Assessment and Learning in Knowledge Spaces (ALEKS) Practice Exam

Which factorization represents the difference of cubes?

1. (a-b)(a²+ab+b²)

2. (a+b)(a²-ab+b²)

3. (a-b)(a+b)

4. (a+b)(c+d)

The correct answer is: (a-b)(a²+ab+b²)

The factorization that represents the difference of cubes is accurately reflected in the first option, which is \( (a-b)(a^2 + ab + b^2) \). This expression is derived from the general formula for the difference of cubes, which is stated as: \[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \] In this formula, \( a \) and \( b \) are the bases of the cubes, and the factor \( (a - b) \) indicates the difference between those two bases. The second factor, \( (a^2 + ab + b^2) \), is a quadratic polynomial that contains the squares of both bases, as well as their product. The other options do not satisfy the criteria for the difference of cubes. The second option, for example, represents the sum of cubes due to the presence of \( (a + b) \) and the accompanying polynomial. The third option simplifies to a product but does not fulfill the requirements to express a difference of cubes. The fourth option is unrelated, involving arbitrary linear terms and not reflecting any cubic relationship. Thus, the first option is indeed the correct representation of the