Extension of The Binomial Theorem (DP IB Analysis & Approaches (AA)): Revision Note

Binomial theorem: fractional & negative indices

What is the extension of the binomial theorem?

• The extension of the binomial theorem is an expansion that allows for the power to be fractional and/or negative

• It is

• This formula

  • has an infinite number of terms

  • works if

• The formula also holds for being a positive integer

  • In this case it's an alternative form of the formula

    • with a finite number of terms

  • thus it holds for all rational powers of

    • 

How do I use the extension of the binomial theorem?

• To use the extension of the binomial theorem

  • first convert the expression into the form

    • this may require index laws

  • then substitute in , and and simplify

• e.g. to find the first three terms in the expansion of

  • Convert it into the form

  • Substitute , and into the extension formula

    • 

    • simplify each term

      •