GCSEFurther MathsAQARevision Notes2. Algebra & FunctionsBinomial ExpansionBinomial Expansion

Binomial Expansion (AQA GCSE Further Maths)

Revision Note

Test Yourself
Jamie W

Author

Jamie W

Expertise

Maths

Pascal's Triangle

What is Pascal's Triangle?

  • Pascal's Triangle is a number pattern laid out in a triangle, which is created by summing pairs of numbers, and writing the answer directly below them
  • We describe the row containing a 2 as the 2nd row, and the row containing a 3 as the 3rd row and so on
  • We describe the first term in a row as the 0th number (always 1), so 4 is the 1st and 3rd number in the 4th row
  • It is symmetrical; the left and right halves are the same
    • For example row 4 is 1, 4, 6, 4, 1
  • There are many interesting patterns in Pascal's Triangle
    • One of which is that it contains the coefficients of each term when expanding brackets in the form open parentheses a plus b close parentheses to the power of n where n is an integer
    • There are other patterns involving triangular numbers, powers of 2, powers of 11, odds & evens, primes, and squares that are all interesting too!
  • To help you when expanding open parentheses a plus b close parentheses to the power of n you should be able to write out the first 6 or so rows yourself

pascals-triangle-no-labels

How does Pascal's Triangle help with expanding (a + b)n?

  • Look at the first few results for open parentheses a plus b close parentheses to the power of n and compare the coefficients to the rows of 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
  • The coefficients of each term for n equals 2 come from the 2nd row
  • The coefficients of each term for n equals 3 come from the 3rd row
  • and so on...

Binomial Expansion

How do I expand brackets using binomial expansion?

  • Understanding the pattern for open parentheses a plus b close parentheses to the power of n will help you expand any binomial with an integer power
    • Binomial just means the sum or difference of two terms, e.g. 2 x minus 1 or 3 plus 4 x
  • To expand, for example, open parentheses a plus b close parentheses to the power of 4
    • The powers of a will start with a to the power of 4 and decrease by 1 in each term, until it reaches a to the power of 0 (which is 1)
    • The powers of b will start with b to the power of 0 (which is 1) and increase by 1 in each term, until it reaches b to the power of 4
    • Notice that the sum of the powers in each term will be 4
    • The coefficient of each term is from the 4th row of Pascal's triangle; 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 plus 4 a cubed b plus 6 a squared b squared plus 4 a b cubed plus 1 b to the power of 4
  • We can use this same pattern to expand something like open parentheses 2 x plus 3 close parentheses to the power of 4 by substituting each term for a and b
    • Using a equals 2 x and b equals 3
      • 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 plus 4 open parentheses 2 x close parentheses cubed open parentheses 3 close parentheses plus 6 open parentheses 2 x close parentheses squared open parentheses 3 close parentheses squared plus 4 open parentheses 2 x close parentheses open parentheses 3 close parentheses cubed plus 1 open parentheses 3 close parentheses to the power of 4
    • Be careful when simplifying; remember that the power is applied to everything in the bracket
      • open parentheses 2 x plus 3 close parentheses to the power of 4 equals 1 open parentheses 2 to the power of 4 x to the power of 4 close parentheses plus 4 open parentheses 2 cubed x cubed close parentheses open parentheses 3 close parentheses plus 6 open parentheses 2 squared x squared close parentheses open parentheses 3 squared close parentheses plus 4 open parentheses 2 x close parentheses open parentheses 3 cubed close parentheses plus 1 open parentheses 3 to the power of 4 close parentheses
      • open parentheses 2 x plus 3 close parentheses to the power of 4 equals 16 x to the power of 4 plus 4 open parentheses 8 x cubed close parentheses open parentheses 3 close parentheses plus 6 open parentheses 4 x squared close parentheses open parentheses 9 close parentheses plus 4 open parentheses 2 x close parentheses open parentheses 27 close parentheses plus open parentheses 81 close parentheses
      • open parentheses 2 x plus 3 close parentheses 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
  • Be careful when finding powers of negative numbers
    • Even powers will be positive
      • open parentheses negative 2 close parentheses to the power of 4 equals 16
    • Odd powers will be negative
      • open parentheses negative 2 close parentheses cubed equals negative 8

Exam Tip

  • Remember that with a bracket like open parentheses 6 x close parentheses to the power of 4 the power of 4 is applied to the 6 and to the x
    • open parentheses 6 x close parentheses to the power of 4 equals 6 to the power of 4 x to the power of 4 equals 1296 x to the power of 4
  • Check that the powers in each term of your expansion sum to the value of n, remembering that a equals a to the power of 1
    • e.g. for open parentheses a plus b close parentheses to the power of 4 equals 1 a to the power of 4 plus 4 a cubed b plus 6 a squared b squared plus 4 a b cubed plus 1 b to the power of 4 the powers in each term sum to 4

Worked example

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

The power is 5 so we will need the 5th row of Pascal's triangle for the coefficients
(You are not expected to remember this, but are expected to be able to write out Pascal's triangle to work out the fifth row)

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

If unsure expand using open parentheses a plus b close parentheses to the power of 5 first, so you can check the powers in each term add up to 5

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

Use 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

Identify that a equals 3 x and b equals negative 2
Applying the basic index laws and substituting for a and b means we can start writing down the expansion
Be careful with negatives - use brackets around that -2

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

Remember to apply the power to each value/letter inside the bracket where necessary

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

Evaluate where possible, again being careful with negatives

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

If expanded correctly, nothing else should simplify at this stage, but double check negatives alternate, powers of x start at 5, descend to 1 and that there is a constant term

You've read 0 of your 0 free revision notes

Get unlimited access

to absolutely everything:

  • Downloadable PDFs
  • Unlimited Revision Notes
  • Topic Questions
  • Past Papers
  • Model Answers
  • Videos (Maths and Science)

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

    Test yourselfNext topic

    Did this page help you?

    Jamie W

    Author: Jamie W

    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.