Rational Expressions: Simplifying

By the end of this lesson, you will be able to simplify rational expressions by factoring and canceling common factors, identify restrictions on the domain (excluded values), and simplify complex fractions.

The Big Idea

A rational expression is a ratio of polynomials. Simplifying means dividing out common factors from numerator and denominator — exactly like reducing a numeric fraction, but the "numbers" are polynomials. The critical difference: you must state any excluded values before simplifying, because the simplified form may not show where the original is undefined.

Domain Restrictions

The domain of a rational expression excludes any values that make the denominator equal to zero.

These are found from the original denominator (before simplification) — even if a factor cancels.

Simplifying: The Process

1. Factor numerator and denominator completely
2. Identify and state excluded values from the denominator
3. Cancel common factors
4. Write the simplified form

Example 1 — Basic Simplification

Simplify: $\dfrac{6x^2 - 12x}{3x}$

Factor: $\dfrac{6x(x-2)}{3x}$

Excluded: $x \neq 0$

Cancel $3x$: $2(x-2) = 2x - 4$ for $x \neq 0$

Example 2 — Quadratic Numerator and Denominator

Simplify: $\dfrac{x^2 - 9}{x^2 + x - 6}$

Factor: $\dfrac{(x+3)(x-3)}{(x+3)(x-2)}$

Excluded: $x \neq -3$ and $x \neq 2$

Cancel $(x+3)$: $\dfrac{x-3}{x-2}$ for $x \neq -3, 2$

Example 3 — Opposite Factors

Simplify: $\dfrac{x - 4}{4 - x}$

Note: $4 - x = -(x-4)$

$$\frac{x-4}{-(x-4)} = \frac{1}{-1} = -1 \quad (x \neq 4)$$

When numerator and denominator are opposites, the simplified value is $-1$.

Example 4 — Degree 3 Polynomials

Simplify: $\dfrac{x^3 - 8}{x^2 - 4}$

$x^3 - 8 = (x-2)(x^2+2x+4)$ (difference of cubes)

$x^2 - 4 = (x-2)(x+2)$ (difference of squares)

Excluded: $x \neq 2$ and $x \neq -2$

Cancel $(x-2)$: $\dfrac{x^2+2x+4}{x+2}$ for $x \neq 2, -2$

Complex Fractions

A has a fraction in its numerator, denominator, or both. Simplify by: