91Ó°ÊÓ

Understanding how to factor polynomials is a critical skill in algebra. Simplified, factoring transforms a complex equation into a product of simpler ones, which can be further manipulated or solved. Let's look at the exercise \(\frac{9 a^{2}-9 a-4}{3 a-4}\). Factoring the numerator, \(9a^2-9a-4\), we search for two numbers that multiply to give us the product of the coefficient of \(a^2\) (which is 9) and the constant term (-4), while also adding up to the coefficient of \(a\) (which is -9).

In this case, the numbers are -12 and +1. When we split the middle term using these numbers and factor by grouping, we obtain \(3a-4\) and \(3a+1\) as the factors. The fact that one of the factors, \(3a-4\), matches the denominator is the keystone. This will allow for simplification in the next steps. Factoring is an invaluable tool when simplifying algebraic expressions or solving equations.

When we deal with rational expressions, we often seek to simplify them; that's reducing them to their lowest terms. In our exercise \(\frac{9 a^{2}-9 a-4}{3 a-4}\), after factoring, we have \(\frac{(3a-4)(3a+1)}{3 a-4}\). Simplification here is straightforward since the same expression \(3a-4\) appears in both the numerator and denominator, suggesting that they can cancel each other out.

To simplify, we divide both parts of the fraction by \(3a-4\), leaving us with \(3a+1\) as the simplified form of the rational expression. It's essential to note that for the cancellation to be valid, \(3a-4\) must not equal zero; this is because division by zero is undefined in mathematics—a vital rule to remember when simplifying rational expressions.

Algebraic division is much like numerical division, but instead of numbers, we're working with expressions involving variables. In the provided exercise, we are dividing a polynomial by a binomial. Once the polynomial is factored, and a common factor with the denominator is identified, the division process is similar to numerical division—the common factors cancel out.

This process is encapsulated in the expression \(\frac{(3a-4)(3a+1)}{3 a-4}\), where by dividing, we essentially 'cancel out' the common \(3a-4\) factor, resulting in the simplified form \(3a+1\). The knowledge of how to perform such algebraic divisions is extremely valuable, especially when solving higher-degree polynomial equations or simplifying complex rational expressions. Remember, always watch out for restrictions like not dividing by zero.