Multiplying, Dividing, Adding and Subtracting Rational Expressions

Operations with rational expressions follow the rules for fractions: multiply numerators and denominators, divide by multiplying by the reciprocal, and add or subtract after finding a common denominator. In brief: (x/(x + 1)) · ((x + 1)/(x − 2)) = x/(x − 2); (a/b) ÷ (c/d) = (a/b) · (d/c); and 1/x + 1/(x + 1) = (2x + 1)/(x(x + 1)). In each case, factor first, simplify and keep track of the excluded values.

These operations are needed to solve rational equations, to combine formulas and to prepare expressions for calculus. The main difficulty is organisation: factoring all denominators, building the least common denominator (LCD) and carefully distributing signs. This guide describes each operation with worked examples, including complex fractions, and gives a table of the rules.

How do you multiply rational expressions?

1. Factor all numerators and denominators.
2. Cancel common factors between any numerator and any denominator.
3. Multiply the remaining numerators and the remaining denominators.

Example 1

(x² − 4)/(x + 3) · (x² + 3x)/(x − 2) = (x − 2)(x + 2)/(x + 3) · x(x + 3)/(x − 2) = x(x + 2), for x ≠ −3 and x ≠ 2.

Example 2

(6a²/(5b)) · (10b³/(9a)) = (60a²b³)/(45ab) = 4ab²/3.

How do you divide rational expressions?

Multiply by the reciprocal of the divisor: (P/Q) ÷ (R/S) = (P/Q) · (S/R). Then follow the steps for multiplication. In addition to the excluded values from the denominators, R ≠ 0 must be added because you cannot divide by zero.

Example 3

(x² − 1)/(x + 4) ÷ (x + 1)/(x² − 16) = (x − 1)(x + 1)/(x + 4) · (x − 4)(x + 4)/(x + 1) = (x − 1)(x − 4). The excluded values are x = −4 and x = 4 (zeros of the denominators) and x = −1 (where the divisor equals zero).

How do you add and subtract rational expressions?

Same denominator: add the numerators and keep the denominator: (a + b)/c. For subtraction, take care to subtract the whole second numerator: (3x + 1)/(x − 2) − (x − 5)/(x − 2) = (3x + 1 − x + 5)/(x − 2) = (2x + 6)/(x − 2).

Different denominators:

1. Factor every denominator.
2. Find the LCD: the product of all different factors, each taken with its highest power.
3. Rewrite each fraction with the LCD by multiplying the numerator and the denominator by the missing factors.
4. Add or subtract the numerators.
5. Simplify the result: factor the numerator and cancel.

Example 4

1/x + 1/(x + 1): the LCD is x(x + 1). The sum is (x + 1)/(x(x + 1)) + x/(x(x + 1)) = (2x + 1)/(x(x + 1)).

Example 5

2/(x − 3) − 5/(x + 2): the LCD is (x − 3)(x + 2). The result is [2(x + 2) − 5(x − 3)]/((x − 3)(x + 2)) = (2x + 4 − 5x + 15)/(…) = (19 − 3x)/((x − 3)(x + 2)).

Example 6: factored denominators

3/(x² − 4) + 1/(x² + 4x + 4) = 3/((x − 2)(x + 2)) + 1/(x + 2)². The LCD is (x − 2)(x + 2)². Then [3(x + 2) + (x − 2)]/((x − 2)(x + 2)²) = (4x + 4)/((x − 2)(x + 2)²) = 4(x + 1)/((x − 2)(x + 2)²). Check with x = 3: left side 3/5 + 1/25 = 16/25 = 0.64. Right side: 4 · 4/(1 · 25) = 16/25 ✓.

Example 7: opposite denominators

x/(x − 2) + 3/(2 − x) = x/(x − 2) − 3/(x − 2) = (x − 3)/(x − 2). Rewriting 2 − x as −(x − 2) avoids a bigger LCD.

How do you simplify complex fractions?

A complex fraction has fractions in its numerator, denominator or both. Method: multiply the numerator and the denominator by the LCD of all the small fractions.

Example 8

(1/x + 1/y)/(1/x − 1/y). Multiply by xy: (y + x)/(y − x). Valid for x, y ≠ 0 and x ≠ y.

Example 9

(1 − 1/x²)/(1 − 1/x). Multiply by x²: (x² − 1)/(x² − x) = (x − 1)(x + 1)/(x(x − 1)) = (x + 1)/x, for x ≠ 0, 1.

This kind of simplification is used in the definition of the derivative.

Summary of the rules

Where do these rules come from?

The rules for fractions of numbers were known in ancient Egypt and Greece, where fractions were handled in unit-fraction form, and in India, where Brahmagupta wrote the rules for addition and multiplication of fractions in the seventh century. They were extended to algebraic fractions as soon as symbolic algebra was developed, in the sixteenth and seventeenth centuries.

Where are they used today?

Two pipes filling a tank in a and b hours work together at the rate 1/a + 1/b = (a + b)/(ab) tanks per hour, so the time is ab/(a + b). For a = 3 and b = 6 hours, the time is 18/9 = 2 hours. Parallel resistors, the combined focal length of lenses and average speeds all use sums of rational expressions.

Which mistakes are common?

Adding denominators

1/x + 1/y is not 2/(x + y). Find a common denominator: (x + y)/(xy).

Forgetting to distribute the minus sign

In a/c − (b − d)/c, the numerator is a − b + d, not a − b − d.

Using a larger denominator than needed

Multiplying all denominators works but creates large expressions. Use the least common denominator.

Dividing without flipping

(a/b) ÷ (c/d) = (a/b) · (d/c), not (a/b) · (c/d).