Algebraic Fractions
2.3 Algebraic Fractions – Cheat Sheet
1. What is an Algebraic Fraction?
- A fraction where the numerator, denominator, or both contain algebraic expressions.
- Examples:
- 3x / (x + 2)
- (a² + b²) / (a − b)
2. Key Rules for Algebraic Fractions
| Rule | Example | Solution |
|---|---|---|
| Cancel common factors (must be factors, not terms added/subtracted) | (x² + 2x) / (x) | (x(x + 2)) / x = x + 2 |
| Multiplication – multiply across | (3a² / b) × (2b³ / a) | (6a²b³) / (ab) = 6ab² |
| Division – multiply by reciprocal | (3a² / b) ÷ (6a / b²) | (3a² / b) × (b² / 6a) = (3a²b²) / (6ab) = (ab) / 2 |
| Addition/Subtraction – same denominator | (x + 2)/y + (x − 3)/y | [(x + 2) + (x − 3)] / y = (2x − 1)/y |
| Addition/Subtraction – different denominators | 1/(x − 2) + (x + 1)/(x − 3) | Common denominator: (x − 2)(x − 3) |
3. Factorising in Algebraic Fractions
| Example | Factorised Form |
|---|---|
| x² − 5x + 6 | (x − 2)(x − 3) |
| 2x³ − 3x² + 10x | x(2x² − 3x + 10) |
| x³ + x² − 4x | x(x² + x − 4) |
4. Simplifying Rational Expressions
Example 1:
( x² − 2x ) / ( x² − 5x + 6 )
- Factorise numerator: x(x − 2)
- Factorise denominator: (x − 2)(x − 3)
- Cancel (x − 2): x / (x − 3)
Example 2:
3a⁴ × 9a¹⁰
- Multiply coefficients: 3 × 9 = 27
- Add powers of a: a⁴ × a¹⁰ = a¹⁴
- Answer: 27a¹⁴
Example 3:
3a⁴ ÷ 9a¹⁰
- Divide coefficients: 3 ÷ 9 = 1/3
- Subtract powers of a: a⁴ ÷ a¹⁰ = a⁻⁶ = 1 / a⁶
- Answer: 1 / (3a⁶)
5. Adding/Subtracting Algebraic Fractions
Example:
1 / (x − 2) + (x + 1) / (x − 3)
- Common denominator: (x − 2)(x − 3)
- First fraction numerator: 1 × (x − 3) = x − 3
- Second fraction numerator: (x + 1) × (x − 2) = x² − 2x + x − 2 = x² − x − 2
- Combine: [x − 3 + x² − x − 2] / [(x − 2)(x − 3)] = (x² − 5) / [(x − 2)(x − 3)]
6. Examples Table for Quick Reference
| Problem | Process | Answer |
|---|---|---|
| (x² + 2x) / x | Factorise numerator | x + 2 |
| (3a² / b) × (2b³ / a) | Multiply across, simplify | 6ab² |
| (3a² / b) ÷ (6a / b²) | Multiply by reciprocal | (ab) / 2 |
| 3a⁴ × 9a¹⁰ | Multiply coeffs, add powers | 27a¹⁴ |
| 3a⁴ ÷ 9a¹⁰ | Divide coeffs, subtract powers | 1 / (3a⁶) |
| (x² − 2x) / (x² − 5x + 6) | Factorise, cancel | x / (x − 3) |
| 1/(x − 2) + (x + 1)/(x − 3) | Common denominator | (x² − 5) / [(x − 2)(x − 3)] |