IB DP Maths: AA HL复习笔记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:

• 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

• • • 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 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?Exam Tip