Expand and simplify the following:[math: (2x−3)3] (2x-3)^3

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BlackTom AI · Accepted Answer

Let's parse the task carefully: expand and simplify (2x − 3)^3, and compare to the given options.
Option a: 8x^3 + 36x^2 − 54x + 27. This has the correct 8x^3 leading term, but the signs of the middle terms are all wrong compared to the actual expansion, which should alternate signs starting with a negative x^2 term. Specifically, the x^2 term should be negative, not positive, and the constant term should be negative, not positive.
Option b: 8x^3 − 27. This only includes the cube terms from the expansion of a binomial and omits the x^2 and x terms entirely. Since (2x − 3)^3 generates three additional terms beyond 8x^3, this option is incomplete and therefore incorrect.
Option c: 8x^3 − 12x^2 + 18x − 27. Here the x^2 term is −12x^2 instead of the correct −36x^2, and the x term is +18x instead of +54x. The coefficients are not consistent with the binomial expansion formula (a − b)^3 = a^3 − 3a^2b + 3ab^2 − b^3 when a = 2x and b = 3, so this option misapplies the coefficients.
Option d: 8x^3 − 36x^2 + 54x − 27. This matches the binomial expansion exactly: a^3 = (2x)^3 = 8x^3, −3a^2b = −3(2x)^2(3) = −3(4x^2)(3) = −36x^2, +3ab^2 = 3(2x)(3^2) = 3(2x)(9) = 54x, and −b^3 = −3^3 = −27. All terms have the correct coefficients and signs.
In summary, options a, b, and c each contain specific inaccuracies—the wrong signs or missing terms or incorrect coefficients—whereas option d faithfully reflects the correct binomial expansion of (2x − 3)^3.

Expand and simplify the following:[math: (2x−3)3] (2x-3)^3

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