Final answer:
An example of a GCF problem involving the difference of cubes is finding the GCF of 8x^3 - 27 and 12x^3 - 54. By factoring both expressions using the difference of cubes formula, we establish that the GCF is 2(2x - 3).
Step-by-step explanation:
An example of a Greatest Common Factor (GCF) problem involving the difference of cubes can be:
Find the GCF of 8x3 - 27 and 12x3 - 54.
First, factor each expression using the difference of cubes formula, a3 - b3 = (a - b)(a2 + ab + b2).
For 8x3 - 27, a = 2x and b = 3, so we have:
(2x - 3)((2x)2 + (2x)(3) + 32) =
(2x - 3)(4x2 + 6x + 9).
For 12x3 - 54, a = 2 × (2x) and b = 2 × 3, so we have:
(2 × (2x - 3))((2x)2 + 2(2x)(3) + (2 × 3)2) = 2(2x - 3)(4x2 + 12x + 18).
The GCF of the two expressions is (2x - 3), which appears in both factored expressions.
Thus, the GCF of 8x3 - 27 and 12x3 - 54 is 2(2x - 3).