Multiple General Binomial Expansions (Cambridge (CIE) A Level Maths): Revision Note

Multiple general binomial expansions

What is the general binomial expansion?

• The general binomial expansion, as given in the formula booklet, is

• If n ∈ ℕ then the expansion is finite (see Binomial Expansion)

• Otherwise the expansion is infinitely long

  • It is only valid for |x| < 1 (-1 < x < 1)

  • Only the first few terms of an expansion are usually needed

What are multiple general binomial expansions? 

• More than one part of an expression can be a binomial expansion

• These may sometimes be called compound expressions

• The expansion will only be valid for the lowest |x| boundary from all the expansions used

How do I expand multiple binomials?

STEP 1        Break the expression down into binomial expansions

STEP 2        Expand each binomial individually, up to a suitable number of terms

• Be careful with negatives and fractions

• Use brackets as appropriate

STEP 3        Collect the expansions together and simplify

• This could be expanding brackets, collecting like terms, etc

• Ignore any terms of degree higher than required

STEP 4        Check the validity of each binomial expansion

• The overall validity is the intersection (∩)

How do I use partial fractions to form multiple general binomial expansions?

• Partial fractions allow rational expressions to be written in a form where the general binomial expansion can then be applied

  

• Validity is an important part of the general binomial expansion