91Ó°ÊÓ

To grasp how to simplify rational expressions, envision a fraction with polynomials in both its numerator and denominator. The goal is to reduce this complex fraction into a simpler form by eliminating common factors. When simplifying, first factor each polynomial.

For the given exercise:
\[ \frac{x^2+9x+18}{x^2+6x+8} \div \frac{x^2-3x-18}{x^2+2x-8} \], both the numerators and denominators of each fraction can be factored into binomials. Factoring is a crucial step because it reveals common factors between the numerators and denominators that can be simplified by cancellation.

After factoring, we get:
\[ \frac{(x+6)(x+3)}{(x+4)(x+2)} \div \frac{(x-6)(x+3)}{(x-4)(x+2)} \].

Look for any common binomial factors between the numerators and denominators across the entire expression, because these will cancel each other out, making the rational expression simpler.

Under the umbrella of algebra, factoring polynomials is like breaking down a number into its prime factors, but instead, we break down expressions into the product of simpler expressions. In the context of simplifying rational expressions, factoring polynomials paves the way for cancellation and, thus, simplification.

For instance, take the polynomial \(x^2 + bx + c\). If there exist two numbers, say \(m\) and \(n\), such that \(m+n=b\) and \(m \times n=c\), then the polynomial can be factored into \((x+m)(x+n)\).

Using this approach for our exercise: you factor \(x^2+9x+18\) by finding two numbers that add up to 9 and multiply to 18, ending up with \(x+3\) and \(x+6\). Factoring is a foundational skill in algebra, vital for mastering rational expressions.

Understanding the multiplication of reciprocals is pivotal when dividing rational expressions. Reciprocals are essentially two numbers whose product is 1, such as 4 and \(\frac{1}{4}\). In the context of fractions, the reciprocal of a fraction \(\frac{a}{b}\) is \(\frac{b}{a}\).

When dividing fractions, you're essentially multiplying by a reciprocal. Returning to our exercise, we flip the second fraction to turn division into multiplication:
\[ \frac{(x+6)(x+3)}{(x+4)(x+2)} \times \frac{(x-4)(x+2)}{(x-6)(x+3)} \].

The concept of reciprocals makes division by fractions straightforward. After this step, look for any terms that appear in both a numerator and a denominator to further simplify through cancellation.