Lesson 2 • 40 min read
Rational Expressions
In This Lesson
1What Are Rational Expressions?
A rational expression is a fraction where the numerator and/or denominator contains polynomials.
(polynomial) / (polynomial)
x/5
Simple
(x+1)/(x-2)
Linear
(x²-4)/(x+2)
Quadratic
⚠️ Restrictions
The denominator can NEVER equal zero! Any value that makes the denominator zero is excluded from the domain.
For (x+1)/(x-2): x ≠ 2 (because x-2=0 when x=2)
2Simplifying Rational Expressions
Steps to Simplify
- Factor the numerator completely
- Factor the denominator completely
- Cancel common factors (not terms!)
- Write the simplified expression
Example 1: Simplify (x² - 9)/(x + 3)
= (x + 3)(x - 3)/(x + 3) ← Factor numerator
= (x + 3)(x - 3)/(x + 3) ← Cancel
= x - 3
Example 2: Simplify (6x²)/(9x³)
= (6x²)/(9x³)
= (2 × 3 × x²)/(3 × 3 × x³)
= 2/(3x)
Example 3: Simplify (x² - 4)/(x² - 4x + 4)
= (x + 2)(x - 2)/(x - 2)² ← Factor both
= (x + 2)(x - 2)/[(x - 2)(x - 2)]
= (x + 2)/(x - 2)
3Multiplication
Multiply Numerators and Denominators
(a/b) × (c/d) = (a × c)/(b × d)
Pro Tip: Factor First, Then Cancel!
Cross-cancel common factors before multiplying to keep numbers smaller.
Example: (x² - 4)/(x + 1) × (x + 1)/(x - 2)
= [(x+2)(x-2)]/(x+1) × (x+1)/(x-2) ← Factor
= [(x+2)(x-2)]/(x+1) × (x+1)/(x-2) ← Cancel
= x + 2
4Division
Multiply by the Reciprocal
(a/b) ÷ (c/d) = (a/b) × (d/c)
Keep the first fraction as is
Change division to multiplication
Flip the second fraction (reciprocal)
Example: (x²)/(x+3) ÷ (x)/(x-1)
= (x²)/(x+3) × (x-1)/(x) ← Flip
= (x² × (x-1))/((x+3) × x)
= (x(x-1))/(x+3) ← Cancel x
= x(x-1)/(x+3)
5Addition & Subtraction
Same Denominator (Easy!)
Add or subtract the numerators, keep the denominator.
a/c + b/c = (a + b)/c
Example: (3x)/(x+1) + (2)/(x+1)
= (3x + 2)/(x + 1)
Different Denominators (Need LCD)
- Find the LCD (Least Common Denominator)
- Rewrite each fraction with the LCD
- Add or subtract numerators
- Simplify if possible
Example: (2)/(x) + (3)/(x+1)
LCD: x(x+1)
= [2(x+1)]/[x(x+1)] + [3x]/[x(x+1)]
= (2x + 2 + 3x)/[x(x+1)]
= (5x + 2)/[x(x+1)]
Example: (x)/(x-2) - (3)/(x+2)
LCD: (x-2)(x+2)
= [x(x+2)]/[(x-2)(x+2)] - [3(x-2)]/[(x-2)(x+2)]
= (x² + 2x - 3x + 6)/[(x-2)(x+2)]
= (x² - x + 6)/[(x-2)(x+2)]