Math101 · Independent Ontario learning library
Math101Multiplying Rational Expressions
Multiplying rational expressions multiplies numerators and denominators, but factoring first exposes common factors that can be cancelled.
Factored multiplication simplifies algebraic rates and prepares rational equations. Domain awareness distinguishes equivalent formulas from identical functions.
Intuition and core definition
Multiplying rational expressions multiplies numerators and denominators, but factoring first exposes common factors that can be cancelled. Cancellation divides the entire numerator and denominator by the same nonzero factor. All values excluded by original denominators remain excluded.
Notation, language, and conditions
$\frac AB\cdot\frac CD=\frac{AC}{BD}$ where $B,D\ne0$. A factor is a multiplicative component; terms connected by addition cannot cancel. Domain restrictions are identified before simplification, including factors that later disappear.
Why this idea matters
Factoring before multiplying rational expressions exposes cancellable factors while a separate restriction list preserves the original domain.
A dependable method
- Factor every numerator and denominator completely.
- Record values that make any original denominator zero.
- Cancel matching nonzero factors across numerators and denominators.
- Multiply the remaining factors; expand only if requested.
- Check restrictions and compare at a legal numerical value.
