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Binomial Theorem 11.7

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**Binomial Expansion of the form (a+b)n**

There are n+1 terms Functions of n Exponent of a in first term Exponent of b in last term Other terms Exponent of a decreases by 1 Exponent of b increases by 1 Sum of exponents in each term is n Coefficients are symmetric (Pascal’sTriangle) At Beginning--increase Towards End---decrease

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Expanding Binomials What if the term in a series is not a constant, but a binomial?

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Pascal’s Triangle The coefficients form a pattern, usually displayed in a triangle Pascal’s Triangle: binomial expansion used to find the possible number of sequences for a binomial pattern features start and end w/ 1 coeff is the sum of the two coeff above it in the previous row symmetric

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Ex 1 Expand using Pascal’s Triangle

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Ex 2 Expand using Pascal’s Triangle

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Binomial Theorem The coefficients can be written in terms of the previous coefficients

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Ex 3 Expand using the binomial theorem

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Ex 4 Expand using the binomial theorem

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Factorials! factorial: a special product that starts with the indicated value and has consecutive descending factors Ex 5 Evaluate

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**Binomial Theorem, factorial form and Sigma Notation**

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Ex 6 Expand using factorial form

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Ex 6 Expand using factorial form

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Pascal’s Triangle Row 1 2 3 4 5 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5.

Pascal’s Triangle Row 1 2 3 4 5 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5.

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