Binomial Theorem
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Transcript of Binomial Theorem
![Page 1: Binomial Theorem](https://reader035.fdocuments.us/reader035/viewer/2022062221/5681365e550346895d9deb20/html5/thumbnails/1.jpg)
11.711.7
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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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na b
What if the term in a series is not a constant, but a binomial?
0a b
1a b
2a b
4a b
3a b
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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 1Expand using Pascal’s Triangle 5p q
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Ex 2Expand using Pascal’s Triangle 6x y
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The coefficients can be written in terms of the previous coefficients
0 1 1 2 2 3 3 01 1 21 ... 1
1 1 2 1 2 3
n n n n n nn n n n nna b a b a b a b a b a b
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Ex 3Expand using the binomial theorem
8x y
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Ex 4Expand using the binomial theorem
45x
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factorial: a special product that starts with the indicated value and has consecutive descending factors
Ex 5Evaluate6!
2!4!
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0 1 1 2 2 0! ! ! !
...!0! 1 !1! 2 !2! 0! !
n n n n nn n n na b a b a b a b a b
n n n n
or
0
!
! !
nn n k k
k
na b a b
n k k
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Ex 6Expand using factorial form 43x y
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Ex 6Expand using factorial form 42 3x