Reviewed by CalcMulti Editorial Team·Last updated: ·← Algebra Hub
A rational expression is a fraction where the numerator and denominator are polynomials — for example, (x² − 4) / (x² − x − 6). Working with rational expressions is the algebraic equivalent of working with numeric fractions, with factoring playing the role that prime factorization plays for number fractions.
This calculator simplifies rational expressions by factoring numerator and denominator, canceling common polynomial factors, and stating any excluded values (values of x that make the denominator zero). It also multiplies and divides pairs of rational expressions step by step.
To simplify a rational expression: (1) Factor the numerator completely. (2) Factor the denominator completely. (3) Cancel factors that appear in both. (4) Note the excluded values where any denominator factor equals zero. For example, (x²−4)/(x²−x−6) = (x+2)(x−2)/((x+3)(x−2)) = (x+2)/(x+3), with x ≠ 2 and x ≠ −3.
Rational expressions appear throughout calculus and algebra. Finding horizontal and oblique asymptotes of rational functions requires dividing the polynomials. Partial fraction decomposition (used in integration) reverses the process of combining rational expressions. Domain restrictions from denominators create vertical asymptotes on graphs. Mastering simplification of rational expressions is prerequisite knowledge for calculus, pre-calculus, and algebra 2.
P(x)/Q(x) = factor both → cancel common factors → state excluded values (Q(x) ≠ 0)
Select an example to see the full simplification with steps.
Original Expression
(x²−4) / (x²−x−6)
Simplified Form
(x + 2) / (x + 3)
Excluded Values
x ≠ 2, x ≠ −3
| Operation | Rule | Example |
|---|---|---|
| Simplify | Factor both; cancel common FACTORS | (x²−4)/(x−2) = (x+2)(x−2)/(x−2) = x+2 |
| Multiply | Factor all; cross-cancel; multiply | (x+2)/(x−1) × (x²−1)/(x+2) = x+1 |
| Divide | Multiply by reciprocal of divisor | (x+2)/x ÷ (x+2)/x² = x |
| Add (same denom) | (P + Q)/R | (x+1)/(x−2) + 3/(x−2) = (x+4)/(x−2) |
| Add (diff denom) | Find LCD, convert, add | 1/x + 1/(x+1) = (2x+1)/(x(x+1)) |
| Cannot cancel | Do NOT cancel terms, only factors | (x+4)/(x+4) = 1; (x+4)/4 ≠ x |
This calculator is for educational purposes only and does not constitute professional advice. Results are based on standard mathematical formulas. Always verify critical calculations with a qualified professional before making important decisions.