In general, a **rational expression **or **algebraic expression** is a quotient of polynomials:

**Rational expression = p (x) / q (x)**

with q (x) ≠ 0. Normally these expressions need to be simplified, that is, transformed into a simpler equivalent expression.

**How to simplify a rational expression**

To simplify rational expressions we must follow the following steps:

- Factor numerator and denominator.
- Simplify.

**Example 1**: Simplify the following rational expression (x² + 8x + 12) / (x² - 36).

We factor the numerator and the denominator, remembering that (a² - 3b²) = (a + b) (a - b):

(x² + 8x + 12) / (x² - 36) = [(x - 6) (x - 2)] / [(x + 6) (x - 6)] =

Simplifying the previous expression:

= (x - 2) / (x + 6)

Final score:

(x² + 8x + 12) / (x² - 36) = (x - 2) / (x + 6)

Since (x - 2) and (x + 6) do not have any factors in common, this expression is irreducible and then it is in its **simplest expression**.

**Example 2**: Simplify the following rational expression (n³ - n) / (n² - 5n - 6).

We factor the numerator and the denominator:

(n³ - n) / (n² - 5n - 6) = n (n² - 1) / [(n - 6) (n - 1)] =

= [n (n - 1) (n + 1)] / [(n - 6) (n - 1)] =

Simplifying the previous expression:

= n (n + 1) / (n - 6)

Final score:

(n³ - n) / (n² - 5n - 6) = n (n + 1) / (n - 6)

Since n (n + 1) and (n - 6) do not have any factors in common, this expression is irreducible and then it is in its **simplest expression**.