# Algebra II: Factoring

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#### 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) .