11.7. there are n+1 terms functions of n exponent of a in first term exponent of b in last term...
TRANSCRIPT
![Page 1: 11.7. 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](https://reader035.vdocuments.site/reader035/viewer/2022070305/55156eaa550346a1418b4fe5/html5/thumbnails/1.jpg)
11.711.7
![Page 2: 11.7. 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](https://reader035.vdocuments.site/reader035/viewer/2022070305/55156eaa550346a1418b4fe5/html5/thumbnails/2.jpg)
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
![Page 3: 11.7. 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](https://reader035.vdocuments.site/reader035/viewer/2022070305/55156eaa550346a1418b4fe5/html5/thumbnails/3.jpg)
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
![Page 4: 11.7. 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](https://reader035.vdocuments.site/reader035/viewer/2022070305/55156eaa550346a1418b4fe5/html5/thumbnails/4.jpg)
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
![Page 5: 11.7. 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](https://reader035.vdocuments.site/reader035/viewer/2022070305/55156eaa550346a1418b4fe5/html5/thumbnails/5.jpg)
Ex 1Expand using Pascal’s Triangle 5p q
![Page 6: 11.7. 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](https://reader035.vdocuments.site/reader035/viewer/2022070305/55156eaa550346a1418b4fe5/html5/thumbnails/6.jpg)
Ex 2Expand using Pascal’s Triangle 6x y
![Page 7: 11.7. 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](https://reader035.vdocuments.site/reader035/viewer/2022070305/55156eaa550346a1418b4fe5/html5/thumbnails/7.jpg)
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
![Page 8: 11.7. 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](https://reader035.vdocuments.site/reader035/viewer/2022070305/55156eaa550346a1418b4fe5/html5/thumbnails/8.jpg)
Ex 3Expand using the binomial theorem
8x y
![Page 9: 11.7. 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](https://reader035.vdocuments.site/reader035/viewer/2022070305/55156eaa550346a1418b4fe5/html5/thumbnails/9.jpg)
Ex 4Expand using the binomial theorem
45x
![Page 10: 11.7. 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](https://reader035.vdocuments.site/reader035/viewer/2022070305/55156eaa550346a1418b4fe5/html5/thumbnails/10.jpg)
factorial: a special product that starts with the indicated value and has consecutive descending factors
Ex 5Evaluate6!
2!4!
![Page 11: 11.7. 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](https://reader035.vdocuments.site/reader035/viewer/2022070305/55156eaa550346a1418b4fe5/html5/thumbnails/11.jpg)
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
![Page 12: 11.7. 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](https://reader035.vdocuments.site/reader035/viewer/2022070305/55156eaa550346a1418b4fe5/html5/thumbnails/12.jpg)
Ex 6Expand using factorial form 43x y
![Page 13: 11.7. 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](https://reader035.vdocuments.site/reader035/viewer/2022070305/55156eaa550346a1418b4fe5/html5/thumbnails/13.jpg)
Ex 6Expand using factorial form 42 3x