Simplify rational expressions and preserve their domains
This rational expression calculator simplifies algebraic fractions in the real variable x. It can also add, subtract, multiply and divide two supported expressions. It expands polynomial arithmetic, finds common polynomial factors and reduces the result using exact rational coefficients rather than rounding them to decimals.
The original domain is part of the answer. For example, (x² − 1)/(x − 1) simplifies to x + 1, but x = 1 remains excluded because it made the original denominator zero. A canceled factor does not turn an originally undefined input into a valid one.
This fills a different role from the Fraction Calculator, which handles numeric fractions, and the Simplifying Expressions Calculator, which expands supported polynomials but excludes variable denominators. The Polynomial Factoring Calculator remains available for a separate factoring task.
How to use the rational expression calculator
- Choose Simplify for a single expression, or select Add, Subtract, Multiply or Divide for two expressions.
- Type the first expression using lowercase x. To reproduce the first example, enter (x^2 − 1)/(x − 1).
- For a two-expression operation, enter the second expression in its separate field. Parenthesize each numerator and denominator clearly.
- Calculate, then read both the reduced expression and the original domain restrictions. The reduced formula alone is not the complete answer.
Use ^ for integer powers, * for explicit multiplication and / for division. Implicit multiplication before x or an opening bracket is accepted, as in 2x and (x + 1)(x − 1). Multiplication and division have the same precedence and are processed left to right; use brackets to remove ambiguity.
Changing an input clears the previous result and disables its result actions. Switching to Simplify hides and disables the second field. Reset returns to the initial single-expression form. Copy, Download and Print include the result, domain restrictions and calculation steps.
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For an optional numerical check, browse scientific calculators (paid link). Confirm the supported functions; a numerical device does not necessarily reproduce this tool’s symbolic working. No purchase is needed.
Common denominators, polynomial GCDs and cancellation
Addition and subtraction first create a common polynomial denominator. Multiplication multiplies numerators and denominators; division multiplies by a reciprocal and additionally requires the divisor to be nonzero. Each original denominator condition is kept even if subsequent arithmetic cancels that factor.
The calculation uses polynomial long division and the Euclidean algorithm to obtain a greatest common polynomial divisor. Only entire common factors are canceled. In (x + 1)/(x + 2), the individual x terms cannot be crossed out, because they are parts of sums rather than common multiplicative factors.
The reduced denominator is normalized to have leading coefficient one. This may place an exact fraction in the numerator: 1/(2x) is displayed as (1/2)/(x). These forms have the same value on the same domain. Accepted decimal inputs are represented exactly as rational coefficients during the calculation.
Worked rational expression examples
Cancellation: In Simplify mode, enter (x^2 − 1)/(x − 1). Factor the numerator as (x − 1)(x + 1), cancel the common x − 1 factor and obtain x + 1. The original condition x − 1 ≠ 0 still excludes x = 1.
Addition: Select Add and enter 1/(x − 1) and 1/(x + 1). The combined numerator is (x + 1) + (x − 1) = 2x, so the result is (2x)/(x² − 1). The original exclusions are x = 1 and x = −1.
Division: Select Divide with 1/x as the first expression and (x − 1)/x as the second. The reduced answer is 1/(x − 1), but both x = 0 and x = 1 are excluded. Zero was forbidden in the original fractions; one makes the divisor equal zero.
A zero result: Subtract 1/x from 1/x. The answer is zero wherever the original expressions were defined, not zero for every real x. The restriction x ≠ 0 must remain even though the reduced result has no visible denominator.
How excluded values are displayed
The domain panel lists exact polynomial conditions that must all hold, such as (x − 1) ≠ 0. These conditions completely describe the retained original restrictions, including those introduced by division or negative integer powers. Repeated identical conditions are combined.
Linear conditions are also converted into exact excluded values. Quadratic conditions use the discriminant: rational roots are reduced fractions, irrational real roots are displayed in an exact quadratic-formula expression, and a negative discriminant has no real excluded roots. The original polynomial condition remains visible as an exact reference.
For degrees above two, the calculator retains the exact condition rather than claiming a complete decimal root list. For instance, the denominator x³ − 2 is reported as a condition that excludes every real zero of x³ − 2. This is mathematically complete domain notation even when no explicit decimal root is displayed.
Supported syntax and calculation limits
This version accepts one real variable x, ordinary numeric literals, parentheses, the four arithmetic operations and integer literal powers from −6 through 6. It does not accept equations, functions, other variables or fractional powers. A negative power adds the requirement that its base be nonzero. Literal 0^0 and division by an identically zero expression are rejected.
Each input is limited to 240 characters, 160 tokens and 20 levels of parentheses. Numeric literals have magnitude at most 1,000,000 and at most six decimal places. Fractions are formed with division, such as (1/3)x. The numerator and denominator of every intermediate polynomial must stay within degree 12. Up to 16 distinct domain conditions are retained, and large intermediate integers have a 256-digit safety limit.
These limits apply to intermediate work as well as the final result; a problem that eventually cancels may still be too large to process. This is an expression tool, not a rational-equation solver or a graph of holes and asymptotes. The Polynomial Long Division Calculator can show a separate quotient-and-remainder procedure.
Rational Expression Calculator FAQs
Sources and methodology
These educational references explain the underlying mathematics. This independently implemented tool is not affiliated with their publisher. Supported methods, numerical precision and restrictions are described above.