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Rational Expressions Calculator

Simplify a factored rational expression, cancel shared roots, and keep all values blocked by the first denominator for algebra work.

Roots of each linear factor, e.g. 2, -3 for (x - 2)(x + 3).

Roots of each linear factor, e.g. 2, 1 for (x - 2)(x - 1).

Result

(x + 3) / (x - 1)

The simplified expression is (x + 3) / (x - 1), canceling the shared factor(s) at x = 2. Excluded values: x = 2, 1.

Quick Answer

A rational expressions calculator cancels the same factor on top and bottom but keeps all old domain limits. For (x - 2)(x + 3) divided by (x - 2)(x - 1), the short form is (x + 3)/(x - 1). Both x = 2 and x = 1 stay blocked because the first denominator was zero there.

Rational Expressions Explained: Factors And Domain

A rational expression is one polynomial over another. This tool starts after both parts have been split into linear factors. Enter each factor by its root. The tool matches roots on the top and bottom, then cancels one pair at a time. The factors left behind form the short result. The domain still comes from the first bottom line. Any root that made that line zero must stay out. Canceling a factor does not bring that input back. The graph often has a hole at a canceled root. A factor left on the bottom can mark a vertical asymptote. Save the full blocked list before you cancel. This split also helps you compare the short form with holes and other blocked inputs on a graph.

Rational Factor Rules And The Domain They Preserve

Simplifying a rational expression cancels shared linear factors while maintaining the original expression's domain restrictions.

  • Cancel factors: [(x − r)A(x)] ÷ [(x − r)B(x)] = A(x) ÷ B(x) for x not equal to r.
  • Excluded domain: every value where the original denominator D(x) = 0.

Top roots: values that form the top factors.

Bottom roots: values that form the first bottom factors.

Shared roots: values found in both lists.

Blocked values: every root from the first bottom list.

Using A Rational Expressions Calculator In 5 Steps

Inputs

  • Numerator roots: enter the root from each top factor (x − r), separated by commas.
  • Denominator roots: enter every root from the first bottom (x − r), separated by commas.

Steps

  1. Factor the top and bottom into linear factors.
  2. Enter each top root.
  3. Enter each first bottom root.
  4. Calculate to cancel matching roots.
  5. Copy the blocked list before using the short form.

Rational Expression Example With Domain Kept Intact

Factored top (x − 2)(x + 3), bottom (x − 2)(x − 1). Top roots list [2, −3], bottom roots list [2, 1].

  1. Match the shared factor: (x − 2) ÷ (x − 2) = 1 for x not equal to 2.
  2. Cancel the shared factor: [(x − 2)(x + 3)] ÷ [(x − 2)(x − 1)] = (x + 3) ÷ (x − 1).
  3. Set first bottom equal to zero: (x − 2)(x − 1) = 0.
  4. Solve each factor for domain exclusions: x = 2 and x = 1.

The simplified form is (x + 3) ÷ (x − 1), with x not equal to 2 or 1. No rounding is needed because the calculation is exact.

Where Rational Expression Simplification Helps Most

Use this tool after factoring the top and bottom. The result helps with the short form, domain limits, and possible holes.

For nearby algebraic, fraction, and graphing tools, see the math calculators hub and look up concepts in the glossary.

Assumptions

  • The expression is in linear factors.
  • Each input is the root r from x − r.
  • One shared copy cancels one shared copy.
  • Domain limits come from the first bottom.

Limitations

  • The tool does not factor expanded polynomials.
  • Nonlinear factors do not fit this root-list input.
  • A canceled bottom root still stays blocked.
  • Full graph behavior may need more work.

In Practice

A common error is removing a canceled root from the blocked list. Write all first bottom roots before you cancel. Keep that list even when the visible factor is gone.

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Related Guides

Rational Expressions Calculator Questions And Answers

Why Does A Canceled Factor Still Exclude A Value?

A canceled factor still blocks its root because the first bottom was zero there. The short form agrees only on the old domain. The missing point does not become a valid input after the factor disappears.

How Do Shared Roots Cancel In Factored Expressions?

Shared roots cancel when the same factor is on the top and bottom. One copy cancels with one copy. If a factor repeats, both lists need equal counts to remove each copy in the result.

What Is The Difference Between A Hole And An Asymptote?

A hole comes from a canceled bottom factor, while an asymptote can come from one that remains. Both roots stay outside the domain. The graph treats the two cases differently, so the factor list matters.

How Can I Check A Simplified Rational Expression?

Check a simplified rational expression by restoring the canceled factors on both top and bottom. The rebuilt form should match the first factored expression. Confirm that every first bottom root stays blocked. A full check needs both form and domain.

Sources

Last updated: . Reviewed for accuracy against the formula shown above.