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Factoring polynomials involves breaking down a complex expression into simpler factors that can be multiplied together to get the original polynomial. This process is crucial for simplifying algebraic expressions, especially when dealing with polynomials in the numerator and denominator.

In our exercise, we start by factoring both the numerator and the denominator.

To factor the numerator, we look for two numbers that multiply to the constant term (64) and add to the coefficient of the middle term (-48). Factoring the numerator \( 9n^2 - 48n + 64 \) gives us:

\( 9n^2 - 48n + 64 = (3n - 8)(3n - 8) = (3n - 8)^2 \)

The denominator can be factored as a difference of squares. The difference of squares formula is \(a^2 - b^2 = (a - b)(a + b) \). For the denominator \( 9n^2 - 64 \), we get:

\( 9n^2 - 64 = (3n)^2 - 8^2 = (3n - 8)(3n + 8) \)

Once both the numerator and denominator are factored, we can simplify the expression.

In any fraction, including algebraic fractions, the numerator is the part of the fraction above the line, and the denominator is the part below the line. Understanding these terms is essential when simplifying complex fractions.

For example, in our exercise:

\( \frac{9n^2 - 48n + 64}{9n^2 - 64} \)

The numerator is \( 9n^2 - 48n + 64 \) and the denominator is \( 9n^2 - 64 \).

After factoring, we see that:

• The numerator becomes \( (3n - 8)^2 \)
• The denominator becomes \( (3n - 8)(3n + 8) \)

These factored forms make it easier to identify common factors, which leads to simplification.

An algebraic fraction is a fraction where the numerator and/or the denominator are algebraic expressions (expressions involving variables as well as constants). Simplifying these fractions involves factoring and canceling common factors.

Take our exercise for example:

After factoring, we have:

\( \frac{(3n - 8)^2}{(3n - 8)(3n + 8)} \)

We notice that both the numerator and the denominator share a common factor of \( 3n - 8 \). Canceling out the common factor from the numerator and the denominator simplifies the fraction to:

\( \frac{3n - 8}{3n + 8} \)

Thus, the simplified form of the given algebraic fraction is \( \frac{3n - 8}{3n + 8} \).

Understanding this process can make working with algebraic fractions much easier and is useful in solving various algebraic problems.