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):

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).

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):

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).

You can remember these two factored forms by remembering that the sign in the binomial is always the same as the sign in the original expression, the first sign in the trinomial is the opposite of the sign in the original expression, and the second sign in the trinomial is always a plus sign.

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 ).

Example 1: Factor 3x³ +6x² + 4x + 8.

3x³ +6x² -4x - 8 = (3x³ +6x²) + (4x + 8) = 3x²(x + 2) + 4(x + 2) = (3x² + 4)(x + 2).

Example 2: Factor 45x³ +18x² - 5x - 2.

45x³ +18x² -5x - 2 = (45x³ +18x²) - (5x + 2) = 9x²(5x + 2) - 1(5x + 2) = (9x² - 1)(5x + 2).
This can be further factored as a difference of squares: (9x² - 1)(5x + 2) = (3x + 1)(3x - 1)(5x + 2).