Problem Solving with Binomial Expansion
How do I find a specific term of a binomial expansion?
- You may be asked to find a specific term or coefficient of a term in an expansion, rather than finding the entire expansion
- We can use several facts to help us
- The powers in each term sum to
- The coefficients for come from the row of Pascal's triangle
- The coefficient of will be the number in that row (and also the number in that row)
- e.g. In the coefficient of will be the 2nd number in the 7th row (or the 5th number in the 7th row)
- Note that this counting system includes the first number in each row
- So the 1st number (of every row) is 1
- This was previously called the 0th number in the row
- However patterns are easier to see this way for the purposes of problem solving
- For example to find the coefficient of in
- The powers in the term will sum to 6, and we need to include , so they must be
- where is the coefficient from Pascal's triangle
- and we are looking at the term with a power of 4 (and 2)
- So we will use the 4th (or 2nd) number in the 6th row of Pascal's triangle, which is 15
- So the term is which we can then expand and simplify
- So the coefficient of is 2160
- The powers in the term will sum to 6, and we need to include , so they must be
How do I find an unknown with binomial expansion?
- Sometimes you may be asked to find an unknown in an expansion
- For example, you may be told that for the expansion , the coefficient of is 2000, and you can then use this information to find
- Use the method described above to find the specific term containing an , which will have its coefficient in terms of
- Then equate this coefficient to 2000 and solve to find
When do powers cancel out in a binomial expansion?
- Sometimes the two terms in the binomial can cause powers to cancel out
- For example when expanding we would find
- Which simplifies to
- You can see that the powers in each term will change, as we have powers of divided by each other
- This expression will reduce to
- or
- You can see that this expansion now looks quite different to "standard" expansions
- We now have a constant term in the middle, rather than at the end
- For this reason you need to be careful with expansions that look like this
- They typically contain a multiple of or etc in the binomial
- e.g. or
- They typically contain a multiple of or etc in the binomial
Exam Tip
- Look out for extra information about unknowns
- they will often say "...where "
- this will help you if you get a solution like and you can then select the positive value
- If you feel unsure about finding a specific term straight away, writing out the first few terms can help you remember or spot the pattern to use
Worked example
and
For the term involving we will need the term involving and the 3rd number in the 5th row from Pascal's triangle
5th row of Pascal's triangle is
Therefore the term required is
So the coefficient of is
The coefficient of the term is 2560
Given that and that the coefficient of in the expansion of is 79 380, find the value of .
and
For the term involving we will need the term involving and the 4th number in the 6th row from Pascal's triangle
The 6th row of Pascal's triangle is
Therefore the term required is
So the coefficient of is
The question gives the coefficient as 79 380 so set up and solve an equation for
so the positive square root is needed
Find the coefficient of in the expansion of .
Spot that this is a question where powers of will cancel
and
The term will be generated when the power of is one greater than the power of
As powers in each term will sum to 5 in this case, this will be the term involving (and
The 3rd number in the 5th row of Pascal's triangle is also needed
Therefore the term required is
So the coefficient of is
The coefficient of the term is 80