GCSEFurther MathsAQARevision Notes2. Algebra & FunctionsBinomial ExpansionBinomial Expansion

Binomial Expansion (AQA GCSE Further Maths): Revision Note

Exam code: 8365

Jamie Wood

Written by: Jamie Wood

Reviewed by: Dan Finlay

Updated on

Pascal's Triangle

What is Pascal's Triangle?

  • Pascal's Triangle is a triangle of number patterns (shown below)

    • where each number is the sum of the two numbers above it

      • It can help to imagine zeros outside the triangle

    • The triangle is symmetric

pascals-triangle-no-labels

Binomial expansion

What is a binomial?

  • A binomial is the sum or difference of two different terms

    • e.g. 3 plus 4 x or p minus q

  • A binomial can be raised to a power, n

    • This has the form open parentheses a plus b close parentheses to the power of n

How do I expand a binomial?

  • You can expand a binomial by multiplying out brackets, but the bigger the power, the longer this takes

    • open parentheses a plus b close parentheses to the power of 1 equals a plus b

    • open parentheses a plus b close parentheses squared equals open parentheses a plus b close parentheses open parentheses a plus b close parentheses

      • This gives a squared plus 2 a b plus b squared

    • open parentheses a plus b close parentheses cubed equals open parentheses a plus b close parentheses open parentheses a plus b close parentheses open parentheses a plus b close parentheses

      • This gives a cubed plus 3 a squared b plus 3 a b squared plus b cubed

    • open parentheses a plus b close parentheses to the power of 4 equals open parentheses a plus b close parentheses open parentheses a plus b close parentheses open parentheses a plus b close parentheses open parentheses a plus b close parentheses

      • This eventually gives a to the power of 4 plus 4 a cubed b plus 6 a squared b squared plus 4 a b cubed plus b to the power of 4

pascals-triangle-no-labels

  • The coefficients of the terms in the expansion of open parentheses a plus b close parentheses to the power of n correspond to a row in Pascal's triangle

    • open parentheses a plus b close parentheses to the power of 0 equals bold 1

    • open parentheses a plus b close parentheses to the power of 1 equals bold 1 a plus bold 1 b

    • open parentheses a plus b close parentheses squared equals bold 1 a squared plus bold 2 a b plus bold 1 b squared

    • open parentheses a plus b close parentheses cubed equals bold 1 a cubed plus bold 3 a squared b plus bold 3 a b squared plus bold 1 b cubed

    • open parentheses a plus b close parentheses to the power of 4 equals bold 1 a to the power of 4 plus bold 4 a cubed b plus bold 6 a squared b squared plus bold 4 a b cubed plus bold 1 b to the power of 4

How do I expand a binomial using Pascal's triangle?

  • Using a to the power of 0 equals b to the power of 0 equals 1, you can write down rules to expand open parentheses a plus b close parentheses to the power of n without having to multiply out brackets

    • It is the sum of terms in the form Pascal space coefficient space cross times a to the power of open parentheses... close parentheses end exponent cross times b to the power of open parentheses... close parentheses end exponent where

      • the powers of a decrease from a to the power of n to a to the power of 0

      • the powers of b increase from b to the power of 0 to b to the power of n

      • the Pascal coefficients come from the row starting with 1 comma space space n comma space space...

    • There should be open parentheses n plus 1 close parentheses terms in total

  • For example, to expand open parentheses a plus b close parentheses to the power of 4, there will be 4 + 1 = 5 terms as follows:

Power of a

a to the power of 4

a cubed

a squared

a to the power of 1

a to the power of 0

Power of b

b to the power of 0

b to the power of 1

b squared

b cubed

b to the power of 4

Pascal's triangle row

1

4

6

4

1

open parentheses a plus b close parentheses to the power of 4 equals

1 a to the power of 4 b to the power of 0

plus 4 a cubed b to the power of 1

plus 6 a squared b squared

plus 4 a to the power of 1 b cubed

plus 1 a to the power of 0 b to the power of 4

  • This simplifies to

    • open parentheses a plus b close parentheses to the power of 4 equals a to the power of 4 plus 4 a cubed b plus 6 a squared b squared plus 4 a b cubed plus b to the power of 4

How do I expand binomials with harder terms?

  • You need to be familiar with index laws, e.g.:

    • open parentheses 2 x close parentheses cubed equals 2 cubed x cubed equals 8 x cubed

    • open parentheses x over 2 close parentheses cubed equals x cubed over 2 cubed equals x cubed over 8

  • For example

    • To expand open parentheses 2 x plus 3 close parentheses to the power of 4

      • Imagine open parentheses a plus b close parentheses to the power of 4 where a equals 2 x and b equals 3

      • Put brackets around open parentheses 2 x close parentheses and open parentheses 3 close parentheses

      • Then use the rules above

    • open parentheses 2 x plus 3 close parentheses to the power of 4 equals 1 open parentheses 2 x close parentheses to the power of 4 open parentheses 3 close parentheses to the power of 0 plus 4 open parentheses 2 x close parentheses cubed open parentheses 3 close parentheses to the power of 1 plus 6 open parentheses 2 x close parentheses squared open parentheses 3 close parentheses squared plus 4 open parentheses 2 x close parentheses to the power of 1 open parentheses 3 close parentheses cubed plus 1 open parentheses 2 x close parentheses to the power of 0 open parentheses 3 close parentheses to the power of 4

    • Apply the index laws carefully

      • equals 2 to the power of 4 x to the power of 4 cross times 1 plus 4 cross times 2 cubed x cubed cross times 3 plus 6 cross times 2 squared x squared cross times 3 squared plus 4 cross times 2 x cross times 3 cubed plus 3 to the power of 4

      • equals 16 x to the power of 4 plus 96 x cubed plus 216 x squared plus 216 x plus 81

  • Note that the final coefficients are not symmetric

    • even though the Pascal coefficients used in the working are

How do I expand binomials with negative terms?

