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**

- Factor each of the two polynomials p (x) and q (x).
- Find the greatest common factor (GCF) of p (x) and q (x).
- If MCD equals one, then the given rational expression p (x) / q (x) is in simplest form.
- 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).
- 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 (x^{2 }- 5x - 6) / (x^{2 }+ 3x + 2)

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

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

(x^{2 }- 5x - 6) / (x^{2 }+ 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 / (x^{2 }- 1) and Q = (x + 1) / (x - 1), determine P × Q and P ÷ Q.

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

Remembering that x^{2 }- 1 = (x - 1) (x + 1), then:

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

- P ÷ Q = 4x / (x
^{2 }- 1) ÷ (x + 1) / (x - 1) = 4x (x - 1) / [(x^{2 }- 1) (x + 1)]

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