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Factoring polynomials is a crucial skill in simplifying rational expressions. In algebra, factoring means breaking down a complex expression into simpler factors, which when multiplied together give the original expression. For instance, consider the polynomial expression \[ c^{2} + 18c + 81 \]. This can be factored as \[ (c + 9)(c + 9) = (c + 9)^2 \], making the expression easier to handle.

Factoring helps to reveal underlying structures and common factors. When simplifying rational expressions, we often deal with quadratic expressions, which can generally be factored into binomials. Remember that recognizing patterns like perfect squares and the difference of squares can expedite the factoring process.

For example, in the given exercise, recognizing that \[ c^{2} - 4c + 4 = (c - 2)(c - 2) = (c - 2)^2 \] allows us to transform the denominator and simplify the overall expression.

After factoring both the numerators and denominators, the next step in simplifying rational expressions is to cancel common factors. This involves identifying and eliminating identical factors present in both the numerator and the denominator.

For our example, after factoring the given rational expressions, we get:

\[ \frac{(c+9)^2}{(c-2)^2} \times \frac{(c-2)(c-3)}{(c+9)(c-3)} \]

Here, you can see that \[ (c+9) \], \[ (c-2) \], and \[ (c-3) \] are common in both the numerator and the denominator.

When we cancel these common factors, we essentially remove them from both the top and bottom of the fraction. This leaves us with:

\[ \frac{c+9}{c-2} \]

Cancelling helps in simplifying expressions by reducing them to their simplest form, making further calculations easier. Remember, we can only cancel factors, not terms that are added or subtracted.

Rational expressions are fractions where the numerator and the denominator are polynomials. Simplifying these expressions often involves factoring and cancelling common factors. Understanding rational expressions helps in manipulating and simplifying complex algebraic fractions.

For the given exercise involving rational expressions:

\[ \frac{c^{2}+18c+81}{c^{2}-4c+4} \times \frac{c^{2}-5c+6}{c^{2}+6c-27} \]

By factoring the polynomials and then canceling the common factors, we transform the complex expression into a simpler one:

\[ \frac{c + 9}{c - 2} \]

Simplifying rational expressions involves dealing with fractions that contain polynomials, requiring skills in both factoring and understanding polynomial functions.

The main goal is to make the expression as simple as possible while keeping the value unchanged. This process is critical in solving equations, graphing functions, and understanding more advanced algebra concepts.

Algebraic fractions are fractions where the numerator and/or the denominator are algebraic expressions. Simplifying algebraic fractions involves a combination of factoring and reducing these fractions to their simplest forms.

Consider the given exercise:

\[ \frac{c^{2}+18 c+81}{c^{2}-4 c+4} \times \frac{c^{2}-5 c+6}{c^{2}+6 c-27} \]

Through the process of factoring both the numerators and denominators, and then canceling common factors, we simplify this to:

\[ \frac{c + 9}{c - 2} \]

The primary goal is to simplify the algebraic fraction while maintaining its value. This requires patience and practice in identifying common factors and reducing fractions.

Simplifying algebraic fractions is an essential skill in algebra that applies to various mathematical problems, including solving rational equations and performing operations with fractions.