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Math101learn.math101.caMultiplying 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.
Worked example
Representations and interpretation
A factor ledger places numerator factors above a line and denominator factors below it. Matching entries may cancel, while a separate restriction column records the original domain. A graph would show the constant $1$ with holes at excluded inputs.
Reasoning about variations
If factors do not match exactly, no cancellation occurs: $x+2$ and $x+3$ are different factors. A negative can be factored to reveal matches, since $2-x=-(x-2)$.
Common mistakes
How to check your work
- Substitute a legal value into original and simplified expressions.
- Multiply uncancelled factors and compare with the factored original.
- Verify every cancellation involved an identical factor and a stated nonzero condition.
Practice
- Simplify $\frac{x^2-4}{x^2+x-6}\cdot\frac{x+3}{x+2}$.
- Can $x$ cancel from $(x+1)/x$?
- What restriction comes from denominator $x^2-4$?
Answers and brief solutions
Show answers
- $1$ Factoring cancels all displayed factors, with original restrictions $x\ne2,-3,-2$ retained.
- No $x$ is not a factor of the entire numerator.
- $x\ne2,-2$ Factor as $(x-2)(x+2)$.
Synthesis and transfer
When two algebraic conversion factors are chained, common units and factors cancel multiplicatively, but any input excluded in an original denominator remains absent from the result.
Take $\frac{x^2-9}{x^2-x-6}\cdot\frac{x-2}{x+3}$. The original denominators exclude $x=3,-2,-3$. Factoring gives $\frac{(x-3)(x+3)}{(x-3)(x+2)}\cdot\frac{x-2}{x+3}$, which simplifies to $(x-2)/(x+2)$ without restoring any excluded input. Cancellation is division by nonzero common factors, so it is justified only after the restriction list is recorded. At $x=0$, both forms evaluate to $-1$, providing a quick legal-point check; at $x=3$, the simplified-looking formula cannot replace the undefined original function.
Related topics
Teaching and accessibility note
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Simplify $\frac{x^2-4}{x^2+x-6}\cdot\frac{x+3}{x+2}$.
- Factoring cancels all displayed factors, with original restrictions $x\ne2,-3,-2$ retained.
End of lesson
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