Applications of Binomial Expansion (Edexcel IGCSE Further Pure Maths)

Applications of Binomial Expansion

How can I use the binomial expansion with more complex expressions?

• You may be asked to find a series expansion for an expression like  

  • Rewrite as a product, 

• Find the binomial expansion of 

• Note this has only been expanded up to the term

• Multiply that expansion by  and simplify

• This is only valid up to the  term

• To get more terms we would have to start with more terms for 

• is not the correct  term for as there are more  terms that were not found

• Use the same process to find the expansion for something like  

  • Rewrite as a product, 

  • Find the binomial expansion of    

  • Multiply the expansion by and simplify

How can I use the binomial expansion to estimate a value?

• The binomial expansion can be used to find estimates or approximations

  • When , higher powers of will be very small

  • So even the first 3 or 4 terms of an expansion can form a good approximation

  • The more terms used the closer the approximation will be to the true value

    • Also the closer to zero is, the better the approximation will be

• For example, find an approximation for  using the expansion of   

  • Compare the value you are approximating to the expression being expanded

    • 

  • Find the value of to use by solving the appropriate equation

    • 

• Substitute this value of into the binomial expansion of 

  • 

  • So

  • The true value of  is 

• On the exam this is often used to approximate square roots

  • It can also be used to approximate other things

  • For example approximate the fraction using the binomial expansion of 

    • 

    • So substitute into the expansion

• Always check that the value of is within the interval of convergence for the expansion

  • If is outside the interval of convergence then the approximation is not reliable

How can I use the binomial expansion with calculus?

• A complete binomial expansion is exactly equal to the function it represents

• This means that it is valid to differentiate or integrate a binomial expansion

  • These will always be powers of  derivatives or integrals

• For example, the function 

  • We saw above that  

  • We can differentiate that:

    • 

  • Or integrate it

    • 

• This can be used to find estimates or approximations

  • For example to estimate

  • Integrate the binomial expansion (as we just did above)

    • 

    • The true value of the integral is 0.178079...

• Always check that any values of  you use are within the interval of convergence for the expansion

  • This includes the integration limits if you are approximating a definite integral 

  • If any  values are outside the interval of convergence then the approximation is not reliable

How can I find the percentage error of an approximation?

• Use the following formula

  • 

    •   is the exact value

    • is the approximated value

  • The exact value must be in the denominator!

• Percentage errors are usually given as positive values

  • If the formula gives you a negative value, you can just remove the minus sign

  • But you will usually get the marks for a correct positive or negative answer