Simplifying rational expressions

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:

  1. Factor numerator and denominator.
  2. 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.