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Syllabus Edition
First teaching 2021
Last exams 2024
Algebraic Fractions (CIE IGCSE Maths: Extended)
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)
Examiner Tip
- If you are asked to simplify an algebraic fraction and have to factorise the top or bottom, it is very likely that one of the factors will be the same on the top and the bottom – you can use this to help you factorise difficult quadratics!
Worked example
Simplify
Factorise the top, by using 2 as a common factor
Factorise the bottom using your preferred method
Using the fact that the top factorised to may help!
The common factors on the top and bottom reduce to 1 (cancel out)
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Adding & Subtracting Algebraic Fractions
How do I add (or subtract) two algebraic fractions?
- The rules are the same as fractions with numbers:
- 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)
- The LCD of x - 2 and x + 5 is found by multiplying them together: LCD = (x - 2)(x + 5)
- Write each fraction over this lowest common denominator
- Multiply the numerators of each fraction by the same amount as the denominators
- Write as a single fraction over the lowest common denominator (by adding or subtracting the numerators, taking care to use brackets when subtracting)
- Check at the end to see if the top factorises and cancels
Examiner Tip
- Leaving the top and bottom of the fraction in factorised form will help you see if anything cancels at the end
Worked example
(a) Express as a single fraction
The lowest common denominator is
Write each fraction over this common denominator, remember to multiply the top of the fractions too
Simplify the numerators
Combine the fractions, as they have the same denominator
Factorise the top
There are no terms which would cancel here, so this is the final answer
(b) Express as a single fraction
The lowest common denominator is (You could also use but this wouldn't be the lowest common denominator)
Write each fraction over this common denominator, remember to multiply the top of the fractions too
Simplify the numerators
Combine the fractions, as they have the same denominator
There is nothing else that can be factorised on the numerator, so this is the final answer
Multiplying & Dividing Algebraic Fractions
How do I multiply algebraic fractions?
- Simplify both fractions first by fully factorising, then cancelling any common brackets on top or bottom (from either fraction)
- Multiply the tops together
- Multiply the bottoms together
- 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
Worked example
Divide by , giving your answer as a simplified fraction
Division is the same as multiplying by the reciprocal (the fraction flipped)
It can often help to factorise first, as there may be factors that cancel out
Multiply the numerators and denominators, and cancel any terms that are the same on the top and bottom
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