How do you divide rational functions?

Dividing rational functions is similar to dividing regular fractions: you multiply by the reciprocal of the second fraction. Here's a breakdown:

Steps to Divide Rational Functions

1. Rewrite as Multiplication: Change the division problem into a multiplication problem by inverting (flipping) the second rational function. This means the numerator becomes the denominator and the denominator becomes the numerator.

2. Factor: Factor all numerators and denominators completely. Factoring allows you to identify common factors that can be canceled.

3. Simplify (Cancel): Look for common factors in the numerators and denominators. Cancel out any common factors. A factor can be canceled from any numerator to any denominator.

4. Multiply: Multiply the remaining numerators together and the remaining denominators together.

5. Simplify (if possible): Ensure the resulting rational function is simplified as much as possible.

Example

Let's illustrate with an example (not from the provided reference, but demonstrative):

(x² - 4) / (x + 1) ÷ (x - 2) / (x + 1)

1. Rewrite as Multiplication:

   (x² - 4) / (x + 1) * (x + 1) / (x - 2)

2. Factor:

   ((x + 2)(x - 2)) / (x + 1) * (x + 1) / (x - 2)

3. Simplify (Cancel):

   ((x + 2)(x - 2)) / (x + 1) * (x + 1) / (x - 2) -> (x + 2) / 1 * 1 / 1

4. Multiply:

   (x + 2)

Therefore, (x² - 4) / (x + 1) ÷ (x - 2) / (x + 1) = x + 2

From the Provided Reference

The reference provided includes an example where, after performing the division, the result is a polynomial. It mentions "2x squared minus x plus 3" being an answer after dividing rational expressions. This implies that after simplifying (canceling common factors), the result can be a simple polynomial rather than a complex rational function. An important step to simplifying rational expressions is to factor all parts of the problem and then reduce common expressions.