AQA A Level Maths: Pure复习笔记4.2.3 General Binomial Expansion - Multiple

General Binomial Expansion - Multiple

 

What is the general binomial expansion?

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

4.2.3-Multiple-GBEs-Notes-Diagram-1

  • 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 (∩)

4.2.3-Multiple-GBEs-Notes-Diagram-4_5

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 applied4.2.3-Multiple-GBEs-EXTRA1

 

  • Validity is an important part of the general binomial expansion

4.2.3-Multiple-GBEs-Notes-Diagram-7

Worked Example

4.2.3-Multiple-GBEs-Example-Diagram-1

4.2.3-Multiple-GBEs-Example-Diagram-2

转载自savemyexams

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