Binomial Expansion
What is the Binomial Expansion?
- The binomial expansion gives a method for expanding a two-term expression in a bracket raised to a power
- For example
- You may also see it referred to as the binomial theorem
- In this note will be a positive integer
- See the 'General Binomial Expansion' revision note for the general case
- To expand a bracket with a two-term expression in it:
- Determine what and are for your example
- Then use the formula for the binomial expansion
-
- in the formula is known as the binomial coefficient
- (" factorial") is defined by
- You may also see written as
- Your calculator should be able to calculate for you
- Or you can use Pascal's triangle (see the next section)
- To get all the terms
- Start with
- Then use , ,... until you get up to
- So there will always be terms in the full expansion
- This version of the binomial expansion formula is not on the exam formula sheet
- But it is a special case of the Binomial Series formula for which is on the formula sheet
- See the 'General Binomial Expansion' revision note
- in the formula is known as the binomial coefficient
- When expanding something like you may only be asked to find the first few terms of an expansion
- Check whether the question wants ascending or descending powers of x
- For ascending powers start with the constant term,
- For descending powers start with the term with ,
- Choosing and appropriately will make it easier to follow the formula above
- Check whether the question wants ascending or descending powers of x
- If you are not writing the full expansion you can either
- show that the series continues by putting an ellipsis (…) after your final term
- or show that the terms you have found are an approximation of the full series by using the 'approximately equals' sign (≈)
Finding binomial coefficients using Pascal's triangle
- Pascal’s triangle is a way of arranging (and finding!) the binomial coefficients
- The first row has just the number 1
- Each row begins and ends with a 1
- Starting in the third row
- Each other terms is the sum of the two terms immediately above it
- Pascal’s triangle is an alternative way of finding the binomial coefficients (also written )
- It can be useful for finding the values of the coefficients without a calculator
- Most useful for smaller values of
- For larger values of it is slow and prone to arithmetic errors
- It can be useful for finding the values of the coefficients without a calculator
- Taking the first row as corresponding to ,
- each row gives the binomial coefficient values for the corresponding value of
- within a row the values run from to
- e.g. from the 6th row of the table ():
How do I find the coefficient of a single term?
- You may just be asked to find the coefficient of a single term, rather than the whole expansion
- Use the formula for the general term
- To find a particular power of term in an expansion
- Choose which value of you will need to use in the formula
- The laws of indices can help you decide which value of to use:
- For , to find the coefficient of let and use
- For , to find the coefficient of let and use
- For something like , you need to consider how the powers will cancel each other
- E.g. for , to find the coefficient of let and use
- Because then
- There are a lot of variations, so practice is better than trying to memorise formulae for !
- If you know the coefficient of a particular term, you can use it to find an unknown in the brackets
- Use the laws of indices to choose the correct term
- Then use the general term formula to form and solve an equation
Exam Tip
- Binomial expansion questions can get messy
- Use separate lines to keep your working clear
- And always put terms in brackets
Worked example
Using the binomial expansion, find the complete expansion of .
Use the formula with , and
will run from 0 to 4, so there will be 5 terms
Now just work out the values of the binomial coefficients
You can use the formula, your calculator or Pascal's triangle
Note that
That's why we usually don't bother writing the binomial coefficients for the first and last terms of an expansion!
Worked example
Find the first three terms, in ascending powers of , in the expansion of .
For ascending powers of we want to start with the constant term
So we want to use the formula with , , and
For the first three terms (constant term, term and term) we want from 0 to 2
Substitute those values into the formula
Find the value of the binomial coefficients and bring the powers inside the brackets
Be careful with the minus signs!
Expand the remaining brackets and write down the final answer