Rational Expressions: Domain and Simplifying
A rational expression is a fraction whose numerator and denominator are polynomials, such as (x2 − 9)/(x − 3), and it is defined for all x except where the denominator is zero. In brief: the domain excludes the zeros of the denominator, and simplifying means factoring and cancelling common factors. (x2 − 9)/(x − 3) = (x − 3)(x + 3)/(x − 3) = x + 3 for x ≠ 3.
Rational expressions behave like fractions of numbers, and the same rules apply: you can cancel common factors, but never terms. Before simplifying, you must find the values excluded from the domain, because they remain excluded after cancelling. This guide explains how to find the domain, how to factor and cancel step by step, how to spot common patterns such as opposites and shows worked examples, with a table of typical simplifications.
What is a rational expression?
A rational expression has the form P(x)/Q(x), where P and Q are polynomials and Q is not the zero polynomial. The name comes from “ratio”. Polynomials themselves are rational expressions with the denominator 1. Examples: 3/(x + 2), (x2 + 1)/(x3 − x), and (2x − 1)/5. An expression with a variable under a root or in an exponent, such as √x/(x + 1), is not rational.
How do you find the domain?
Division by zero is not defined, so the domain consists of all real numbers except the zeros of the denominator.
- Set the denominator equal to zero.
- Solve the equation by factoring or by the quadratic formula.
- Exclude these values.
Example 1
5/(x − 4): the denominator is zero for x = 4, so the domain is x ≠ 4.
Example 2
(x + 1)/(x2 − 5x + 6): x2 − 5x + 6 = (x − 2)(x − 3), so x ≠ 2 and x ≠ 3.
Example 3
3x/(x2 + 4): x2 + 4 is never zero for real x, so the domain is all real numbers.
Find the domain before cancelling. For (x2 − 9)/(x − 3), the exclusion x ≠ 3 comes from the original denominator, and it remains after simplification to x + 3.
How do you simplify a rational expression?
- Factor the numerator and the denominator completely.
- Cancel factors that appear in both, since a/a = 1 for a ≠ 0.
- Write the result, with the excluded values noted.
Only factors can be cancelled, not terms. In (x + 3)/(x + 5), you cannot cancel x. For numbers: (2 + 3)/(2 + 5) = 5/7, and cancelling 2 would give 3/5, which is wrong.
Example 4
(x2 − 4)/(x2 + 5x + 6) = (x − 2)(x + 2)/((x + 2)(x + 3)) = (x − 2)/(x + 3), for x ≠ −2 and x ≠ −3.
Example 5
(6x2 − 12x)/(3x2 − 3x − 6) = 6x(x − 2)/(3(x2 − x − 2)) = 6x(x − 2)/(3(x − 2)(x + 1)) = 2x/(x + 1), for x ≠ 2 and x ≠ −1.
Example 6: with powers
12x3y/(18x2y4) = 2x/(3y3).
What about opposite factors?
The factors a − b and b − a are opposites: b − a = −(a − b). So (a − b)/(b − a) = −1. Example: (x − 5)/(5 − x) = −1, for x ≠ 5. And (2 − x)(x + 1)/((x − 2)(x − 1)) = −(x + 1)/(x − 1).
Worked examples
Example 7: grouping
(x3 + x2 − 4x − 4)/(x2 − 4). Numerator by grouping: x2(x + 1) − 4(x + 1) = (x + 1)(x − 2)(x + 2). Denominator: (x − 2)(x + 2). The result is x + 1, for x ≠ ±2.
Example 8: a numerical check
Does (x2 − 9)/(x − 3) = x + 3? At x = 5: 16/2 = 8 and 5 + 3 = 8 ✓. At x = 3: the left side is 0/0 (not defined), but the right side is 6. The two expressions agree except at x = 3, so they are equal only on the common domain.
Example 9: sum and difference of cubes
(x3 − 8)/(x2 − 4) = (x − 2)(x2 + 2x + 4)/((x − 2)(x + 2)) = (x2 + 2x + 4)/(x + 2), for x ≠ ±2. The quadratic x2 + 2x + 4 does not factor over the reals.
Table of simplifications
| Expression | Factored form | Simplified | Excluded x |
|---|---|---|---|
| (x2 − 9)/(x − 3) | (x − 3)(x + 3)/(x − 3) | x + 3 | 3 |
| (x2 − 4)/(x2 + 5x + 6) | (x − 2)(x + 2)/((x + 2)(x + 3)) | (x − 2)/(x + 3) | −2, −3 |
| (x2 − x − 6)/(x2 − 9) | (x − 3)(x + 2)/((x − 3)(x + 3)) | (x + 2)/(x + 3) | 3, −3 |
| (x − 5)/(5 − x) | −(5 − x)/(5 − x) | −1 | 5 |
| (x3 − 8)/(x2 − 4) | (x − 2)(x2 + 2x + 4)/((x − 2)(x + 2)) | (x2 + 2x + 4)/(x + 2) | 2, −2 |
| (2x2 + x)/(4x2 − 1) | x(2x + 1)/((2x − 1)(2x + 1)) | x/(2x − 1) | 1/2, −1/2 |
Where does this topic come from?
Fractions and ratios are among the oldest mathematical ideas. Algebraic fractions were treated in medieval Arabic algebra and in the sixteenth-century works of Viète, who introduced the symbolic manipulation of expressions with letters. The idea that cancelling a common factor changes the domain of the expression is a more modern way of thinking that came with the function concept.
Where are rational expressions used today?
Many formulas are rational: the combined resistance of two resistors in parallel R = R1R2/(R1 + R2), the average cost per item (fixed cost + variable cost)/(number of items) and the lens formula in optics. Simplifying these expressions shows how a quantity behaves when a parameter changes. Engineers also use rational functions for transfer functions in control theory.
Which mistakes are common?
Cancelling terms instead of factors
(x + 2)/(x + 4) cannot be simplified. Only factors, expressions multiplied together, can be cancelled.
Forgetting excluded values
After cancelling, the excluded values of the original denominator remain excluded. Write them next to the answer.
Missing a sign with opposite factors
(a − b)/(b − a) = −1, not 1.
Not factoring completely
Look for common factors first, and then for differences of squares, trinomials and grouping.