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# Pre-Cal 40S Slides April 25, 2007

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Introduction to the Binomial Theorem.

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### Pre-Cal 40S Slides April 25, 2007

1. 1. The Binomial Theorem or "one of the ways G-d built the universe" ... Zhu Shijiei 1261 Pascal 1653
2. 2. Expand and simplify ... a+b
3. 3. Find a pattern, add two more rows to the triangle ... 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1
4. 4. Evaluate each term ...
5. 5. Pascal's Triangle How many different patterns can you find in the triangle? SYMMETRY Counting Numbers Sum of COUNTING numbers triangular numbers Sum of Triangular numbers
6. 6. Pascal's Triangle How many different patterns can you find in the triangle?
7. 7. Pascal's Triangle How many different patterns can you find in the triangle?
8. 8. The Binomial Theorem ... Algebraically Combinatorically Notice the patterns ... (1) The coefficient of the term is: (2) The exponent on a is given by: [n - (i - 1)] (3) The exponent on b is given by: i (4) This relation holds for each term in the expansion: [exponent on a] + [exponent on b] = n (5) The number of terms in any binomial expansion is: n + 1