Introduction to Rational Expressions
Learn what rational expressions are, how to identify them, and understand their key properties including domain restrictions.
Definition
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Worked Examples
Which of these are rational expressions? a) b) c)
Check expression (a)
Numerator: (polynomial). Denominator: (polynomial) → YES - rational expression
Check expression (b)
Numerator: (NOT a polynomial - has a radical) → NO - not a rational expression
Check expression (c)
Numerator: (constant polynomial). Denominator: (polynomial) → YES - rational expression
Answer: Expressions (a) and (c) are rational expressions. Expression (b) is not because is not a polynomial.
Common Mistakes
Setting the numerator equal to zero instead of the denominator
Why it's wrong: When finding restrictions, we need to find when the expression is undefined. Division by zero is undefined, not division into zero.
Correct: Always set the DENOMINATOR equal to zero to find restrictions. When the numerator is zero, the expression simply equals zero.
Forgetting to factor the denominator completely
Why it's wrong: A quadratic denominator like has TWO roots, so there are TWO restrictions.
Correct: Always factor the denominator completely before solving. gives restrictions at AND .
Thinking and are the same
Why it's wrong: (zero divided by anything is zero), but is undefined (cannot divide by zero).
Correct: Zero in the numerator gives zero. Zero in the denominator is undefined.
Interactive Sandbox
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Practice Problems
18 problemsWhich of the following is a rational expression?
Why It Matters
- Physics: The formula for lens magnification is
- Chemistry: Concentration calculations use ratios of polynomials
- Economics: Cost per unit is often expressed as
- Engineering: Electrical resistance in parallel circuits uses
Real World Applications
Average Speed Problems
When traveling different distances at different speeds, the average speed is a rational expression.
Example:
If you drive 100 km at speed and return at speed , the average speed for the whole trip is
Work Rate Problems
When two people work together, their combined rate involves rational expressions.
Example:
If Alice completes a job in hours and Bob in hours, working together they complete it in hours.
Electrical Circuits
Parallel resistance formulas use rational expressions.
Example:
For two resistors in parallel:
Key Takeaways
- 1A rational expression is a fraction with polynomials in both numerator and denominator
- 2The denominator can NEVER equal zero (division by zero is undefined)
- 3To find domain restrictions, set the denominator equal to zero and solve
- 4Factor the denominator completely to find ALL restrictions
- 5Always check for restrictions before evaluating a rational expression
Frequently Asked Questions
Glossary
- Rational expression
- A fraction where both the numerator and denominator are polynomials
- Domain restriction
- A value of the variable that makes the expression undefined (when denominator equals zero)
- Excluded value
- Another term for domain restriction; a value that must be excluded from the domain
- Undefined
- When an expression has no valid numerical value, typically due to division by zero