1. The Binomial Theorem OBJECTIVES: Evaluate a Binomial Coefficient
Expand a Binomial raised to a power
Find a particular term in a binomial expansion

3. The Binomial Theorem
Let be real numbers. For any positive integer , we have
So let’s expand using the Binomial Theorem

4. Observe the following patterns:
1. The first term in the expansion is The exponents on decrease by 1 in each successive term
2. The exponents on in the expansion increase by 1 in each successive term. In the first term, the exponent on is 0 and the last term is .
3. The sum of the exponents on the variables in any term in the expansion is equal to .
4. The number of terms in the polynomial expansion is one greater than the power of the binomial There are terms in the expanded form.

5. But what does mean or equal? It is the Binomial Coefficient
For nonnegative integers , the expression (read “n above r”) is called a Binomial Coefficient and is defined by
EX: Evaluate each expression

6. Four useful formulas involving the symbol
Now suppose we arrange the values of the symbol in a triangular display. This display is called the Pascal Triangle and is an interesting and organized display of the symbol.

7. EX: Expand the expression using the Binomial Theorem1. 2.

8. EX: Finding a particular term in a Binomial Expansion
3. Find the fourth term in the expansion of
4. Find the coefficient of in the expansion of