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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. The Binomial Theorem. Let be real numbers. For any positive integer , we have

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The Binomial Theorem

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

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

  3. 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.

  4. 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.

  5. 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.

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

  7. 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

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