91Ó°ÊÓ

When simplifying rational expressions, the first step is to factor both the numerator and the denominator. Factoring involves breaking down the polynomial into simpler expressions, or factors, that multiply together to give the original polynomial.

In our example, the numerator is \(8x^2 - 10x - 3\). When factored, it becomes \( (4x + 1)(2x - 3) \). The denominator, \(8x^2 - 6x - 9\), factors to \( (4x + 3)(2x - 3) \).

By expressing the polynomials as a product of their factors, we set the stage for the next steps: identifying and canceling common factors to simplify the expression further.

In any fraction, the top part is called the numerator, and the bottom part is called the denominator. These terms are crucial when simplifying rational expressions.

In our specific problem, the entire rational expression is presented as a single fraction. Before simplifying it, we break down both the numerator and the denominator into their factorized forms: \[ \frac{ (4x + 1)(2x - 3)}{ (4x + 3)(2x - 3)} \]

Understanding this distinction helps in identifying common factors that exist in both the numerator and the denominator. This identification allows us to simplify the expression by canceling out these common factors, making the expression easier to understand and work with.

Once you have factored the numerator and the denominator, the next step in simplifying the rational expression is to cancel out any common factors that appear in both. These common factors are terms that are present both in the numerator and in the denominator.

In our exercise, we notice that \(2x-3\) is a common factor in both the numerator \( (4x + 1)(2x - 3) \) and the denominator \( (4x + 3)(2x - 3) \). By canceling out \(2x - 3\), we simplify the original expression:

\[ \frac{ (4x + 1)(2x - 3)}{ (4x + 3)(2x - 3)} \rightarrow \frac{4x + 1}{4x + 3} \]

Always ensure that the factors you cancel out are identical and appear in both the numerator and the denominator; otherwise, you could end up altering the expression incorrectly.