Summe Addition & Integration
Free online algebra tool

Rational Expression Simplifier

Simplify algebraic fractions by finding and canceling common polynomial factors. View the reduced expression, factoring information, worked steps, and restrictions imposed by the original denominator.

Simplify a rational expression

Enter the numerator and denominator as polynomial expressions in x.

Use x, numbers, parentheses, +, -, *, / by a constant, and ^ for powers.
The denominator cannot be the zero polynomial.

What is a rational expression?

A rational expression is a quotient of two polynomials. The denominator must not equal zero, so some values of x may be excluded from the domain. Simplifying a rational expression means rewriting it in an equivalent form with common factors removed.

Step 1: Find common factors

Consider the expression (x^2 - 9) / (x^2 - 3x). Factor the numerator as a difference of squares and factor x from the denominator:

x^2 - 9 = (x - 3)(x + 3)
x^2 - 3x = x(x - 3)

Step 2: Cancel the shared factor

The factor x - 3 appears in both the numerator and denominator. Cancel that common factor to obtain:

((x - 3)(x + 3)) / (x(x - 3)) = (x + 3) / x

Step 3: Preserve the domain restrictions

The original denominator x(x - 3) equals zero when x = 0 or x = 3. Therefore, both values remain excluded, even though the factor x - 3 no longer appears in the simplified expression.

Simplified expression: (x + 3) / x
Restrictions: x ≠ 0 and x ≠ 3

Why terms cannot be canceled

Cancellation is valid only when the same nonzero factor multiplies the entire numerator and denominator. A term joined to another term by addition or subtraction is not automatically a common factor.

For example, (x + 3) / x is not equal to 3. The x in the numerator is only part of a sum, so it cannot be canceled against the denominator.

Frequently asked questions

Does this calculator factor every polynomial completely?

The calculator uses polynomial greatest-common-divisor reduction to remove common factors. The displayed factor information emphasizes the common factor and reduced polynomials; it does not promise a complete irreducible factorization of every polynomial.

Why do excluded values matter after cancellation?

The simplified expression agrees with the original only where the original denominator was nonzero. Canceling a factor does not restore values that made the original expression undefined.

Can I enter fractions in the numerator or denominator?

You can divide a polynomial by a nonzero constant, such as (x^2 - 1)/2. Division by an expression involving x is not supported inside an individual numerator or denominator field.

What are the input limits?

Each expression can contain up to 500 characters and 200 tokens. The maximum polynomial degree is 20. The calculator uses floating-point arithmetic, so results involving decimal coefficients may be affected by numerical rounding.