12X1 T08 02 general binomial expansions (2011)

General Expansion of Binomials

General Expansion of Binomials

kkk

n xxC 1in oft coefficien theis

General Expansion of Binomials

kkk

n xxC 1in oft coefficien theis

kn

Ckn

General Expansion of Binomials

kkk

n xxC 1in oft coefficien theis

nn

nnnnn xCxCxCCx 22101

kn

Ckn

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

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

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

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

Pascal’s Triangle Relationships

Pascal’s Triangle Relationships

11 where 1 11

1

nkCCC kn

kn

kn

Pascal’s Triangle Relationships

11 where 1 11

1

nkCCC kn

kn

kn

1111 nn xxx

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

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

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

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

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

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

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

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

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

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