Question 16

x³ - y³ is equivalent to (x - y)(x² + xy + y²)

The expression x³ - y³ can be factored into (x - y)(x² + xy + y²), which demonstrates the difference of cubes formula.

A) (x - y)(x ^ 2 - xy - y ^ 2)

This option is incorrect because the second factor, (x² - xy - y²), does not match the correct factorization of x³ - y³. The proper factorization requires the sum of xy, not a subtraction.

B) (x - y)(x ^ 2 - xy + y ^ 2)

This option is also incorrect. While the first factor (x - y) is correct, the second factor (x² - xy + y²) does not conform to the standard form required for the difference of cubes.

C) (x - y)(x ^ 2 + xy + y ^ 2)

This is the correct option. It accurately represents the factorization of x³ - y³ according to the difference of cubes formula, where the second factor is (x² + xy + y²).

D) (x + y)(x ^ 2 - xy - y ^ 2)

This option is incorrect because it uses (x + y) as the first factor, which is not appropriate for the difference of cubes. Additionally, the second factor does not align with the correct formula.

E) (x + y)(x ^ 2 - xy + y ^ 2)

This option is incorrect for similar reasons as option D. The use of (x + y) is inappropriate for the factorization of x³ - y³, and the second factor does not match the required form.

Conclusion

Option C is definitively correct as it correctly follows the difference of cubes formula for factoring x³ - y³. All other options fail to meet the necessary criteria either by misidentifying the first factor or by incorrectly stating the second factor, thus confirming that C is the only valid expression.