Math101 · Independent Ontario learning library
Math101Simplifying Rational Expressions
Simplifying a rational expression factors numerator and denominator and cancels common nonzero factors. It does not cancel terms joined by addition.
Simplification exposes behaviour, makes operations easier, and distinguishes holes from asymptotes. Factor-based cancellation is fundamental across algebra and calculus.
Intuition and core definition
Simplifying a rational expression factors numerator and denominator and cancels common nonzero factors. It does not cancel terms joined by addition. The simplified expression agrees with the original only on the original domain, so excluded values must be retained.
Notation, language, and conditions
$\frac{P(x)}{Q(x)}$ requires $Q(x)\ne0$. If $P=RF$ and $Q=RG$, then $P/Q=F/G$ where $R\ne0$. A cancelled factor creates a removable discontinuity, or hole, rather than making the original expression defined there.
Why this idea matters
Factoring rational expressions distinguishes removable common factors from denominator factors that remain and create vertical asymptotes.
A dependable method
- Factor numerator and denominator completely.
- List zeros of the original denominator as restrictions.
- Cancel identical factors, not individual terms.
- Leave the result factored or expand according to purpose.
- Verify at legal inputs and keep excluded values beside the answer.
