# Division of rational expressions

Let p (x) and q (x) be two polynomials, with q (x) ≠ 0, then the quotient p (x) / q (x) is a rational expression or algebraic expression. Now let p (x) / q (x) and r (x) / s (x) be two rational expressions with q (x) ≠ 0 and s (x) ≠ 0, then:

[p (x) / q (x)] ÷ [r (x) / s (x)] = p (x) ∙ s (x) / q (x) ∙ (r (x)

In general, to divide rational expressions, we recommend following these steps:

## Steps to divide rational expressions

1. Multiply the numerator of the first term with the denominator of the second and place that result in the numerator after equality.
2. Multiply the denominator of the first term by the numerator of the second and place the result in the denominator after equality.
3. Factor the terms of the numerator and denominator, and then simplify. Example 1: Solve the following division of rational expressions [(x² - 9) / (6x + 18)] ÷ [(x - 3) / 6].

We multiply the numerator of the first term with the denominator of the second:

(x² - 9) × 6 = 6 (x² - 9)

This result will be the numerator of our final rational expression:

[(x² - 9) / (6x + 18)] ÷ [(x - 3) / 6] = 6 (x² - 9) / ¿?

We multiply the denominator of the first term with the numerator of the second:

(6x + 18) × (x - 3) = (6x + 18) (x - 3)

This result will be the numerator of our final rational expression:

[(x² - 9) / (6x + 18)] ÷ [(x - 3) / 6] = 6 (x² - 9) / [(6x + 18) (x - 3)]

We factor the terms of the numerator and denominator:

Since (a² - b²) = (a + b) (a - b), we have:

= 6 (x² - 9) / [(6x + 18) (x - 3)] = [6 (x + 3) (x - 3)] / [(6x + 18) (x - 3)] = 1

Final score:

[(x² - 9) / (6x + 18)] ÷ [(x - 3) / 6] = 1

Example 2: Solve the following division of rational expressions [(x² + 3x) / (x² + 2x - 3) ÷ [x / (x + 1)].

Steps 1 and 2:

[(x² + 3x) / (x² + 2x - 3) ÷ [x / (x + 1)] = [(x² + 3x) (x + 1)] / [x (x² + 2x - 3)] =

Step 3:

= [x (x + 3) (x + 1)] / [x (x + 3) (x - 1)] = (x + 1) / (x - 1)

Final score:

[(x² + 3x) / (x² + 2x - 3) ÷ [x / (x + 1)] = (x + 1) / (x - 1)

Example 3: Solve the following division of rational expressions (n + 1) ÷ [(n² + 4n + 3) / 5].

Step 1 and 2:

(n + 1) ÷ [(n² + 4n + 3) / 5] = 5 (n + 1) / (n² + 4n + 3) =

Step 3:

= (5) (n + 1) / [(n + 1) (n + 3)] = 5 / (n + 3)

Final score:

(n + 1) ÷ [(n² + 4n + 3) / 5] = 5 / (n + 3)