Binomial Theorem (Cambridge O Level Additional Maths)

What is the Binomial Expansion?

• The binomial theorem (also known as the binomial expansion) gives a method for expanding a two-term expression in a bracket raised to a power
• To expand a bracket with a two-term expression in:
  • First choose the most appropriate parts of the expression to assign to a and b
  • Then use the formula for the binomial theorem:

• • where  is a positive integer and  
    
    • You may also see   written as or 
• You will usually be asked to find the first three or four terms of an expansion

What is Pascal’s Triangle?

• Pascal’s triangle is a way of arranging the binomial coefficients and neatly shows how they are formed
  • Each term is formed by adding the two terms above it
  • The first row has just the number 1
  • Each row begins and ends with a number 1
  • From the third row the terms in between the 1s are the sum of the two terms above it

How does Pascal’s Triangle relate to the binomial expansion?

• Pascal’s triangle is an alternative way of finding the binomial coefficients, 
  • It can be useful for finding for smaller values of without a calculator
  • However for larger values of it is slow and prone to arithmetic errors

• • Taking the first row as the 0th row, , each row corresponds to the row and the term within that row corresponds to the term
    • The first term, which will be a 1, will be the 0th term
• In the non-calculator exam Pascal's triangle can be helpful if you need to get the coefficients of an expansion quickly, provided the value of n is not too big 

How do I find the coefficient of a single term?

• Most of the time you will be asked to find the coefficient of a term, rather than carry out the whole expansion
• Use the formula for the general term

• The question will give you the power of x of the term you are looking for
  • Use this to choose which value of r  you will need to use in the formula
  • This will depend on where the x is in the bracket
  • The laws of indices can help you decide which value of r  to use:
    • For to find the coefficient of use
    • For to find the coefficient of use  
    • For look at how the powers will cancel out to decide which value of to use
  • So for  to find the coefficient of  use the term with and to find the constant term use the term with
    • There are a lot of variations of this so it is usually easier to see this by inspection of the powers
• You may also be given the coefficient of a particular term and asked to find an unknown in the brackets
  • Use the laws of indices to choose the correct term and then use the binomial theorem formula to form and solve and equation

What are binomial expansions used for?

• Binomial expansions are used to expand brackets
  • Normally in the form (a + b)ⁿ 
• You will most likely be asked to find the first few terms
• Look out for whether you should give your answer in ascending or descending powers of x
  • For ascending powers start with the constant term, aⁿ
  • For descending powers start with the term with x in
    • You may wish to swap a and b over so that you can follow the general formula given in the formula book
• If you are not writing the full expansion you can either
  • show that the sequence continues by putting an ellipsis (…) after your final term
  • or show that the terms you have found are an approximation of the full sequence by using the sign for approximately equals to (≈)  

• You may be asked to find the coefficient of a particular term

• You may be asked to solve a problem to find an unknown