91Ó°ÊÓ

Factoring is a key component when simplifying rational expressions. Essentially, factoring means breaking down a number or an algebraic expression into its basic building blocks, or factors, which when multiplied together give the original number or expression.
For example, in the expression \(8n - 96\), both terms share a common factor, which is 8. So, you can factor 8 out from both terms: \[ 8n - 96 = 8(n - 12) \]
Similarly, the denominator \(3n - 36\) can be factored by pulling out the common factor, which is 3: \[ 3n - 36 = 3(n - 12) \]
Factoring is essentially about making expressions simpler so that you can easily recognize common factors between the numerator and the denominator.

When factoring expressions, finding the Greatest Common Factor (GCF) is crucial. The GCF is the largest number, or most extensive polynomial, that divides two or more numbers or terms without leaving a remainder.
For example, in the numerator \(8n - 96\), the GCF is 8. Both 8n and 96 can be divided by 8. Likewise, in the denominator \(3n - 36\), the GCF is 3.
Finding the GCF helps simplify each term in the polynomial by factoring the greatest shared factor out, like so: \[ \text{Numerator: } 8n - 96 = 8(n - 12) \ \text{Denominator: } 3n - 36 = 3(n - 12) \]
This step is crucial because it prepares the rational expression for further simplification.

A rational expression is a fraction where the numerator and the denominator are polynomials. Simplifying them is like simplifying numerical fractions but with some algebra involved.
Once you have factored both the numerator and the denominator, you can cancel out the common terms.
In the example: \[ \frac{8n - 96}{3n - 36} = \frac{8(n - 12)}{3(n - 12)} \]
The \(n - 12\) terms in both the numerator and the denominator cancel each other out, leaving you with: \[ \frac{8}{3} \]
The rational expression is now simplified. Always check for common factors to cancel out to get the simplest form of the expression.