Factor the difference of two cubes with the pattern (a^3 - b^3) = (a - b)(a^2 + ab + b^2). For example, to factor K^3 - 125 let a = K and b = 5 and applying the pattern, K^3 - 125 = (K - 5)(K^2 -5K + 25). Another example is factoring 8N^3 = M^6. Here, a = 2N and b = M^2, and 8N^3 = M^6 = (2N - M^2)(4N^2 -2NM^2 + M^4). If the binomial cannot be put into the form (a^3 - b^3), this pattern cannot be applied. In these cases K^3 - 125 = K^3 - 5^3 and 8N^3 = M^6 = (2N)^3 - (M^2)^3.
Use the pattern (a^3 + b^3) = (a + b)(a^2 - ab + b^2) to factor the sum of two cubes. For an example of how useful the sum and difference of cubes patterns can be, consider the problem (X + Y) / (X^(1/3) + Y^(1/3)). This difficult looking problem becomes easy if you use the sum of cubes pattern. Let a = X^(1/3) and b = Y^(1/3). The solution is (X + Y) / (X^(1/3) + Y^(1/3)) = (X^(1/3) + Y^(1/3))(X^(2/3) - (XY)^(1/3) + Y^(2/3)) / (X^(1/3) + Y^(1/3)) = (X^(2/3) - (XY)^(1/3) + Y^(2/3)).
Generalize the sum or difference of two cubes with the pattern (a^3 sign b^3) = (a sign b)(a^2 opposite sign ab + b^2). If you substitute -b for b in either pattern, you get the other pattern. (a^3 + (-b)^3) = (a + (-b))(a^2 - a(-b) + (-b)^2) = (a - b)(a^2 + ab + b^2), but (a^3 + (-b)^3) = (a^3 - b^3) so (a^3 - b^3) = (a - b)(a^2 + ab + b^2). Also (a^3 - (-b)^3) = (a - (-b))(a^2 + a(-b) + (-b)^2) = (a + b)(a^2 - ab + b^2), but (a^3 - (-b)^3) = (a^3 + b^3) so (a^3 + b^3) = (a + b)(a^2 - ab + b^2).