What are Sum and Difference of Cubes?
Expressions of the form a³ + b³ (sum of cubes) and a³ - b³ (difference of cubes) can be factorised using special formulas.
Sum of Cubes Formula:
\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\]
Difference of Cubes Formula:
\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]
How to Remember the Formulas:
| Formula | Signs in Factors |
|---|---|
| a³ - b³ | (a - b)(a² + ab + b²) |
Memory Trick (SOAP):
Same sign, Opposite sign, Always Positive
- For a³ + b³: (a + b)(a² - ab + b²)
- For a³ - b³: (a - b)(a² + ab + b²)
Recognizing Cubes:
| Number | Cube | Variable | Cube |
|---|---|---|---|
| 2 | 8 | x² | x⁶ |
| 3 | 27 | x³ | x⁹ |
| 4 | 64 | x⁴ | x¹² |
| 5 | 125 | ||
| 6 | 216 | ||
| 7 | 343 | ||
| 8 | 512 | ||
| 9 | 729 | ||
| 10 | 1000 |