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Factoring is a technique often used to simplify algebraic expressions. It involves breaking down an expression into simpler terms or factors that can be multiplied together to get the original expression. Some common factoring techniques include finding common factors, factoring by grouping, and using special patterns such as the difference of squares.

For instance, consider the expression in the exercise, "\(3 - 9\sqrt{2}\)". Both terms in the expression have the common factor of 3. By factoring out the 3, we simplify the terms inside the parentheses, creating "3(1 - 3\sqrt{2})".

This step is key because factoring simplifies further processes such as solving equations or simplifying expressions like rational expressions.

Rational expressions are similar to fractions but with polynomials in the numerator, the denominator, or both. In any rational expression, simplifying is crucial to ease operations like addition, subtraction, multiplication, and division.

In the solution, once we've factored the numerator, we obtain a rational expression of the form \(\frac{3(1 - 3\sqrt{2})}{6}\). This simplification allows us to see and remove common factors from both the numerator and the denominator efficiently.

Keeping rational expressions in their simplest form is essential. This helps prevent mistakes during further calculations and gives the result in the most reduced and comprehensible form possible.

Identifying and canceling out common factors is a crucial step in simplifying expressions. A common factor is a factor that is shared by all terms of an expression or a numerator and denominator in a fraction.

In our example, the numerator "3(1 - 3\sqrt{2})" and the denominator "6" both share the common factor of 3. By canceling this factor out, the expression simplifies from \(\frac{3(1 - 3\sqrt{2})}{6}\) to \(\frac{1 - 3\sqrt{2}}{2}\).

Recognizing common factors can greatly ease the simplification process, reducing complexity and making expressions more manageable. Clean mathematical expressions are easier to interpret and work with in various applications.