Rational Expressions Calculator — Simplify, Add, Multiply, Divide
Simplify, add, subtract, multiply, or divide rational expressions with full factoring steps — domain restrictions are always shown, even when they cancel out.
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Full bio →Expression
- x ≠ 2 — no longer visible in the simplified result, but still excluded because it made an original denominator (or, for division, the divisor) zero
Steps
- Factor the numerator: (x + 2)(x - 2)
- Factor the denominator: (x - 2)
- Cancel the common factor(s) tied to x = 2.
This calculator handles linear and quadratic numerators/denominators entered as coefficients (ax² + bx + c), factoring over the rationals to simplify, combine, multiply, or divide rational expressions. Excluded values are always the zeros of the ORIGINAL denominator(s) — they stay excluded even when a factor cancels during simplification. For differentiating composite expressions instead, see the Chain Rule Calculator.
Reference Values
Last verified:| Category | Range | What It Means | Status |
|---|---|---|---|
| Simplify ★ | Factor numerator & denominator, cancel common factors | Factor both polynomials completely, then cancel any factor that appears in both the numerator and the denominator. What's left is the reduced expression — but every zero of the ORIGINAL denominator stays excluded from the domain, even the ones that cancel out. | ★ Best |
| Add / Subtract | Factor each denominator, build the LCD, combine numerators | Factor each denominator, form the least common denominator (LCD) from the union of all distinct factors, rewrite each fraction over that LCD, then add or subtract the numerators. Domain restrictions come from BOTH original denominators, not just the final one. | Good |
| Multiply | Multiply numerators together and denominators together, then simplify | Multiply straight across — numerator × numerator over denominator × denominator — then cancel any common factors in the result. Domain restrictions come from both original denominators (never from the numerators). | Good |
| Divide | Multiply by the reciprocal of the second expression | Flip the second fraction (swap its numerator and denominator) and multiply. Domain restrictions come from three places: the first denominator, the second numerator, and the second denominator — because the second expression's numerator becomes a new denominator once you flip it. | Okay |
| Excluded values (domain restrictions) | Every zero of an ORIGINAL denominator, always shown | Worked example: (x²-4)/(x-2) simplifies to (x+2) after canceling the shared (x-2) factor. But x=2 made the ORIGINAL denominator zero, so x=2 stays excluded from the domain even though the simplified expression x+2 is perfectly defined at x=2. The excluded value describes the original expression's domain, not the simplified one's. | Poor |
Source: Domain-restriction convention per CK-12 Flexbook "Excluded Values for Rational Expressions" and Varsity Tutors "Rational Expressions and Domain Restrictions"; operation methods reflect standard Algebra II / College Algebra rational-expression rules.
Worked Examples
Simplify — Common Factor Cancels But Stays Excluded
- Operation
- Simplify
- Expression
- (x² - 4) / (x - 2)
Factor the numerator: x²-4 = (x-2)(x+2). Cancel the shared (x-2) factor with the denominator, leaving x+2. But x=2 made the ORIGINAL denominator zero, so x=2 remains excluded from the domain even though x+2 is defined everywhere, including at x=2.
Add Two Rational Expressions With Different Denominators
- Operation
- Add
- Expression A
- (x² - 4) / (x - 2)
- Expression B
- (x + 1) / (x - 3)
LCD = (x-2)(x-3). Rewriting and adding gives [x³-2x²-5x+10] / [(x-2)(x-3)], which reduces to (x²-5)/(x-3) because (x-2) happens to divide out of the combined numerator too. x=2 is still excluded — it was a zero of Expression A's original denominator — even though the final (x²-5)/(x-3) no longer shows an (x-2) factor.
Subtract Two Rational Expressions
- Operation
- Subtract
- Expression A
- x / (x - 1)
- Expression B
- 1 / (x + 1)
LCD = (x-1)(x+1) = x²-1. New numerators: x(x+1) = x²+x and 1(x-1) = x-1. Subtracting: (x²+x) - (x-1) = x²+1. Nothing cancels here, so the result stays over the full LCD, and both original denominator zeros remain excluded.
Multiply — Denominator Factor Cancels But Stays Excluded
- Operation
- Multiply
- Expression A
- (x + 2) / (x - 3)
- Expression B
- (x - 3) / (x + 5)
Multiply straight across: numerator (x+2)(x-3), denominator (x-3)(x+5). The (x-3) factors cancel, leaving (x+2)/(x+5). x=3 is still excluded — it zeroed Expression A's original denominator — even though (x-3) no longer appears anywhere in the simplified result.
Divide Two Rational Expressions
- Operation
- Divide
- Expression A
- (x + 1) / (x - 2)
- Expression B
- (x + 1) / (x + 3)
Dividing flips Expression B and multiplies: (x+1)/(x-2) × (x+3)/(x+1). The (x+1) factors cancel, leaving (x+3)/(x-2). Three values stay excluded: x=2 (Expression A's denominator), x=-1 (Expression B's numerator, which becomes the new denominator after flipping), and x=-3 (Expression B's denominator) — even though x=-1 and x=-3 don't appear in the final simplified expression at all.
How to Use This Calculator
- 1
Pick an operation
Choose Simplify, Add, Subtract, Multiply, or Divide. Simplify works on one expression; the other four compare two expressions, Expression A and Expression B.
- 2
Enter Expression A's numerator and denominator
Toggle "quadratic" on if you need an x² term, then fill in the a, b, and c coefficients for both the numerator and denominator.
- 3
Enter Expression B (if the operation needs it)
Add/Subtract/Multiply/Divide need a second expression. Fill in its numerator and denominator coefficients the same way.
- 4
Read the result and excluded values
The result box shows the combined/simplified expression in both expanded and factored form. The Excluded Values box below it always lists every x-value that made an original denominator zero — including ones that disappeared during simplification.
- 5
Check the step-by-step work
The Steps panel shows how each denominator was factored, how the LCD (for add/subtract) or product (for multiply/divide) was built, and which factors canceled.
What Each Value Means
- Excluded Value / Domain Restriction (x-value)
- An x-value that makes an ORIGINAL denominator (or, for division, the divisor) equal zero. Excluded values must be declared even when the factor that caused them cancels out during simplification, because the simplified expression and the original are only equivalent away from those values.
- LCD (Least Common Denominator) (polynomial)
- The smallest polynomial that both denominators divide into evenly, built by factoring each denominator and taking every distinct factor at its highest multiplicity across the two. Used to rewrite fractions with a common denominator before adding or subtracting.
- Factored Form (expression)
- An expression written as a product of its irreducible factors, e.g. (x-2)(x+2) instead of x²-4. Factored form is what makes common-factor cancellation and domain-restriction identification possible.