Final answer:
The correct formula for the sum of two cubes is A) x³ + y³ = (x + y)(x² - xy + y²), which can be verified by expanding the right-hand side and showing that it simplifies to x³ + y³.
Step-by-step explanation:
The correct formula for the sum of two cubes is A) x³ + y³ = (x + y)(x² - xy + y²). To verify this, we can expand the right-hand side of option A and check if it simplifies to the left-hand side of the equation. Expansion: (x + y)(x² - xy + y²), x(x²) - x(xy) + x(y²) + y(x²) - y(xy) + y(y²), x³ - x²y + xy² + x²y - y²x + y³, x³ + y³. The terms '- x²y' and 'x²y' cancel out, as do 'xy²' and '- xy²', leaving us with x³ + y³, which matches the left-hand side of our original equation, confirming that option A is correct.
The correct formula for the sum of two cubes is option C: x³ + y³ = (x + y)(x² + xy + y²). To verify this formula, we can multiply the factors (x + y) and (x² + xy + y²) using the distributive property: (x + y)(x² + xy + y²) = x(x² + xy + y²) + y(x² + xy + y²). Expanding these two terms: x³ + x²y + xy² + xy² + y³ + y²x. Adding like terms: x³ + y³ + x²y + 2xy² + y²x. And rearranging the terms: x³ + y³ + (x²y + xy² + y²x). Since (x²y + xy² + y²x) can be simplified as xy(x + y), we have: x³ + y³ + xy(x + y). Therefore, the formula x³ + y³ = (x + y)(x² + xy + y²) is verified.