# Rational expressions If p (x) and q (x) are two polynomials, with q (x) ≠ 0, then the quotient: p (x) / q (x) is called rational expression.

A rational expression or algebraic expression is a quotient of polynomials.

## Simplifying rational expressions

1. Factor each of the two polynomials p (x) and q (x).
2. Find the greatest common factor (GCF) of p (x) and q (x).
3. If MCD equals one, then the given rational expression p (x) / q (x) is in simplest form.
4. If the GCF is different from one, then the numerator p (x) and the denominator q (x) must be divided by the GCF of p (x) and q (x).
5. The rational expression p (x) / q (x) obtained in 3 or 4 is in its simplest form.

Example: Simplify the following rational expression to its simplest expression (x2 - 5x - 6) / (x2 + 3x + 2)

Let: p (x) = x2 - 5x - 6 = (x - 6) · (x + 1) and q (x) = x2 + 3x + 2 = (x + 2) · (x + 1).

The greatest common factor of both is GCD = (x + 1).

(x2 - 5x - 6) / (x2 + 3x + 2) = [(x - 6) · (x + 1)] / [(x + 2) · (x + 1)] = (x - 6) / (x + 2)

## Adding and subtracting rational expressions

Let A = p (x) / q (x) and B = r (x) / s (x) be two rational expressions with q (x) ≠ 0 and s (x) ≠ 0, then:

• A + B = [p (x) ∙ s (x) + r (x) ∙ q (x)] / [q (x) ∙ s (x)]
• A - B = [p (x) ∙ s (x) - r (x) ∙ q (x)] / [q (x) ∙ s (x)]

## Multiplication and division of rational expressions

Let C = p (x) / q (x) and D = r (x) / s (x) be two rational expressions with q (x) ≠ 0 and s (x) ≠ 0, then:

• C × D = [p (x) ∙ r (x)] / [q (x) ∙ s (x)]
• C ÷ D = [p (x) ∙ s (x)] / [q (x) ∙ r (x)]

Example: Let P = 4x / (x2 - 1) and Q = (x + 1) / (x - 1), determine P × Q and P ÷ Q.

• P × Q = 4x / (x2 - 1) × (x + 1) / (x - 1) = 4x (x + 1) / [(x2 - 1) (x - 1)]

Remembering that x2 - 1 = (x - 1) (x + 1), then:

P × Q = 4x (x + 1) / [(x - 1) (x + 1) (x - 1)] = 4x / (x-1)2

• P ÷ Q = 4x / (x2 - 1) ÷ (x + 1) / (x - 1) = 4x (x - 1) / [(x2 - 1) (x + 1)]

= 4x (x - 1) / [(x - 1) (x + 1) (x + 1)] = 4x / (x-1)2