Assessment and Learning in Knowledge Spaces (ALEKS) Practice Exam

What is the factored form of a³ + b³?

1. (a+b)(a²-ab+b²)

2. (a-b)(a²+ab+b²)

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

4. (y₂-y₁)/(x₂-x₁)

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

To factor the expression \(a^3 + b^3\), you can apply the sum of cubes formula. The sum of cubes states that for any two terms \(x\) and \(y\), the expression \(x^3 + y^3\) can be factored as: \[ x^3 + y^3 = (x + y)(x^2 - xy + y^2). \] In this case, let \(x = a\) and \(y = b\). Applying the sum of cubes formula gives: \[ a^3 + b^3 = (a + b)(a^2 - ab + b^2). \] The first part of the factorization \( (a + b) \) indicates that when the cubes \(a^3\) and \(b^3\) are summed, they can be grouped based on their linear factors, leading to \( a + b \). The second part \( (a^2 - ab + b^2) \) is a quadratic expression representing the remaining factors after factoring out \( (a + b) \). This expression ensures that the factors will return to the original cubic form when multiplied out. Thus, the