1.6.1 Binomial Theorem

What is the Binomial Theorem?

• The binomial theorem (sometimes known as the binomial expansion) gives a method for expanding a two-term expression in a bracket raised to a power
  • A binomial expression is in fact any two terms inside the bracket, however in IB the expression will usually be linear
• 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 
    • See below for more information on  
    • You may also see   written as or 
• You will usually be asked to find the first three or four terms of an expansion
• 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 (≈)

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 exponents
• 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 is ?

• If we want to find the number of ways to choose r items out of n different objects we can use the formula for 
  • The formula for r combinations of n items is 
  • This formula is given in the formula booklet along with the formula for the binomial theorem
  • The function can be written  or and is often read as ‘n choose r’
    • Make sure you can find and use the button on your GDC

 

How does  relate to the binomial theorem?

• The formula  is also known as a binomial coefficient
• For a binomial expansion the coefficients of each term will be ,  and so on up to 
  • The coefficient of the  term will be 
• 
• The binomial coefficients are symmetrical, so 
  • This can be seen by considering the formula for 
  • 

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 theorem?

• 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 zero, , each row corresponds to the row and the term within that row corresponds to the term