Syllabus Edition

First teaching 2021

Last exams 2024

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Algebraic Fractions (CIE IGCSE Maths: Extended)

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Mark

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Mark

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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 fraction numerator 4 x plus 6 over denominator 2 x squared minus 7 x minus 15 end fraction

Factorise the top, by using 2 as a common factor

fraction numerator 2 open parentheses 2 x plus 3 close parentheses over denominator 2 x squared minus 7 x minus 15 end fraction

Factorise the bottom using your preferred method
Using the fact that the top factorised to open parentheses 2 x plus 3 close parentheses may help!

fraction numerator 2 open parentheses 2 x plus 3 close parentheses over denominator open parentheses 2 x plus 3 close parentheses open parentheses x minus 5 close parentheses end fraction

The common factors on the top and bottom reduce to 1 (cancel out)

fraction numerator 2 up diagonal strike open parentheses 2 x plus 3 close parentheses end strike over denominator up diagonal strike open parentheses 2 x plus 3 close parentheses end strike open parentheses x minus 5 close parentheses end fraction

bold equals fraction numerator bold 2 over denominator begin bold style stretchy left parenthesis x minus 5 stretchy right parenthesis end style end fraction

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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:
  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 + 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

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 fraction numerator x over denominator x plus 4 end fraction minus fraction numerator 3 over denominator x minus 1 end fraction as a single fraction

The lowest common denominator is open parentheses x plus 4 close parentheses open parentheses x minus 1 close parentheses
Write each fraction over this common denominator, remember to multiply the top of the fractions too

fraction numerator x open parentheses x minus 1 close parentheses over denominator open parentheses x plus 4 close parentheses open parentheses x minus 1 close parentheses end fraction minus fraction numerator 3 open parentheses x plus 4 close parentheses over denominator open parentheses x minus 1 close parentheses open parentheses x plus 4 close parentheses end fraction

Simplify the numerators

fraction numerator x squared minus x over denominator open parentheses x plus 4 close parentheses open parentheses x minus 1 close parentheses end fraction minus fraction numerator 3 x plus 12 over denominator open parentheses x minus 1 close parentheses open parentheses x plus 4 close parentheses end fraction

Combine the fractions, as they have the same denominator

fraction numerator x squared minus x minus open parentheses 3 x plus 12 close parentheses over denominator open parentheses x plus 4 close parentheses open parentheses x minus 1 close parentheses end fraction equals fraction numerator x squared minus x minus 3 x minus 12 over denominator open parentheses x plus 4 close parentheses open parentheses x minus 1 close parentheses end fraction equals fraction numerator x squared minus 4 x minus 12 over denominator open parentheses x plus 4 close parentheses open parentheses x minus 1 close parentheses end fraction

Factorise the top

fraction numerator open parentheses x plus 2 close parentheses open parentheses x minus 6 close parentheses over denominator open parentheses x plus 4 close parentheses open parentheses x minus 1 close parentheses end fraction

There are no terms which would cancel here, so this is the final answer

(b) Express fraction numerator x minus 4 over denominator 2 open parentheses x minus 3 close parentheses end fraction minus fraction numerator x minus 1 over denominator 2 x end fraction as a single fraction

The lowest common denominator is 2 x open parentheses x minus 3 close parentheses (You could also use 4 x open parentheses x minus 3 close parentheses 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

fraction numerator x open parentheses x minus 4 close parentheses over denominator 2 x open parentheses x minus 3 close parentheses end fraction minus fraction numerator open parentheses x minus 1 close parentheses open parentheses x minus 3 close parentheses over denominator 2 x open parentheses x minus 3 close parentheses end fraction

Simplify the numerators

fraction numerator x squared minus 4 x over denominator 2 x open parentheses x minus 3 close parentheses end fraction minus fraction numerator x squared minus 4 x plus 3 over denominator 2 x open parentheses x minus 3 close parentheses end fraction

Combine the fractions, as they have the same denominator

fraction numerator x squared minus 4 x minus open parentheses x squared minus 4 x plus 3 close parentheses over denominator 2 x open parentheses x minus 3 close parentheses end fraction equals fraction numerator x squared minus 4 x minus x squared plus 4 x minus 3 over denominator 2 x open parentheses x minus 3 close parentheses end fraction equals fraction numerator negative 3 over denominator 2 x open parentheses x minus 3 close parentheses end fraction

There is nothing else that can be factorised on the numerator, so this is the final answer

Error converting from MathML to accessible text.

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 divided by a over b becomes cross times b over a
  • Then follow the same rules for multiplying two fractions

Worked example

Divide fraction numerator x plus 3 over denominator x minus 4 end fraction by fraction numerator 2 x plus 6 over denominator x squared minus 16 end fraction, giving your answer as a simplified fraction

Division is the same as multiplying by the reciprocal (the fraction flipped)

fraction numerator x plus 3 over denominator x minus 4 end fraction divided by fraction numerator 2 x plus 6 over denominator x squared minus 16 end fraction equals fraction numerator x plus 3 over denominator x minus 4 end fraction cross times fraction numerator x squared minus 16 over denominator 2 x plus 6 end fraction

It can often help to factorise first, as there may be factors that cancel out

fraction numerator x plus 3 over denominator x minus 4 end fraction cross times fraction numerator x squared minus 16 over denominator 2 x plus 6 end fraction equals fraction numerator x plus 3 over denominator x minus 4 end fraction cross times fraction numerator open parentheses x minus 4 close parentheses open parentheses x plus 4 close parentheses over denominator 2 open parentheses x plus 3 close parentheses end fraction

Multiply the numerators and denominators, and cancel any terms that are the same on the top and bottom

equals fraction numerator open parentheses x plus 3 close parentheses open parentheses x minus 4 close parentheses open parentheses x plus 4 close parentheses over denominator 2 open parentheses x minus 4 close parentheses open parentheses x plus 3 close parentheses end fraction equals fraction numerator up diagonal strike open parentheses x plus 3 close parentheses end strike up diagonal strike open parentheses x minus 4 close parentheses end strike open parentheses x plus 4 close parentheses over denominator 2 up diagonal strike open parentheses x minus 4 close parentheses end strike up diagonal strike open parentheses x plus 3 close parentheses end strike end fraction

bold equals fraction numerator bold italic x bold plus bold 4 over denominator bold 2 end fraction

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Mark

Author: Mark

Expertise: Maths

Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.