12x1 t08 02 general binomial expansions
TRANSCRIPT
![Page 1: 12X1 T08 02 general binomial expansions](https://reader033.vdocuments.us/reader033/viewer/2022052906/55895d36d8b42a643f8b464e/html5/thumbnails/1.jpg)
General Expansion of Binomials
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General Expansion of Binomials
kkk
n xxC 1in oft coefficien theis
![Page 3: 12X1 T08 02 general binomial expansions](https://reader033.vdocuments.us/reader033/viewer/2022052906/55895d36d8b42a643f8b464e/html5/thumbnails/3.jpg)
General Expansion of Binomials
kkk
n xxC 1in oft coefficien theis
kn
Ckn
![Page 4: 12X1 T08 02 general binomial expansions](https://reader033.vdocuments.us/reader033/viewer/2022052906/55895d36d8b42a643f8b464e/html5/thumbnails/4.jpg)
General Expansion of Binomials
kkk
n xxC 1in oft coefficien theis
nn
nnnnn xCxCxCCx 22101
kn
Ckn
![Page 5: 12X1 T08 02 general binomial expansions](https://reader033.vdocuments.us/reader033/viewer/2022052906/55895d36d8b42a643f8b464e/html5/thumbnails/5.jpg)
General Expansion of Binomials
kkk
n xxC 1in oft coefficien theis
nn
nnnnn xCxCxCCx 22101
which extends to; n
nnn
nnnnnnnnn bCabCbaCbaCaCba
1
122
21
10
kn
Ckn
![Page 6: 12X1 T08 02 general binomial expansions](https://reader033.vdocuments.us/reader033/viewer/2022052906/55895d36d8b42a643f8b464e/html5/thumbnails/6.jpg)
General Expansion of Binomials
kkk
n xxC 1in oft coefficien theis
nn
nnnnn xCxCxCCx 22101
432.. xge
which extends to; n
nnn
nnnnnnnnn bCabCbaCbaCaCba
1
122
21
10
kn
Ckn
![Page 7: 12X1 T08 02 general binomial expansions](https://reader033.vdocuments.us/reader033/viewer/2022052906/55895d36d8b42a643f8b464e/html5/thumbnails/7.jpg)
General Expansion of Binomials
kkk
n xxC 1in oft coefficien theis
nn
nnnnn xCxCxCCx 22101
432.. xge
which extends to; n
nnn
nnnnnnnnn bCabCbaCbaCaCba
1
122
21
10
4443
3422
243
144
04 33232322 xCxCxCxCC
kn
Ckn
![Page 8: 12X1 T08 02 general binomial expansions](https://reader033.vdocuments.us/reader033/viewer/2022052906/55895d36d8b42a643f8b464e/html5/thumbnails/8.jpg)
General Expansion of Binomials
kkk
n xxC 1in oft coefficien theis
nn
nnnnn xCxCxCCx 22101
432.. xge
which extends to; n
nnn
nnnnnnnnn bCabCbaCbaCaCba
1
122
21
10
4443
3422
243
144
04 33232322 xCxCxCxCC
432 812162169616 xxxx
kn
Ckn
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Pascal’s Triangle Relationships
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Pascal’s Triangle Relationships
11 where 1 11
1
nkCCC kn
kn
kn
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Pascal’s Triangle Relationships
11 where 1 11
1
nkCCC kn
kn
kn
1111 nn xxx
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Pascal’s Triangle Relationships
11 where 1 11
1
nkCCC kn
kn
kn
1111 nn xxx 1
1111
11
11
011
n
nnk
knk
knnn xCxCxCxCCx
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Pascal’s Triangle Relationships
11 where 1 11
1
nkCCC kn
kn
kn
1111 nn xxx 1
1111
11
11
011
n
nnk
knk
knnn xCxCxCxCCx
