Work with algebraic fractions like (x² − 9)/(x² + 5x + 6). Simplify one expression or add, subtract, multiply and divide two of them; the calculator factors everything, finds the LCD, cancels common factors and lists the excluded values.
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Examples:
Write each expression with x, using / for fraction bars and parentheses around numerators and denominators: (x^2 - 1)/(x^2 + 2x + 1). Products like (x+1)(x-2) are fine.
Result
Factored form
Expanded form
Excluded values
Step by step
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How to use the Rational Expressions Calculator
Enter expression A, for example 2/(x - 1).
Enter expression B for operations with two expressions.
Choose an operation: Simplify A, A + B, A − B, A × B or A ÷ B.
Read the result in factored and expanded form with its excluded values.
Follow the steps, then copy the result.
Rules for working with rational expressions
Operation
Rule
Example
Simplify
Factor, then cancel common factors
(x² − 4)/(x² + 2x) = (x − 2)/x, x ≠ −2, 0
Add
a/b + c/d = (ad + bc)/bd, using the LCD
1/x + 1/(x + 1) = (2x + 1)/(x(x + 1))
Subtract
a/b − c/d = (ad − bc)/bd, distribute the minus
1/(x − 1) − 1/x = 1/(x(x − 1))
Multiply
(a/b)(c/d) = ac/bd
(x/(x + 1))·((x + 1)/2) = x/2, x ≠ −1
Divide
(a/b) ÷ (c/d) = (a/b)(d/c)
(x/3) ÷ (x/6) = 2, x ≠ 0
Common mistakes to avoid
Cancelling terms instead of factors. (x + 3)/(x + 5) cannot be simplified: the x's are terms inside sums, not common factors.
Forgetting excluded values. Write them down from the original denominators before you cancel anything.
Using the product as the denominator. Multiplying the denominators always works, but the LCD keeps numbers smaller and makes the final simplification easier.
Sign errors when subtracting. Put the second numerator in parentheses so the minus sign reaches every term.
Opposite factors. (2 − x) equals −(x − 2), so these factors cancel to −1.
Frequently asked questions
What is a rational expression?
A rational expression is a fraction whose numerator and denominator are polynomials, such as (x + 1)/(x² − 4). It works like a numeric fraction, except that it is undefined wherever the denominator equals zero.
How do you simplify a rational expression?
Factor the numerator and denominator completely, then cancel factors that appear in both. For example (x² − 9)/(x² + 5x + 6) = (x − 3)(x + 3)/((x + 2)(x + 3)) = (x − 3)/(x + 2), with x ≠ −3 and x ≠ −2. Only whole factors can be cancelled, never individual terms.
What are excluded values?
Excluded values are the x-values that make a denominator zero in the original problem. They stay excluded even if the factor cancels during simplification. When dividing, the zeros of the divisor's numerator are excluded too, because you cannot divide by zero.
How do you add or subtract rational expressions?
Factor the denominators, find the least common denominator (each factor to its highest power), rewrite each fraction over the LCD, combine the numerators, and simplify. Remember to distribute the minus sign when subtracting.
How do you multiply and divide rational expressions?
To multiply, multiply the numerators and the denominators and cancel common factors (cancelling before multiplying out is easier). To divide, multiply the first expression by the reciprocal of the second.
What is a hole?
When a factor such as (x + 3) cancels from both the numerator and the denominator, the graph has a hole at x = −3 instead of a vertical asymptote. The simplified expression matches the original everywhere except at that point.
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