  • You need to be familiar with powers of negatives: 

    • open parentheses negative 2 close parentheses squared equals 4

      • Even powers are positive

    • open parentheses negative 2 close parentheses cubed equals negative 8

      • Odd powers are negative

  • For example, open parentheses 2 x minus 3 close parentheses to the power of 4 is open parentheses a plus b close parentheses to the power of 4 with a equals open parentheses 2 x close parentheses and b equals open parentheses negative 3 close parentheses

    • open parentheses 2 x minus 3 close parentheses to the power of 4 equals 1 open parentheses 2 x close parentheses to the power of 4 open parentheses negative 3 close parentheses to the power of 0 plus 4 open parentheses 2 x close parentheses cubed open parentheses negative 3 close parentheses to the power of 1 plus 6 open parentheses 2 x close parentheses squared open parentheses negative 3 close parentheses squared plus 4 open parentheses 2 x close parentheses to the power of 1 open parentheses negative 3 close parentheses cubed plus 1 open parentheses 2 x close parentheses to the power of 0 open parentheses negative 3 close parentheses to the power of 4

      • equals 2 to the power of 4 x to the power of 4 cross times 1 plus 4 cross times 2 cubed x cubed cross times open parentheses negative 3 close parentheses plus 6 cross times 2 squared x squared cross times 9 plus 4 cross times 2 x cross times open parentheses negative 27 close parentheses plus 81

      • equals 16 x to the power of 4 minus 96 x cubed plus 216 x squared minus 216 x plus 81

    • The signs alternate between positive and negative

Examiner Tips and Tricks

  • Check that the pairs of powers in each term of your working sum to the power of the binomial

    • e.g. for open parentheses a plus b close parentheses to the power of 4 equals 1 a to the power of 4 b to the power of 0 plus 4 a cubed b to the power of 1 plus 6 a squared b squared plus 4 a to the power of 1 b cubed plus 1 a to the power of 0 b to the power of 4 the sums are 4+0, 3+1, 2+2, etc.

Worked Example

Expand and simplify open parentheses 3 x minus 2 close parentheses to the power of 5.

As the power of the binomial is 5, you need the row from Pascal's triangle that starts with 1, 5, ...
(You are not expected to remember this, but you are expected to be able to write out Pascal's triangle to work out the fifth row)

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table row bold 1 bold space bold 5 bold space bold 10 bold space bold 10 bold space bold 5 bold space bold 1 end table

Write out the expansion of open parentheses a plus b close parentheses to the power of 5 with decreasing powers of a and increasing power of b

left parenthesis a plus b right parenthesis to the power of 5 equals 1 a to the power of 5 end exponent b to the power of 0 plus 5 a to the power of 4 b to the power of 1 plus 10 a cubed b squared plus 10 a squared b cubed plus 5 a to the power of 1 b to the power of 4 plus 1 a to the power of 0 b to the power of 5

Remember that b to the power of 0 equals a to the power of 0 equals 1 and b to the power of 1 equals b comma space a to the power of 1 equals a

Substitute in a equals open parentheses 3 x close parentheses and b equals open parentheses negative 2 close parentheses

open parentheses 3 x minus 2 close parentheses to the power of 5 equals 1 open parentheses 3 x close parentheses to the power of 5 plus 5 open parentheses 3 x close parentheses to the power of 4 open parentheses negative 2 close parentheses plus 10 open parentheses 3 x close parentheses cubed open parentheses negative 2 close parentheses squared plus 10 open parentheses 3 x close parentheses squared open parentheses negative 2 close parentheses cubed plus 5 open parentheses 3 x close parentheses open parentheses negative 2 close parentheses to the power of 4 plus 1 open parentheses negative 2 close parentheses to the power of 5

Use index laws to simplify each term (remember to apply the power to the number as well as the letter)

open parentheses 3 x minus 2 close parentheses to the power of 5 equals open parentheses 3 to the power of 5 x to the power of 5 close parentheses plus 5 open parentheses 3 to the power of 4 x to the power of 4 close parentheses open parentheses negative 2 close parentheses plus 10 open parentheses 3 cubed x cubed close parentheses open parentheses 4 close parentheses plus 10 open parentheses 3 squared x squared close parentheses open parentheses negative 8 close parentheses plus 5 open parentheses 3 x close parentheses open parentheses 16 close parentheses plus open parentheses negative 32 close parentheses

Calculate the numerical values, being careful with negative numbers

open parentheses 3 x minus 2 close parentheses to the power of 5 equals 243 x to the power of 5 minus 810 x to the power of 4 plus 1080 x cubed minus 720 x squared plus 240 x minus 32

Check that the signs alternate between positive and negative, and that powers of x decrease from 5 to 1 to 0 (a constant term)

Alternative text not available

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    Author
    Reviewer
    Jamie Wood

    Author: Jamie Wood

    Expertise: Maths Content Creator

    Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.