Binomial Theorem and Negative Exponentials

1. Binomial Theorem and Negative Exponentials

2. What is a Negative Exponential? (1+x)-1

3. Problems using Binomial Theorem and Negative or Fractional Exponents • The binomial theorem already mentioned only deals with Finite expansion. • If for instance we wished to use Negative or Fractional exponents it would not be possible to expand. • Also the ncr button can only be used for positive integers. (1+x)-1.5=???

4. Newton’s Influences • At around 1665 Newton generalised the formula to allow the use of negative and fractional exponents. • Newton’s first results concerning Binomial Series were given by Sir Isaac Newton in the study of areas enclosed under a curve. • The Binomial series is sometimes referred to Newton’s Binomial Theorem. Newton gives no proof and is not explicit about the nature of the series.

5. How it works • Newton’s Binomial Theorem allows us to expand binomial expressions for any rational valued exponent. What is a Rational Number??? 1/2 √3 A rational number is a number which can be expressed as a ratio of two integers

6. Newton’s Binomial Theorem