Understanding the Expansion of (a  b)(a² + ab + b²)
The expression (a  b)(a² + ab + b²) is a classic example of a difference of cubes factorization. This particular pattern appears frequently in algebra and can be expanded using a simple rule.
The Difference of Cubes Formula
The general formula for the difference of cubes is:
a³  b³ = (a  b)(a² + ab + b²)
This formula shows that the difference of two cubes can be factored into a binomial (a  b) and a trinomial (a² + ab + b²).
Expanding the Expression
To expand (a  b)(a² + ab + b²), we simply apply the distributive property:

Multiply the first term of the binomial (a) by each term in the trinomial:
 a * a² = a³
 a * ab = a²b
 a * b² = ab²

Multiply the second term of the binomial (b) by each term in the trinomial:
 b * a² = a²b
 b * ab = ab²
 b * b² = b³

Combine the resulting terms:
 a³ + a²b + ab²  a²b  ab²  b³ = a³  b³
Conclusion
Therefore, expanding (a  b)(a² + ab + b²) using the difference of cubes formula results in a³  b³. This pattern is crucial for simplifying expressions and solving equations involving the difference of two cubes. Understanding this factorization will prove invaluable for various mathematical operations and problemsolving situations.