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

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

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.

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 1. 2. 3.

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.

EX: Expand the expression using the Binomial Theorem 1. 2.

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