Algebra 2 · Unit 4 · Topic 4.1
Simplifying Rational Expressions and Preserving Domain
A rational expression is a quotient of polynomials with a nonzero denominator.
Definition and explanation
A rational expression is a quotient of polynomials with a nonzero denominator. Factoring and canceling shared nonzero factors can simplify it, but excluded inputs from the original denominator remain excluded.
Worked example
Simplify (x² − 9)/(x² + x − 6). Factoring gives (x − 3)(x + 3)/[(x + 3)(x − 2)] = (x − 3)/(x − 2), with x ≠ −3 and x ≠ 2. The canceled factor does not restore x = −3.
Common mistake
Cancel common factors, not separate terms inside a sum.
Key takeaway
A rational expression is a quotient of polynomials with a nonzero denominator.
Explain each step of the worked example, then check the units, signs, and conditions that apply.
Further reading
Topic coverage was checked against the openly available OpenStax Intermediate Algebra 2e, Algebra and Trigonometry 2e, and Introductory Statistics 2e textbooks. The explanations and examples on this site are independently written; textbook prose, exercises, and artwork are not reproduced.