Expand (x + y)^3.

• A. x^3 + x^2 y + x y^2 + y^3
• B. x^3 + 3x y^2 + y^3
• C. x^3 + 3x^2 y + 3x y^2 + y^3
• D. x^2 + 3x y + y^2

Answer: (C)

When expanding a binomial to the third power, the coefficients follow 1, 3, 3, 1. Using the binomial theorem, (x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3. You can get this by applying the formula or by multiplying step by step: first square (x + y) to get x^2 + 2xy + y^2, then multiply by (x + y) to obtain x^3 + 3x^2y + 3xy^2 + y^3.

This form is the complete cubic expansion, with the two middle terms carrying coefficient 3. The other options omit one of the middle terms or mix in a quadratic pattern (as in the square of a binomial), so they don’t match the correct expansion.