Sum and Difference of Cubes: Formulas, the SOAP Trick and Examples
Learn the sum and difference of cubes formulas, remember the signs with SOAP, and factor two worked examples step by step.
The two patterns
- a³ + b³ = (a + b)(a² − ab + b²)
- a³ − b³ = (a − b)(a² + ab + b²)
Spot them: two perfect cubes joined by a plus or a minus.
Remember it with soap
- a³ + b³ = (a + b)(a² − ab + b²)
- SSame: the binomial keeps the original sign
- OOpposite: the trinomial's middle term flips it
- APAlways Positive: the trinomial's last term is +
Example 1
Factor x³ + 27
- Rewrite: x³ + 3³
- So a = x and b = 3
(x + 3)(x² − 3x + 9)
Example 2
Factor 8x³ − 27
- Rewrite: (2x)³ − 3³
- So a = 2x and b = 3
(2x − 3)(4x² + 6x + 9)
Common mistake
a³ + b³ = (a + b)³
(a + b)³ expands to four terms, so it is not the same thing.
The trinomial factor does not factor any further. Stop there.
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