18 = 9$$ ### Exam Strategy & Shortcut While the formula is fast, for small integers you can quickly guess the roots. What two numbers add to 3 and multiply to 2? The numbers are 2 and 1. Therefore, $x = 2$ and $y = 1$ (or vice versa). Calculate $2^3 + 1^3 = 8 + 1 = 9$. This is even faster than writing out the formula. ### Common Pitfall A frequent mistake is using the formula for $(x+y)^3$ and forgetting to subtract the middle terms to isolate $x^3 + y^3$, leading students to incorrectly select $27$ if it were an option. ### Final Answer Therefore, the correct answer is **9**.