### Factoring a3 - b3

An expression of the form a3 - b3 is called a difference of cubes. The factored form of a3 - b3 is (a - b)(a2 + ab + b2):

(a - b)(a2 + ab + b2) = a3 - a2b + a2b - ab2 + ab2 - b3 = a3 - b3

For example, the factored form of 27x3 - 8 (a = 3x, b = 2) is (3x - 2)(9x2 + 6x + 4).

Similarly, the factored form of 125x3 -27y3 (a = 5x, b = 3y) is (5x - 3y)(25x2 +15xy + 9y2).

To factor a difference of cubes, find a and b and plug them into (a - b)(a2 + ab + b2).

### Factoring a3 + b3

An expression of the form a3 + b3 is called a sum of cubes. The factored form of a3 + b3 is (a + b)(a2 - ab + b2):

(a + b)(a2 - ab + b2) = a3 + a2b - a2b - ab2 + ab2 + b3 = a3 - b3.

For example, the factored form of 64x3 + 125 (a = 4x, b = 5) is (4x + 5)(16x2 - 20x + 25).

Similarly, the factored form of 343x3 + y3 (a = 7x, b = y) is (7x + y)(49x2 -7xy + y2).

To factor a sum of cubes, find a and b and plug them into (a + b)(a2 - ab + b2).

ax3 + bx2 + cx + d can be easily factored if = First, group the terms: (ax3 + bx2) + (cx + d ). Next, factor x2 out of the first group of terms: x2(ax + b) + (cx + d ). Factor the constants out of both groups. This should leave an expression of the form d1x2(ex + f )+ d2(ex + f ). We can add these two terms by adding their "coefficients": (d1x2 + d2)(ex + f ).