12 x1 t08 01 binomial expansions (2012)
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Binomial Theorem
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Binomial TheoremBinomial ExpansionsA binomial expression is one which contains two terms.
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Binomial TheoremBinomial ExpansionsA binomial expression is one which contains two terms. 11 0 x
![Page 4: 12 x1 t08 01 binomial expansions (2012)](https://reader034.fdocuments.us/reader034/viewer/2022042700/5597df3c1a28ab64388b4763/html5/thumbnails/4.jpg)
Binomial TheoremBinomial ExpansionsA binomial expression is one which contains two terms. 11 0 x
xx 111 1
![Page 5: 12 x1 t08 01 binomial expansions (2012)](https://reader034.fdocuments.us/reader034/viewer/2022042700/5597df3c1a28ab64388b4763/html5/thumbnails/5.jpg)
Binomial TheoremBinomial ExpansionsA binomial expression is one which contains two terms. 11 0 x
xx 111 1
22 1211 xxx
![Page 6: 12 x1 t08 01 binomial expansions (2012)](https://reader034.fdocuments.us/reader034/viewer/2022042700/5597df3c1a28ab64388b4763/html5/thumbnails/6.jpg)
Binomial TheoremBinomial ExpansionsA binomial expression is one which contains two terms. 11 0 x
xx 111 1
22 1211 xxx
32
322
23
331221
12111
xxxxxxxx
xxxx
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Binomial TheoremBinomial ExpansionsA binomial expression is one which contains two terms. 11 0 x
xx 111 1
22 1211 xxx
32
322
23
331221
12111
xxxxxxxx
xxxx
432
43232
324
464133331
33111
xxxxxxxxxxx
xxxxx
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Blaise Pascal saw a pattern which we now know as Pascal’s Triangle
![Page 9: 12 x1 t08 01 binomial expansions (2012)](https://reader034.fdocuments.us/reader034/viewer/2022042700/5597df3c1a28ab64388b4763/html5/thumbnails/9.jpg)
Blaise Pascal saw a pattern which we now know as Pascal’s Triangle
1
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Blaise Pascal saw a pattern which we now know as Pascal’s Triangle
11 1
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Blaise Pascal saw a pattern which we now know as Pascal’s Triangle
11 1
1 2 1
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Blaise Pascal saw a pattern which we now know as Pascal’s Triangle
11 1
1 2 11 3 3 1
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Blaise Pascal saw a pattern which we now know as Pascal’s Triangle
11 1
1 2 11 3 3 1
1 4 6 4 1
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Blaise Pascal saw a pattern which we now know as Pascal’s Triangle
11 1
1 2 11 3 3 1
1 4 6 4 11 5 10 10 5 1
![Page 15: 12 x1 t08 01 binomial expansions (2012)](https://reader034.fdocuments.us/reader034/viewer/2022042700/5597df3c1a28ab64388b4763/html5/thumbnails/15.jpg)
Blaise Pascal saw a pattern which we now know as Pascal’s Triangle
11 1
1 2 11 3 3 1
1 4 6 4 11 5 10 10 5 1
1 6 15 20 15 6 1
![Page 16: 12 x1 t08 01 binomial expansions (2012)](https://reader034.fdocuments.us/reader034/viewer/2022042700/5597df3c1a28ab64388b4763/html5/thumbnails/16.jpg)
Blaise Pascal saw a pattern which we now know as Pascal’s Triangle
11 1
1 2 11 3 3 1
1 4 6 4 11 5 10 10 5 1
1 6 15 20 15 6 11 7 21 35 35 21 7 1
![Page 17: 12 x1 t08 01 binomial expansions (2012)](https://reader034.fdocuments.us/reader034/viewer/2022042700/5597df3c1a28ab64388b4763/html5/thumbnails/17.jpg)
Blaise Pascal saw a pattern which we now know as Pascal’s Triangle
11 1
1 2 11 3 3 1
1 4 6 4 11 5 10 10 5 1
1 6 15 20 15 6 11 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
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Blaise Pascal saw a pattern which we now know as Pascal’s Triangle
11 1
1 2 11 3 3 1
1 4 6 4 11 5 10 10 5 1
1 6 15 20 15 6 11 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 11 9 36 84 126 126 84 36 9 1
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Blaise Pascal saw a pattern which we now know as Pascal’s Triangle
11 1
1 2 11 3 3 1
1 4 6 4 11 5 10 10 5 1
1 6 15 20 15 6 11 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 11 9 36 84 126 126 84 36 9 1
1 10 45 120 210 252 210 120 45 10 1
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7
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xige
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7
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xige
76
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32171
xx
xxxxx
![Page 22: 12 x1 t08 01 binomial expansions (2012)](https://reader034.fdocuments.us/reader034/viewer/2022042700/5597df3c1a28ab64388b4763/html5/thumbnails/22.jpg)
7
321..
