Rational Expressions- Unit 6 Study Guide
What Are Rational Expressions?
A rational expression is a fraction where both the numerator and denominator are polynomials. That's it. Nothing fancy. If you can handle fractions with numbers, you can handle these—you just need to be careful with algebra.
The general form is:
P(x) / Q(x)
where P and Q are polynomials, and Q(x) ≠ 0. That last part matters. You cannot divide by zero. Ever.
Domain Restrictions
Before you do anything with a rational expression, find the values that make the denominator zero. These are excluded from the domain.
Example: For (x + 3) / (x² - 4)
- Set denominator equal to zero: x² - 4 = 0
- Solve: x² = 4
- x = 2 or x = -2
- Domain excludes x = 2 and x = -2
Write your final answer with these restrictions noted. Forgetting this is one of the most common mistakes students make.
Simplifying Rational Expressions
Factor both numerator and denominator, then cancel any common factors.
Step-by-Step Process
- Factor the numerator completely
- Factor the denominator completely
- Cancel matching factors (not terms—factor first)
- State any domain restrictions lost in cancellation
Example
Simplify (x² - 9) / (x² + 5x + 6)
- Factor numerator: (x + 3)(x - 3)
- Factor denominator: (x + 3)(x + 2)
- Cancel (x + 3): 1 / (x + 2)
- Restriction: x ≠ -3
Multiplying Rational Expressions
Multiply numerators together. Multiply denominators together. Simplify by canceling common factors before multiplying—it's easier.
Example: [(x + 2) / (x - 4)] × [(x + 5) / (x + 3)]
- Multiply: (x + 2)(x + 5) / (x - 4)(x + 3)
- Check for common factors: none exist
- Answer: (x + 2)(x + 5) / (x - 4)(x + 3)
Dividing Rational Expressions
Flip the second fraction (take the reciprocal), then multiply. Remember: keep, change, flip.
Example: [(x + 2) / (x - 4)] ÷ [(x + 5) / (x + 3)]
- Flip: (x + 3) / (x + 5)
- Multiply: (x + 2)(x + 3) / (x - 4)(x + 5)
- Answer: (x + 2)(x + 3) / (x - 4)(x + 5)
Adding and Subtracting Rational Expressions
You need a common denominator. Two cases:
Same Denominator
Easy. Add or subtract the numerators, keep the denominator, simplify.
(3x / (x+2)) + (5 / (x+2)) = (3x + 5) / (x+2)
Different Denominators
Find the Least Common Denominator (LCD).
- Factor each denominator
- LCD = product of highest powers of each factor
- Rewrite each fraction with LCD
- Combine numerators
- Simplify
Example
(3/x) + (5/(x+2))
- Denominators: x and (x+2)
- LCD: x(x+2)
- Rewrite: (3(x+2))/(x(x+2)) + (5x)/(x(x+2))
- Combine: (3x + 6 + 5x) / x(x+2)
- Simplify: (8x + 6) / x(x+2)
- Factor if possible: 2(4x + 3) / x(x+2)
Complex Rational Expressions
A complex rational expression has a fraction in the numerator, denominator, or both.
[1/(x-1)] / [2/(x+1)]
Two methods to simplify:
Method 1: Division
- Rewrite as: (numerator) ÷ (denominator)
- Example: [1/(x-1)] ÷ [2/(x+1)]
- Flip second fraction: [1/(x-1)] × [(x+1)/2]
- Multiply: (x+1) / 2(x-1)
Method 2: Clear Inner Fractions
- Find LCD of all inner fractions
- Multiply entire expression by LCD
- Simplify
Both methods work. Use whichever feels faster for the problem in front of you.
Solving Rational Equations
Rational equations have rational expressions set equal to each other. The goal: find x values that make the equation true.
The Process
- Find LCD of all denominators
- Multiply both sides by LCD
- This clears all denominators
- Solve the resulting polynomial equation
- Check every answer in the original equation
That last step is non-negotiable. Multiplying by expressions containing variables can create extraneous solutions—answers that look right but aren't.
Example
2/x = 3/(x-1)
- LCD: x(x-1)
- Multiply: 2(x-1) = 3x
- Simplify: 2x - 2 = 3x
- Solve: -2 = x
- Check: 2/(-2) = -1 and 3/(-3) = -1 ✓
- Solution: x = -2
Word Problems and Applications
Rational expressions show up in rate problems, work problems, and proportional relationships.
Work Problems
If person A completes a task in a hours and person B in b hours:
Combined rate = 1/a + 1/b
Time to complete together = 1 / (1/a + 1/b)
Rate Problems
Distance = rate × time. When combining rates:
- Same direction: add rates
- Opposite directions: subtract rates
Common Mistakes to Avoid
| Mistake | Why It's Wrong |
|---|---|
| Canceling terms instead of factors | Can only cancel factors: (x+2)/(2+5) = can't cancel |
| Forgetting domain restrictions | Always identify values that make denominator zero |
| Not checking solutions | Extraneous solutions are real—verify everything |
| Incorrectly finding LCD | Must use all factors at highest power |
| Sign errors when subtracting | Distribute negatives carefully |
Quick Reference Summary
- Simplify: Factor → Cancel common factors
- Multiply: Multiply across, then simplify
- Divide: Keep first fraction, flip second, multiply
- Add/Subtract: Find LCD, rewrite, combine numerators
- Solve equations: Clear denominators, solve, check answers
Practice Tips
Work through problems until factoring polynomials becomes automatic. If you're slow at factoring, rational expressions will feel impossible. Practice GCF factoring, difference of squares, and trinomial factoring until they're reflex.
When stuck on a complex problem, back up. Can you simplify any factors first? Can you rewrite a complex fraction as a division problem? Most problems have a cleaner path—they just hide it.