Rational Expression Calculator
Simplify, add, subtract, multiply, or divide rational expressions (fractions with polynomial numerators and denominators) in one step. Ideal for algebra students checking homework or teachers preparing worked examples.
About this calculator
A rational expression is a fraction of the form P(x)/Q(x) where P and Q are polynomials and Q ≠ 0. To simplify, you find the greatest common divisor (GCD) of the numerator and denominator using the Euclidean algorithm and divide both by it: simplified = (n/GCD) / (d/GCD). To add two fractions a/b + c/d, you compute (a·d + c·b) / (b·d) and then simplify. Multiplication follows (a/b) × (c/d) = (a·c)/(b·d), and division follows (a/b) ÷ (c/d) = (a·d)/(b·c). The critical rule when working with rational expressions is that the denominator can never equal zero — values of x that make Q(x) = 0 are excluded from the domain. Mastering these operations is essential for solving rational equations and understanding limits in calculus.
How to use
Example — add 3/8 and 5/12. The calculator computes numerator = 3×12 + 5×8 = 36 + 40 = 76 and denominator = 8×12 = 96, giving 76/96. It then finds GCD(76, 96) = 4 and simplifies to 19/24. Enter Operation = Add, First Numerator = 3, First Denominator = 8, Second Numerator = 5, Second Denominator = 12, and click Calculate. The result 19/24 appears immediately. For simplify mode, enter just the single fraction's numerator and denominator and the tool reduces it to lowest terms automatically.
Frequently asked questions
How do you simplify a rational expression step by step?
First, factor both the numerator and the denominator completely into their prime or polynomial factors. Next, identify any factors that appear in both the numerator and denominator — these are the common factors. Cancel (divide out) those common factors, since any non-zero quantity divided by itself equals one. Finally, write the remaining factors as the simplified expression and state any restrictions on the variable (values that would make the original denominator zero). For purely numeric fractions, the GCD computed by the Euclidean algorithm achieves the same result instantly.
Why is it important to find restrictions when simplifying rational expressions?
Restrictions are values of the variable that make any denominator in the original expression equal to zero, which is undefined in mathematics. Even after you cancel a common factor, the restriction from that factor still applies to the simplified form. For example, (x² − 4)/(x − 2) simplifies to x + 2, but x = 2 remains excluded because it made the original denominator zero. Ignoring restrictions can lead to incorrect solutions when you set simplified expressions equal to values or substitute into equations. Stating restrictions is standard practice in algebra and required on most standardized tests.
What is the difference between multiplying and dividing rational expressions?
Multiplying rational expressions is straightforward: multiply numerators together and denominators together, then simplify — (a/b) × (c/d) = (ac)/(bd). Division requires one extra step: flip (take the reciprocal of) the second fraction and then multiply — (a/b) ÷ (c/d) = (a/b) × (d/c) = (ad)/(bc). A common mistake is forgetting to flip before multiplying during division. It also helps to cross-cancel common factors before multiplying, which keeps the numbers smaller and the arithmetic easier. Both operations follow the same rules whether the expressions contain numbers, variables, or full polynomials.