How do you factor a^3 - b^3?

To factor \(a^3 - b^3\), we use the difference of cubes formula:

\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]

Here is the step-by-step explanation:

1. **Recognize the pattern**: The expression \(a^3 - b^3\) is a difference of cubes since both \(a^3\) and \(b^3\) are perfect cubes.

2. **Apply the formula**: The difference of cubes formula states that:
\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]
- The first factor, \(a - b\), represents the difference of the cube roots.
- The second factor, \(a^2 + ab + b^2\), is a trinomial that cannot be factored further over the real numbers.

3. **Verify the factorization**: Expand \((a - b)(a^2 + ab + b^2)\) to confirm:
\[
(a - b)(a^2 + ab + b^2) = a(a^2 + ab + b^2) - b(a^2 + ab + b^2)
\]
Expanding both terms:
\[
a(a^2) + a(ab) + a(b^2) - b(a^2) - b(ab) - b(b^2)
\]
Combine like terms:
\[
a^3 + a^2b + ab^2 - a^2b - ab^2 - b^3
\]
Simplify:
\[
a^3 - b^3
\]
This confirms the factorization is correct.

**Final Answer**:
The factorization of \(a^3 - b^3\) is:
\[
(a - b)(a^2 + ab + b^2)
\]