Binomial Theorem (DP IB Analysis & Approaches (AA)) : Revision Note

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

The Binomial Coefficient nCr

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 

  • 

Pascal's Triangle

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