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When it comes to polynomials, the difference of cubes is a handy concept that enables us to factor certain expressions. A difference of cubes has the form \(a^3 - b^3\), where both terms are perfect cubes. This expression can be factored using the formula:

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

Applying this to our original problem \(a^3 - 27\), we identify that 27 is the perfect cube \(3^3\). Therefore, the expression becomes \(a^3 - 3^3\), which allows us to factor it as:

• \((a - 3)(a^2 + 3a + 9)\)

Understanding the difference of cubes makes factoring easier and paves the way for further simplifications in polynomial algebra.

Factoring polynomials is the process of breaking down a complex polynomial into simpler 'factor' polynomials that can be multiplied together to give the original polynomial. This process is crucial in simplifying polynomial expressions or solving polynomial equations.

In the given problem, after recognizing our polynomial as a difference of cubes, we used the special formula to factor the numerator, \(a^3 - 27\), into the product \((a - 3)(a^2 + 3a + 9)\).

Here, identifying and extracting the common factor (\(a - 3\)) helped not only in simplifying but also in making further calculations manageable. Factoring reduces the risk of errors by simplifying complex expressions piece by piece.

• This method is used extensively in algebra to solve equations and to simplify expressions, making it a foundational skill for students.

A rational expression is essentially a ratio of two polynomials. Simplifying these expressions involves reducing them to their simplest form by canceling common factors from the numerator and the denominator.

For our exercise, after factoring the numerator \(a^3 - 27\) into \((a - 3)(a^2 + 3a + 9)\), we notice that \(a - 3\) is common to both the numerator and the denominator, which is \(a - 3\).

• We cancel the common factor \(a - 3\) from both top and bottom, simplifying the expression to just \(a^2 + 3a + 9\).

Simplifying rational expressions in this way streamlines algebraic manipulations and helps to clearly see the underlying parts of an expression. Not only does this help in finding solutions faster, but it also strengthens the understanding of the relationships between algebraic terms.