Algebraic Fractions (AQA GCSE Further Maths) : Revision Note

Simplifying Algebraic Fractions

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)

Adding & Subtracting Algebraic Fractions

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

Multiplying & Dividing Algebraic Fractions

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

Solving Algebraic Fractions

How do I solve an equation that contains algebraic fractions?

• There are two methods for solving equations that contain algebraic fractions

• One method is to deal with the algebraic fractions by adding or subtracting them first and then solving the equation 

  • Follow the rules for solving a linear equation containing a fraction on one or both sides

    • Remove the fractions first by multiplying both sides by everything on the denominator

    • Remember to put brackets around any expression that you multiply by

• The second method is to begin by multiplying everything in the fraction by each of the expressions on the denominator

  • This will remove the denominators of the fractions, leaving you with either a linear or a quadratic equation to solve   

  • Multiplying everything in the fraction by the common denominator is a way of carrying out this process in one go

• For example, to solve the equation you will need to multiply every term in the equation by both and  

  • STEP 1
    Multiply every term by

• STEP 2
  Multiply every term by

• STEP 3
  Expand the brackets on both sides and simplify

• STEP 4
  Rearrange the equation so that it is in a form that can be solved

• STEP 5
  Solve the equation

  • You can swap the sides if it makes solving the equation easier