kx of tscoefficienat looking
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Pascal’s Triangle Relationships
11 where 1 11
1
nkCCC kn
kn
kn
1111 nn xxx 1
1111
11
11
011
n
nnk
knk
knnn xCxCxCxCCx
kx of tscoefficienat looking
knCLHS
![Page 15: 12X1 T08 02 general binomial expansions](https://reader033.vdocuments.us/reader033/viewer/2022052906/55895d36d8b42a643f8b464e/html5/thumbnails/15.jpg)
Pascal’s Triangle Relationships
11 where 1 11
1
nkCCC kn
kn
kn
1111 nn xxx 1
1111
11
11
011
n
nnk
knk
knnn xCxCxCxCCx
kx of tscoefficienat looking
knCLHS
![Page 16: 12X1 T08 02 general binomial expansions](https://reader033.vdocuments.us/reader033/viewer/2022052906/55895d36d8b42a643f8b464e/html5/thumbnails/16.jpg)
Pascal’s Triangle Relationships
11 where 1 11
1
nkCCC kn
kn
kn
1111 nn xxx 1
1111
11
11
011
n
nnk
knk
knnn xCxCxCxCCx
kx of tscoefficienat looking
knCLHS k
nk
n CCRHS 11
1 11
![Page 17: 12X1 T08 02 general binomial expansions](https://reader033.vdocuments.us/reader033/viewer/2022052906/55895d36d8b42a643f8b464e/html5/thumbnails/17.jpg)
Pascal’s Triangle Relationships
11 where 1 11
1
nkCCC kn
kn
kn
1111 nn xxx 1
1111
11
11
011
n
nnk
knk
knnn xCxCxCxCCx
kx of tscoefficienat looking
knCLHS k
nk
n CCRHS 11
1 11
kn
kn CC 1
11
![Page 18: 12X1 T08 02 general binomial expansions](https://reader033.vdocuments.us/reader033/viewer/2022052906/55895d36d8b42a643f8b464e/html5/thumbnails/18.jpg)
Pascal’s Triangle Relationships
11 where 1 11
1
nkCCC kn
kn
kn
1111 nn xxx 1
1111
11
11
011
n
nnk
knk
knnn xCxCxCxCCx
kx of tscoefficienat looking
knCLHS k
nk
n CCRHS 11
1 11
kn
kn CC 1
11
k
nk
nk
n CCC 11
1
![Page 19: 12X1 T08 02 general binomial expansions](https://reader033.vdocuments.us/reader033/viewer/2022052906/55895d36d8b42a643f8b464e/html5/thumbnails/19.jpg)
Pascal’s Triangle Relationships
11 where 1 11
1
nkCCC kn
kn
kn
1111 nn xxx 1
1111
11
11
011
n
nnk
knk
knnn xCxCxCxCCx
kx of tscoefficienat looking
knCLHS k
nk
n CCRHS 11
1 11
kn
kn CC 1
11
k
nk
nk
n CCC 11
1
l"symmetrica is trianglesPascal'"
11 where 2 nkCC knn
kn
![Page 20: 12X1 T08 02 general binomial expansions](https://reader033.vdocuments.us/reader033/viewer/2022052906/55895d36d8b42a643f8b464e/html5/thumbnails/20.jpg)
Pascal’s Triangle Relationships
11 where 1 11
1
nkCCC kn
kn
kn
1111 nn xxx 1
1111
11
11
011
n
nnk
knk
knnn xCxCxCxCCx
kx of tscoefficienat looking
knCLHS k
nk
n CCRHS 11
1 11
kn
kn CC 1
11
k
nk
nk
n CCC 11
1
l"symmetrica is trianglesPascal'"
11 where 2 nkCC knn
kn
1 3 0 nnn CC
![Page 21: 12X1 T08 02 general binomial expansions](https://reader033.vdocuments.us/reader033/viewer/2022052906/55895d36d8b42a643f8b464e/html5/thumbnails/21.jpg)
Pascal’s Triangle Relationships
11 where 1 11
1
nkCCC kn
kn
kn
1111 nn xxx 1
1111
11
11
011
n
nnk
knk
knnn xCxCxCxCCx
kx of tscoefficienat looking
knCLHS k
nk
n CCRHS 11
1 11
kn
kn CC 1
11
k
nk
nk
n CCC 11
1
l"symmetrica is trianglesPascal'"
11 where 2 nkCC knn
kn
1 3 0 nnn CC
Exercise 5B; 2ace, 5, 6ac,10ac, 11, 14