# How to simplify rational expressions?

Simplifies, adds, subtracts, multiplies and divides rational expressions in one variable: one fraction in lowest terms, with the excluded values.

- Page: https://www.acalculator.org/math/rational-expressions-calculator
- JSON spec: https://www.acalculator.org/math/rational-expressions-calculator.json
- Version: 34473d966fe5

## Default answer

Example with the default inputs (Rational expression (x^2 - 9)/(x^2 + 4x + 3)): (x^2 - 9)/(x^2 + 4x + 3) simplifies to (x - 3)/(x + 1).

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| expr | Rational expression | Polynomials divided by polynomials, joined by + − × ÷. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| simplified | Simplified | One fraction in lowest terms. |
| excluded | Excluded values | Where a denominator of the typed expression is 0; the expression is undefined there. |
| top | Numerator, factored | The numerator after combining, factored. |
| bottom | Denominator, factored | The denominator after combining, factored. |

## Method

Combine into one fraction (multiply across, or over the product of the denominators), factor the top and the bottom, and cancel the common factors.

## Assumptions

- One variable; whole-number powers.
- The simplified form equals the expression wherever the expression is defined.

## Worked examples

1. expr = (x^2 - 9)/(x^2 + 4x + 3) gives simplified = (x - 3)/(x + 1), excluded = x ≠ -3, x ≠ -1. Source: OpenStax, College Algebra 2e, section 1.6 Rational Expressions (https://openstax.org/books/college-algebra-2e/pages/1-6-rational-expressions), Example 1: (x − 3)/(x + 1).
2. expr = (x^2 + 4x - 5)/(3x + 18) * (2x - 1)/(x + 5) gives simplified = (x - 1) (2x - 1)/(3 (x + 6)), excluded = x ≠ -6, x ≠ -5. Source: OpenStax, College Algebra 2e, section 1.6 Rational Expressions (https://openstax.org/books/college-algebra-2e/pages/1-6-rational-expressions), Example 2: (x − 1)(2x − 1)/(3(x + 6)).
3. expr = (2x^2 + x - 6)/(x^2 - 1) / ((x^2 - 4)/(x^2 + 2x + 1)) gives simplified = (2x - 3) (x + 1)/((x - 1) (x - 2)), excluded = x ≠ -2, x ≠ -1, x ≠ 1, x ≠ 2. Source: OpenStax, College Algebra 2e, section 1.6 Rational Expressions (https://openstax.org/books/college-algebra-2e/pages/1-6-rational-expressions), Example 3: (2x − 3)(x + 1)/((x − 1)(x − 2)).
4. expr = 6/(x^2 + 4x + 4) - 2/(x^2 - 4) gives simplified = 4 (x - 4)/((x - 2) (x + 2)^2), excluded = x ≠ -2, x ≠ 2. Source: OpenStax, College Algebra 2e, section 1.6 Rational Expressions (https://openstax.org/books/college-algebra-2e/pages/1-6-rational-expressions), Example 5: 4(x − 4)/((x + 2)²(x − 2)).

## FAQ

### What is a rational expression?

A fraction whose top and bottom are polynomials, such as (x² − 9)/(x² + 4x + 3). It is undefined where the bottom is 0.

### How do I simplify a rational expression?

Factor the numerator and the denominator, then cancel the factors they share. (x² − 9)/(x² + 4x + 3) = (x − 3)(x + 3)/((x + 1)(x + 3)) = (x − 3)/(x + 1), for x ≠ −3 and x ≠ −1.

### How do I multiply or divide rational expressions?

To multiply, multiply the tops and the bottoms, then simplify. To divide, multiply by the reciprocal of the second expression. Type them with * and /, and put the second fraction in brackets when dividing: (2x^2 + x - 6)/(x^2 - 1) / ((x^2 - 4)/(x^2 + 2x + 1)).

### How do I add or subtract rational expressions?

Write each over a common denominator, then add or subtract the numerators. The page multiplies the denominators together and then cancels, so the result is the same as with the least common denominator: 6/(x² + 4x + 4) − 2/(x² − 4) = 4(x − 4)/((x − 2)(x + 2)²).

### What are the excluded values?

The values of the variable that make a denominator of the original expression 0. They stay excluded after simplifying, even when the factor that caused them has cancelled: (x − 3)/(x + 1) above still needs x ≠ −3.

### Can I use more than one variable?

No. The page works in one letter, such as x or t. For 5/x + 6/y, combine by hand: (5y + 6x)/(xy).

### How is the answer checked?

The simplified form must equal the typed expression at 15 test points where both are defined. Each excluded value is a solution from the algebra, checked by putting it back into the denominator.

## Sources

- OpenStax, College Algebra 2e, section 1.6 Rational Expressions (retrieved 2026-10-03): https://openstax.org/books/college-algebra-2e/pages/1-6-rational-expressions