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 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