Multiplying Rational Expressions
Learn to multiply rational expressions by multiplying numerators and denominators, then simplifying.
Definition
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Worked Examples
Multiply:
Write as a single fraction
→
Factor out common terms
→ Identify common factors: 12 and
Cancel common factors
and →
Write final answer
Combine the simplified parts → , where
Answer: , where
Common Mistakes
Forgetting to factor before multiplying
Why it's wrong: Multiplying unfactored polynomials creates larger expressions that are harder to simplify later.
Correct: Always factor each numerator and denominator completely BEFORE multiplying. This makes cancellation much easier.
Canceling terms instead of factors
Why it's wrong: You can only cancel factors (things being multiplied), not terms (things being added).
Correct: cannot be simplified by canceling . But because is a factor.
Forgetting domain restrictions
Why it's wrong: Even after cancellation, values that made the original expression undefined are still excluded.
Correct: State all restrictions from the ORIGINAL denominators, not just the simplified form.
Adding instead of multiplying
Why it's wrong: Multiplication and addition of fractions follow different rules.
Correct: For multiplication: multiply across.
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Practice Problems
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Why It Matters
- Solving equations: Many algebraic equations require multiplying fractions to isolate variables
- Physics formulas: Combining rates, like velocity and time, often involves multiplying expressions
- Engineering: Calculating combined effects in circuits or mechanical systems
- Economics: Computing compound growth rates and combined probabilities
Real World Applications
Combined Rates in Physics
When calculating compound effects like resistance or velocity changes, multiplying rational expressions models how quantities combine.
Example:
If a car travels km/h for the first leg and its speed changes by a factor of , the effective rate is km/h.
A factory produces units per hour, where is the number of workers. If efficiency increases by a factor of , what is the new production rate?
Multiply the expressions and simplify.
Step 1: Write the mathematical expression
Calculate:
Probability Calculations
Finding the probability of combined independent events requires multiplying rational expressions.
Example:
If the probability of event A is and event B is , then .
A game has two independent stages. The probability of passing stage 1 is and stage 2 is .
What is the probability of passing both stages?
Step 1: Write the mathematical expression
Calculate:
Key Takeaways
- 1To multiply rational expressions: multiply numerators together and denominators together
- 2Always factor completely BEFORE multiplying to identify common factors
- 3Cancel common factors between any numerator and any denominator
- 4State all domain restrictions from the original expressions
- 5The product of rational expressions is also a rational expression
Frequently Asked Questions
Glossary
- Rational expression
- A fraction where the numerator and/or denominator are polynomials
- Factor
- A number or expression that divides evenly into another
- Domain restriction
- Values of the variable that make the expression undefined (denominator equals zero)
- Cancel
- Divide out common factors from numerator and denominator
Formula Card
Multiplying Rational Expressions
Multiply numerators together and denominators together, then simplify
Simplified Process
Factor all parts, cancel common factors, then multiply remaining terms