Final answer:
The correct long division problem to prove the formula for factoring the difference of two perfect cubes is option c) a^3 +0a^2b+0ab^2 - b^3. The formula for factoring the difference of two perfect cubes is (a - b) = (a^2 + ab + b^2), where a represents the cube root of the first term, and b represents the cube root of the second term. By substituting the values according to the formula, you can verify that it is correct.
Step-by-step explanation:
The correct long division problem that can be used to prove the formula for factoring the difference of two perfect cubes is option c) a^3 +0a^2b+0ab^2 - b^3.
To factor the difference of two perfect cubes, you can use the formula (a - b) = (a^2 + ab + b^2). In this case, a represents the cube root of the first term, and b represents the cube root of the second term. By substituting the values according to the formula, you can verify that it is correct.
For example, if a = 3 and b = 2, the equation becomes (3 - 2) = (3^2 + 3(3)(2) + 2^2), which simplifies to 1 = 9 + 18 + 4, and the equation holds true.