Algebraic Fractions (Cambridge O Level Maths)

What is an algebraic fraction?

• An algebraic fraction is a fraction with an algebraic expression on the top (numerator) and/or the bottom (denominator)

How do you simplify an algebraic fraction?

• Factorise fully top and bottom
• Cancel common factors (including common brackets)

How do I add (or subtract) two algebraic fractions?

• The rules are the same as fractions with numbers:

1. Find the lowest common denominator (LCD)
   
   • The LCD of x - 2 and x + 5 is found by multiplying them together: LCD = (x - 2)(x + 5)
     • this is the same as with numbers, where the LCD of 2 and 9 is 2 × 9 = 18
   • The LCD of x and 2x is not found by multiplying them together, as 2x already includes an x , so the LCD is just 2x
     • this is the same as with numbers, where the LCD of 2 and 4 is just 4, not 2 × 4 = 8
   • The LCD of x + 2 and (x + 2)(x - 1) is just (x + 2)(x - 1), as this already includes an (x + 2)
   • The LCD of x + 1 and (x + 1)² is just (x + 1)², as this already includes an (x + 1)
   • The LCD of (x + 3)(x - 1) and (x + 4)(x - 1) is three brackets: (x + 3)(x - 1)(x + 4), without repeating the (x - 1)
2. Write each fraction over this lowest common denominator
3. Multiply the numerators of each fraction by the same amount as the denominators
4. Write as a single fraction over the lowest common denominator (by adding or subtracting the numerators, taking care to use brackets when subtracting)
5. Check at the end to see if the top factorises and cancels

How do I multiply algebraic fractions?

1. Simplify both fractions first by fully factorising, then cancelling any common brackets on top or bottom (from either fraction)
2. Multiply the tops together
3. Multiply the bottoms together
4. Check for any further factorising and cancelling

How do I divide algebraic fractions?

• Flip ("reciprocate") the second fraction and replace ÷ with ×
  • So becomes 

• Then follow the same rules for multiplying two fractions