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When factoring polynomials, the first and often the most important step is finding the greatest common factor (GCF). The GCF is essentially the largest expression that can evenly divide each term of the polynomial. To find it, you need to look at all terms of the polynomial and identify the smallest power for each variable present across all the terms.

For example, let's consider the polynomial given in the exercise: \(r^{3} s^{6} t^{9} + r^{2} s^{2} t^{2}\). Here, you'll check the powers of each variable in each term:

• For the variable \(r\), the smallest exponent is 2. So, the GCF for \(r\) is \(r^{2}\).
• Similarly, for \(s\), it's \(s^{2}\) as it is the smallest exponent of \(s\) present in both terms.
• Finally, for \(t\), the smallest exponent is 2, giving us \(t^{2}\) for the GCF.

Together, these give the GCF \(r^{2}s^{2}t^{2}\), which you'll factor out from each term.

Polynomial division is a technique used to simplify polynomials by dividing each term by a common factor, often the GCF. It's a straightforward step that follows the determination of the GCF.

In our exercise, after determining the GCF as \(r^{2}s^{2}t^{2}\), you divide each term of the polynomial by this expression:

• For the term \(r^{3} s^{6} t^{9}\), dividing it by \(r^{2} s^{2} t^{2}\) results in \(rs^{4}t^{7}\).
• The second term \(r^{2} s^{2} t^{2}\), when divided by itself, naturally gives 1.

After performing these divisions, the polynomial becomes \(rs^{4}t^{7} + 1\). This simplification is crucial as it paves the way to easily factor the polynomial further, positioning you for the factorized form.

Algebraic expressions are a combination of numbers, variables, and mathematical operations. They form the foundation of algebra and are pivotal in developing equations, inequalities, and polynomial functions.

In the context of factoring, algebraic expressions like \(r^{3} s^{6} t^{9} + r^{2} s^{2} t^{2}\) are simplified by extracting shared factors, as shown in this exercise. By rewriting the expression into a product of a GCF and a simplified polynomial (like \(r^{2}s^{2}t^{2} (rs^{4}t^{7} + 1)\)), you gain a clearer understanding of its structure.

This form highlights how terms can be broken down, giving you insights into their components and relationships. Such an understanding is crucial for solving more complex equations and understanding the nature of polynomials in algebra.