4. The Sum and Difference of Cubes

We came across these before:

[Sum of two cubes]: x3 + y3 = (x + y)(x2 − xy + y2)

[Difference of 2 cubes]: x3 − y3 = (x − y)(x2 + xy + y2)

We can use these to factor expressions involving cubes, as in the following examples.

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Example

Factor 64x3 + 125

Answer:

We use the Sum of 2 Cubes formula above.

64x3 + 125

= (4x)3 + (5)3

= (4x + 5)[(4x)2 − (4x)(5) + (5)2]

= (4x + 5)(16x2 − 20x + 25)

Exercises

Factor:

(1) x3 + 27

(2) 3m3 − 81

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