Operations with Rational Expressions

By the end of this lesson, you will be able to multiply, divide, add, and subtract rational expressions, find the LCD for rational expressions with polynomial denominators, and simplify the results completely.

The Big Idea

Operations on rational expressions follow the same rules as operations on numeric fractions — factor, find common denominators, combine, simplify. The difference is that the "numbers" are polynomials. Fluency with these operations is essential for solving rational equations and working with partial fractions in calculus.

Multiplication

$$\frac{A}{B} \cdot \frac{C}{D} = \frac{AC}{BD}$$

Process: Factor all numerators and denominators first, cancel common factors, then multiply.

Division

$$\frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \cdot \frac{D}{C} = \frac{AD}{BC}$$

Multiply by the reciprocal of the divisor.

Example 1 — Multiplication

$\dfrac{x^2-4}{x+3} \cdot \dfrac{x^2+6x+9}{x-2}$

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

Cancel $(x-2)$ and $(x+3)$: $(x+2)(x+3)$ for $x \neq 2, -3$

Example 2 — Division

$\dfrac{x^2-9}{x+1} \div \dfrac{x-3}{x^2-1}$

$= \dfrac{x^2-9}{x+1} \cdot \dfrac{x^2-1}{x-3}$

Factor: $\dfrac{(x+3)(x-3)}{x+1} \cdot \dfrac{(x+1)(x-1)}{x-3}$

Cancel $(x-3)$ and $(x+1)$: $(x+3)(x-1)$ for $x \neq 3, -1$

Adding and Subtracting

Same denominator: $$\frac{A}{B} + \frac{C}{B} = \frac{A+C}{B}$$

Different denominators: find the LCD, rewrite each fraction, then combine.

Finding the LCD

1. Factor each denominator completely
2. LCD = product of each distinct factor raised to its highest power

Example: LCD of $\dfrac{1}{x-3}$ and $\dfrac{1}{x^2-9}$:

$x^2-9 = (x-3)(x+3)$

LCD = $(x-3)(x+3)$

Example 3 — Addition with Different Denominators

$\dfrac{3}{x-2} + \dfrac{5}{x+3}$

LCD = $(x-2)(x+3)$

$$= \frac{3(x+3)}{(x-2)(x+3)} + \frac{5(x-2)}{(x-2)(x+3)} = \frac{3x+9+5x-10}{(x-2)(x+3)} = \frac{8x-1}{(x-2)(x+3)}$$

Example 4 — Subtraction Requiring Factoring

$\dfrac{x}{x^2-1} - \dfrac{1}{x+1}$

$x^2-1 = (x+1)(x-1)$; LCD = $(x+1)(x-1)$

$$= \frac{x}{(x+1)(x-1)} - \frac{x-1}{(x+1)(x-1)} = \frac{x-(x-1)}{(x+1)(x-1)} = \frac{1}{(x+1)(x-1)} = \frac{1}{x^2-1}$$

Example 5 — Full Workflow