# Algebra II: Factoring

### Factoring Polynomials of Degree 3

#### Factoring a 3 - b 3

An expression of the form a 3 - b 3 is called a difference of cubes. The factored form of a 3 - b 3 is (a - b)(a 2 + ab + b 2) :

(a - b)(a 2 + ab + b 2) = a 3 - a 2 b + a 2 b - ab 2 + ab 2 - b 3 = a 3 - b 3

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

Similarly, the factored form of 125x 3 -27y 3 ( a = 5x, b = 3y ) is (5x - 3y)(25x 2 +15xy + 9y 2) .

To factor a difference of cubes, find a and b and plug them into (a - b)(a 2 + ab + b 2) .

#### Factoring a 3 + b 3

An expression of the form a 3 + b 3 is called a sum of cubes. The factored form of a 3 + b 3 is (a + b)(a 2 - ab + b 2) :

(a + b)(a 2 - ab + b 2) = a 3 + a 2 b - a 2 b - ab 2 + ab 2 + b 3 = a 3 - b 3 .

For example, the factored form of 64x 3 + 125 ( a = 4x, b = 5 ) is (4x + 5)(16x 2 - 20x + 25) .

Similarly, the factored form of 343x 3 + y 3 ( a = 7x, b = y ) is (7x + y)(49x 2 -7xy + y 2) .

To factor a sum of cubes, find a and b and plug them into (a + b)(a 2 - ab + b 2) .

#### Factoring ax 3 + bx 2 + cx + d

ax 3 + bx 2 + cx + d can be easily factored if = First, group the terms: (ax 3 + bx 2) + (cx + d ) . Next, factor x 2 out of the first group of terms: x 2(ax + b) + (cx + d ) . Factor the constants out of both groups. This should leave an expression of the form d 1 x 2(ex + f )+ d 2(ex + f ) . We can add these two terms by adding their "coefficients": (d 1 x 2 + d 2)(ex + f ) .

Example 1: Factor 3x 3 +6x 2 + 4x + 8 .

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

Example 2: Factor 45x 3 +18x 2 - 5x - 2 .

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

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