What is the general binomial expansion?

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

• If?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 is meant by 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 use general binomial expansions in complicated expressions?

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 work with partial fractions and the general binomial expansion?

• 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