xige
76
52
43
34
2567
32
3217
32121
32135
32135
32121
32171
xx
xxxxx
2187128
729448
243672
81560
27280
984
3141
765432 xxxxxxx
![Page 23: 12 x1 t08 01 binomial expansions (2012)](https://reader034.fdocuments.us/reader034/viewer/2022042700/5597df3c1a28ab64388b4763/html5/thumbnails/23.jpg)
7
321..
xige
76
52
43
34
2567
32
3217
32121
32135
32135
32121
32171
xx
xxxxx
2187128
729448
243672
81560
27280
984
3141
765432 xxxxxxx
dps 8 to0.998 of value thefind to1 ofexpansion the Use 1010xii
![Page 24: 12 x1 t08 01 binomial expansions (2012)](https://reader034.fdocuments.us/reader034/viewer/2022042700/5597df3c1a28ab64388b4763/html5/thumbnails/24.jpg)
7
321..
xige
76
52
43
34
2567
32
3217
32121
32135
32135
32121
32171
xx
xxxxx
2187128
729448
243672
81560
27280
984
3141
765432 xxxxxxx
dps 8 to0.998 of value thefind to1 ofexpansion the Use 1010xii
109
876543210
1045120210252210120451011
xxxxxxxxxxx
![Page 25: 12 x1 t08 01 binomial expansions (2012)](https://reader034.fdocuments.us/reader034/viewer/2022042700/5597df3c1a28ab64388b4763/html5/thumbnails/25.jpg)
7
321..
xige
76
52
43
34
2567
32
3217
32121
32135
32135
32121
32171
xx
xxxxx
2187128
729448
243672
81560
27280
984
3141
765432 xxxxxxx
dps 8 to0.998 of value thefind to1 ofexpansion the Use 1010xii
109
876543210
1045120210252210120451011
xxxxxxxxxxx
3210 002.0120002.045002.0101998.0
![Page 26: 12 x1 t08 01 binomial expansions (2012)](https://reader034.fdocuments.us/reader034/viewer/2022042700/5597df3c1a28ab64388b4763/html5/thumbnails/26.jpg)
7
321..
xige
76
52
43
34
2567
32
3217
32121
32135
32135
32121
32171
xx
xxxxx
2187128
729448
243672
81560
27280
984
3141
765432 xxxxxxx
dps 8 to0.998 of value thefind to1 ofexpansion the Use 1010xii
109
876543210
1045120210252210120451011
xxxxxxxxxxx
3210 002.0120002.045002.0101998.0
98017904.0
![Page 27: 12 x1 t08 01 binomial expansions (2012)](https://reader034.fdocuments.us/reader034/viewer/2022042700/5597df3c1a28ab64388b4763/html5/thumbnails/27.jpg)
42 5432in oft coefficien theFind xxxiii
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42 5432in oft coefficien theFind xxxiii 432234
4
5544546544432
5432
xxxxx
xx
![Page 29: 12 x1 t08 01 binomial expansions (2012)](https://reader034.fdocuments.us/reader034/viewer/2022042700/5597df3c1a28ab64388b4763/html5/thumbnails/29.jpg)
42 5432in oft coefficien theFind xxxiii 432234
4
5544546544432
5432
xxxxx
xx
![Page 30: 12 x1 t08 01 binomial expansions (2012)](https://reader034.fdocuments.us/reader034/viewer/2022042700/5597df3c1a28ab64388b4763/html5/thumbnails/30.jpg)
42 5432in oft coefficien theFind xxxiii 432234
4
5544546544432
5432
xxxxx
xx
![Page 31: 12 x1 t08 01 binomial expansions (2012)](https://reader034.fdocuments.us/reader034/viewer/2022042700/5597df3c1a28ab64388b4763/html5/thumbnails/31.jpg)
42 5432in oft coefficien theFind xxxiii 432234
4
5544546544432
5432
xxxxx
xx
54435462 oft coefficien 3222 x
![Page 32: 12 x1 t08 01 binomial expansions (2012)](https://reader034.fdocuments.us/reader034/viewer/2022042700/5597df3c1a28ab64388b4763/html5/thumbnails/32.jpg)
42 5432in oft coefficien theFind xxxiii 432234
4
5544546544432
5432
xxxxx
xx
54435462 oft coefficien 3222 x
96038404800
![Page 33: 12 x1 t08 01 binomial expansions (2012)](https://reader034.fdocuments.us/reader034/viewer/2022042700/5597df3c1a28ab64388b4763/html5/thumbnails/33.jpg)
42 5432in oft coefficien theFind xxxiii 432234
4
5544546544432
5432
xxxxx
xx
54435462 oft coefficien 3222 x
Exercise 5A; 2ace etc, 4, 6, 7, 9ad, 12b, 13ac, 14ace, 16a, 22, 23
96038404800