Binomial Theorem and Pascal’s Triangle

Binomial Theorem

Pascals Triangle 1 2 3 4 6 5 10 15 20 7 21 35

Binomial Theorem Notice each expression has n + 1 terms The degree of each term is equal to n The exponent of each a decreases by 1 and the exponent of each b increases by 1 for each succeeding term in the series The coefficients come from Pascal’s Triangle In subtraction alternate signs starting with positive then negative

Expand using the Binomial Theorem and Pascal’s Triangle

Binomial Theorem Write the general rule for the binomial using Pascal’s Triangle Substitute into the general rule Simplify your expression

Expand using the Binomial Theorem and Pascal’s Triangle

Use the previous term method to determine each of the following

Factorial If n > 0 is an integer, the factorial symbol n! is defined as follows: 0! = 1 and 1! = 1 n! = n(n – 1) •… • 3 • 2 • 1 if n > 2 4! = 4 • 3 • 2 • 1 = 24 6! = 6 • 5 • 4 • 3 • 2 • 1 = 720

Factorial

Evaluate the following expressions

We can use the Binomial Theorem to find a particular term in an expression without writing the entire expansion.