1. Factor completely
Rewrite the numerator and denominator as products. Difference of squares, trinomials, and a greatest common factor are common patterns.
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Enter an expression to find restrictions.
Rational Expression Simplifier — Factor, Cancel and Find Restrictions
Enter the numerator and denominator separately to see their factored forms, canceled common factor, simplified expression, and all real values excluded by the original denominator.
Enter a rational expression
Simplified result
Equivalent expression on the allowed domain
Factored numerator—
Factored denominator—
Common factor—
Discontinuities—
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Step-by-step simplification
The factoring and cancellation steps will appear here.
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How to simplify rational expressions
2. Cancel factors only
A factor multiplies the whole expression. You cannot cancel individual terms joined by addition or subtraction.
3. Keep restrictions
Solve the original denominator equal to zero. Those inputs remain excluded after cancellation.
4. Classify discontinuities
A canceled real factor creates a hole. A real denominator zero that remains can be a vertical asymptote.
Core rule
If N(x)=C(x)A(x) and D(x)=C(x)B(x), then
N(x)/D(x) = A(x)/B(x), provided C(x) ≠ 0 and B(x) ≠ 0.
The restriction from C(x) does not disappear: the simplified expression matches the original only on their shared allowed domain.
Worked examples
Difference of squares
(x²−9)/(x²−x−6) factors to (x−3)(x+3)/[(x−3)(x+2)]. Cancel x−3 to get (x+3)/(x+2), with x ≠ 3, −2.
Cancel a binomial
(2x²+7x+3)/(2x²+5x+3) becomes (2x+1)(x+3)/[(2x+3)(x+1)]. There is no common factor, so it is already reduced.
Expression simplifies to 1
(x²+1)/(x²+1)=1, but over the real numbers there is no restriction because x²+1 never equals zero.
Zero numerator
0/[x(x−4)]=0 wherever the original expression exists. The restrictions remain x ≠ 0, 4.
Calculation method and limits
Polynomial coefficients, division, greatest common divisors, and cancellations use reduced BigInt fractions, so no rounded decimals are used in the simplification itself. Factored forms extract rational linear factors and monic integer quadratic factors; any remaining factor stays in polynomial form. Irrational real zeros of quadratic factors are shown exactly, while unresolved real zeros of higher-degree factors are labeled as numerical approximations.
The tool supports one-variable polynomials through degree 8. It does not simplify nested rational functions, radicals, negative powers, or expressions containing variables other than x. Last reviewed: 4 August 2026.
Rational expression FAQs
How do you simplify a rational expression?
Factor the numerator and denominator, cancel only factors common to the entire numerator and denominator, then state every value that made the original denominator zero.
Why do canceled factors still create restrictions?
Cancellation gives an equivalent expression only where the original expression was defined. If a value made the original denominator zero, it stays excluded even when its factor cancels.
Can you cancel terms across addition?
No. Cancellation applies to factors, not terms. Factor sums or differences first; for example, x²−9=(x−3)(x+3).
What inputs does this simplifier accept?
It accepts polynomials in x through degree 8, parentheses, implicit multiplication, nonnegative integer powers, and integer, finite-decimal, or fractional coefficients.
What if the denominator has no real zeros?
Then there are no real-number restrictions. For example, x²+1 is positive for every real x.
Is a hole the same as a vertical asymptote?
No. A canceled real denominator factor usually produces a removable discontinuity, or hole. A denominator factor that remains after simplification can produce a vertical asymptote.