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The difference of cubes is a special algebraic expression. It involves subtracting one cubed term from another. In simple terms, it's like taking away a three-dimensional block from another. The general form is:

\[a^3 - b^3 \]

The good news is that there's a standard formula to factor these types of expressions: \[a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]

Whenever you come across a difference of cubes, you can directly apply this formula. It breaks the expression into a simpler form that's easier to work with. Notice how it always results in one binomial \[ (a - b) \] and one trinomial \[(a^2 + ab + b^2) \].

Factoring algebraic expressions means breaking down a complex expression into simpler parts (factors). Think of it like splitting a piece of bread into manageable slices.

To factor the given expression \(\frac{a^3 - b^3}{a - b} \), identify the type of algebraic expression. Here, the numerator \(a^3 - b^3 \) is a difference of cubes. Using the formula from the previous section, we can factor it:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]

So the rational expression becomes:
\[\frac{(a - b)(a^2 + ab + b^2)}{a - b} \]

Notice how \((a - b) \) appears in both the numerator and the denominator. They can cancel each other out, simplifying the expression.

Simplifying fractions in algebra is similar to simplifying regular fractions. When you have a term in both the numerator and denominator, you can cancel them out.

In our example: \[\frac{(a - b)(a^2 + ab + b^2)}{a - b} \]

The factor \((a - b)\) cancels out because it is present in both parts of the fraction, as long as \(a eq b\). What remains is:
\[a^2 + ab + b^2 \]

This is the simplest form of the original rational expression. Learning to identify common terms and cancel them is a key part of simplifying fractions in algebra.

Always remember to check that the values you cancel do not make the original expression undefined. In this case, cancelling \(a - b \) is valid as long as \(a eq b \).