The perfect cube forms (x+y)3 and (x−y)3 come up a lot in algebra. We will go over how to expand them in the examples below, but you should also take some time to store these forms in memory, since you'll see them often:
(x+y)3(x−y)3=x3+3x2y+3xy2+y3=x3−3x2y+3xy2−y3.
We will also look at the sum and difference of cubes, and see various ways in which we can apply them:
We recognize that this is the sum of x3 and 23. Hence, by the sum of cubes factorization, we obtain
x3+8=(x+2)(x2−2x+22)=(x+2)(x2−2x+4).□
False
True
True or False?
x3−y3=(x−y)(x2−xy+y2)
Challenging Examples
What is (x+y)3+(x−y)3?
Solution 1: If we look at the expansion of these terms, we see that the second and fourth terms will cancel out, and the first and third terms will be combined. Thus, we obtain
(x+y)3+(x−y)3=2x3+6xy2.□
Solution 2: If we treat this as the sum of 2 cubes, we will obtain