Which of the following is equivalent to x³-y³?
(x - y)(x ^ 2 + xy + y ^ 2)
The expression (x^3 - y^3) can be factored using the difference of cubes formula, which states that (a^3 - b^3 = (a - b)(a^2 + ab + b^2)). Here, substituting (a) with (x) and (b) with (y) results in the correct factorization being ((x - y)(x^2 + xy + y^2)).
This option incorrectly applies the difference of cubes formula. The second factor should include (+xy) instead of (-xy). Therefore, although it starts with the correct ((x - y)) term, the second part does not correspond with the proper factorization of (x^3 - y^3).
While this option maintains the correct ((x - y)) factor, the second factor is also incorrect. The term (+y^2) should be (+xy) to accurately reflect the factorization of the difference of cubes. Thus, it does not equate to (x^3 - y^3).
This choice correctly represents the factorization of (x^3 - y^3) according to the difference of cubes formula. The structure ((x - y)) followed by ((x^2 + xy + y^2)) confirms the proper components of the formula, making it the equivalent expression.
This option incorrectly starts with ((x + y)) instead of ((x - y)), which is essential for the difference of cubes. Moreover, the second factor does not match the correct form. This misalignment renders it a non-equivalent expression for (x^3 - y^3).
Similar to option D, this choice incorrectly uses ((x + y)) instead of the required ((x - y)). Even though it has a polynomial structure, it cannot represent (x^3 - y^3) accurately due to both the incorrect leading term and the wrong sign in the last part of the second factor.
The expression (x^3 - y^3) can be effectively factored into (x - y)(x ^ 2 + xy + y ^ 2)
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