Understanding Algebraic Fractions
Algebraic fractions are fractions where the numerator, the denominator, or both contain algebraic expressions involving variables. For example, \( \frac{x}{x-4} \) and \( \frac{3a}{9a} \) are algebraic fractions because they include variables \( x \) and \( a \). Working with these requires careful manipulation to simplify expressions and perform operations such as addition, subtraction, multiplication, and division.
The key difference from numerical fractions is that variables can affect domain restrictions (values for which the expression is defined) and require factorisation to simplify or combine terms.
- Variables can appear in numerator, denominator, or both.
- Expressions must be simplified carefully to avoid undefined values.
- Factorisation helps to identify common factors for simplification.
Simplifying Algebraic Fractions
Simplification involves rewriting the fraction in its simplest form by factorising numerator and denominator and cancelling common factors. This reduces complexity and helps in further operations.
For example, consider \( \frac{3a}{9a} \). Both numerator and denominator contain the factor \( a \), which can be cancelled (provided \( a \neq 0 \)). Also, 3 and 9 share a common factor 3.
Step-by-step: 1) Factor numerator and denominator: numerator is \( 3 \times a \), denominator is \( 9 \times a = 3 \times 3 \times a \). 2) Cancel common factors \( 3 \) and \( a \). 3) Remaining fraction is \( \frac{1}{3} \). Always state domain restrictions such as \( a \neq 0 \) to avoid division by zero.
- Factorise numerator and denominator fully.
- Cancel common factors carefully.
- State any restrictions on variables to avoid division by zero.
Adding and Subtracting Algebraic Fractions
Addition and subtraction require a common denominator. Unlike numerical fractions, the denominators often contain algebraic expressions that must be factorised to find the least common denominator (LCD).
Example: Add \( \frac{x}{x-4} + \frac{3}{2x} \). 1) Identify denominators: \( x-4 \) and \( 2x \). 2) The LCD is \( (x-4)(2x) \). 3) Rewrite each fraction with the LCD denominator: \( \frac{x}{x-4} = \frac{x \times 2x}{(x-4)(2x)} = \frac{2x^2}{(x-4)(2x)} \), \( \frac{3}{2x} = \frac{3(x-4)}{(x-4)(2x)} \). 4) Add numerators: \( 2x^2 + 3(x-4) = 2x^2 + 3x - 12 \). 5) Final expression: \( \frac{2x^2 + 3x - 12}{(x-4)(2x)} \). 6) Factor numerator if possible and simplify further.
Always check for factorisation opportunities after combining fractions.
- Find the least common denominator by factorising denominators.
- Rewrite each fraction with the LCD denominator.
- Add or subtract numerators only, keep denominator the same.
- Simplify the resulting expression by factorising numerator and denominator.
Multiplying Algebraic Fractions
Multiplication is more straightforward than addition or subtraction. Multiply numerators together and denominators together, then simplify the resulting fraction.
Example: Multiply \( \frac{3a}{9a} \times \frac{4}{10} \). 1) Multiply numerators: \( 3a \times 4 = 12a \). 2) Multiply denominators: \( 9a imes 10 = 90a \). 3) The fraction is \( \frac{12a}{90a} \). 4) Cancel \( a \) (assuming \( a \neq 0 \)) and simplify \( \frac{12}{90} = \frac{2}{15} \). 5) Final answer: \( \frac{2}{15} \).
Always factorise numerator and denominator before cancelling common factors to simplify efficiently.
- Multiply numerators together.
- Multiply denominators together.
- Factorise and cancel common factors.
- State any variable restrictions.
Dividing Algebraic Fractions
Division of algebraic fractions involves multiplying by the reciprocal of the divisor. For \( \frac{A}{B} \div \frac{C}{D} \), rewrite as \( \frac{A}{B} \times \frac{D}{C} \).
Example: Divide \( \frac{x}{x-4} \div \frac{3}{2x} \). 1) Rewrite division as multiplication by reciprocal: \( \frac{x}{x-4} \times \frac{2x}{3} \). 2) Multiply numerators: \( x imes 2x = 2x^2 \). 3) Multiply denominators: \( (x-4) imes 3 = 3(x-4) \). 4) Expression: \( \frac{2x^2}{3(x-4)} \). 5) Simplify if possible and state restrictions \( x \neq 0, x \neq 4 \).
Always check domain restrictions to avoid division by zero.
- Rewrite division as multiplication by reciprocal.
- Multiply numerators and denominators.
- Simplify the resulting fraction.
- State variable restrictions to avoid undefined expressions.
At-a-glance summary
Comparison table
| Step | Detail | Example | Exam Use |
|---|---|---|---|
| Simplification | Factorise numerator and denominator, then cancel common factors | \( \frac{3a}{9a} = \frac{3 \times a}{3 \times 3 \times a} = \frac{1}{3} \) | Always state restrictions on variables to avoid division by zero |
| Addition/Subtraction | Find common denominator by factorising denominators, rewrite fractions, add/subtract numerators | \( \frac{x}{x-4} + \frac{3}{2x} = \frac{2x^2 + 3x - 12}{(x-4)(2x)} \) | Show all steps clearly, factorise numerator after addition |
| Multiplication | Multiply numerators and denominators directly, then simplify | \( \frac{3a}{9a} \times \frac{4}{10} = \frac{12a}{90a} = \frac{2}{15} \) | Cancel common factors before multiplying to simplify calculations |
| Division | Multiply by reciprocal of divisor fraction | \( \frac{x}{x-4} \div \frac{3}{2x} = \frac{x}{x-4} \times \frac{2x}{3} = \frac{2x^2}{3(x-4)} \) | State domain restrictions and simplify final expression |
Active recall
Quick check
1. Simplify \( \frac{6x}{9x} \) assuming \( x \neq 0 \).
Cancel \( x \) and divide numerator and denominator by 3: \( \frac{6x}{9x} = \frac{6}{9} = \frac{2}{3} \). Final answer: \( \frac{2}{3} \).
2. Add \( \frac{2}{x} + \frac{3}{x+1} \).
Common denominator is \( x(x+1) \). Rewrite: \( \frac{2(x+1)}{x(x+1)} + \frac{3x}{x(x+1)} = \frac{2x + 2 + 3x}{x(x+1)} = \frac{5x + 2}{x(x+1)} \).
3. Multiply \( \frac{x}{x-1} \times \frac{x-1}{x+2} \).
Cancel \( x-1 \) (assuming \( x \neq 1 \)): \( \frac{x}{\cancel{x-1}} \times \frac{\cancel{x-1}}{x+2} = \frac{x}{x+2} \).
4. Divide \( \frac{3x}{4} \div \frac{x}{2} \) assuming \( x \neq 0 \).
Multiply by reciprocal: \( \frac{3x}{4} \times \frac{2}{x} = \frac{3x \times 2}{4 \times x} = \frac{6x}{4x} \). Cancel \( x \): \( \frac{6}{4} = \frac{3}{2} \).