Develop the following binomials:

Exercise 1

1. { (2 + 3x) }^{ 4 }

{ (2 + 3x) }^{ 4 } = { _{ 4 }{ C }_{ 0 } } { (2) }^{ 4 } { (3x) }^{ 0 } + { _{ 4 }{ C }_{ 1 } } { (2) }^{ 4 - 1 } { (3x) }^{ 1 } + { _{ 4 }{ C }_{ 2 } } { (2) }^{ 4-2 } { (3x) }^{ 2 } + { _{ 4 }{ C }_{ 3 } } { (2) }^{ 4-3 } { (3x) }^{ 3 } + { _{ 4 }{ C }_{ 4 } } { (2) }^{ 4-4 } { (3x) }^{ 4 }

{ (2 + 3x) }^{ 4 } = 1 . ( 16 ) + 4 . ( 8 )( 3x ) + 6 . ( 4 )( 9{ x }^{ 2 } ) + 4 . ( 2 )( 27 { x }^{ 3 }) + 1 . ( 81 ) ({ x }^{ 4 } )

{ (2 + 3x) }^{ 4 } = 16 + 96x + 216{ x }^{ 2 } + 216{ x }^{ 3 } + 81{ x }^{ 4 }

Exercise 2

2. { (2 - 3y) }^{ 4 }

{ (2 - 3y) }^{ 4 } = { _{ 4 }{ C }_{ 0 } } { (2) }^{ 4 } { (3y) }^{ 0 } - { _{ 4 }{ C }_{ 1 } } { (2) }^{ 4 - 1 } { (3y) }^{ 1 } + { _{ 4 }{ C }_{ 2 } } { (2) }^{ 4-2 } { (3y) }^{ 2 } - { _{ 4 }{ C }_{ 3 } } { (2) }^{ 4-3 } { (3y) }^{ 3 } + { _{ 4 }{ C }_{ 4 } } { (2) }^{ 4-4 } { (3y) }^{ 4 }

{ (2 - 3y) }^{ 4 } = 1 . ( 16 ) - 4 . ( 8 )( 3y ) + 6 . ( 4 )( 9{ y }^{ 2 } ) - 4 . ( 2 )( 27 { y }^{ 3 }) + 1 . ( 81) ( { y }^{ 4 } )

{ (2 - 3y) }^{ 4 } = 16 - 96y + 216{ y }^{ 2 } - 216{ y }^{ 3 } + 81{ y }^{ 4 }

Exercise 3

3. { (4 - x) }^{ 7 }

{ (4-x) }^{ 7 }={ _{ 7 }{ C }_{ 0 } }{ (4) }^{ 7 }{ (x) }^{ 0 }-{ _{ 7 }{ C }_{ 1 } }{ (4) }^{ 7-1 }{ (x) }^{ 1 }+{ _{ 7 }{ C }_{ 2 } }{ (4) }^{ 7-2 }{ (x) }^{ 2 }-{ _{ 7 }{ C }_{ 3 } }{ (4) }^{ 7-3 }{ (x) }^{ 3 }+{ _{ 7 }{ C }_{ 4 } }{ (4) }^{ 7-4 }{ (x) }^{ 4 }-{ _{ 7 }{ C }_{ 5 } }{ (4) }^{ 7-5 }{ (x) }^{ 5 }+{ _{ 7 }{ C }_{ 6 } }{ (4) }^{ 7-6 }{ (x) }^{ 6 }-{ _{ 7 }{ C }_{ 7 } }{ (4) }^{ 7-7 }{ (x) }^{ 7 }

{ (4 - x) }^{ 7 } = 1 . ( 16,374 ) - 7 . ( 4,096 )( x ) + 21 . ( 1,024 )( { x }^{ 2 } ) - 35 . ( 256 )( { x }^{ 3 } ) + 35 . ( 64 ) ( { x }^{ 4 } ) - 21 . ( 16 ) ( { x }^{ 5 } )  + 7 ( 4 ) ( { x }^{ 6 } ) - 1 ( 1 ) ( { x }^{ 7 } )

{ (4 - x) }^{ 7 } = 16,374 - 28,672x + 21,504{ x }^{ 2 } - 8,960{ x }^{ 3 } + 2,240{ x }^{ 4 } - 336{ x }^{ 5 } + 28{ x }^{ 6 } - { x }^{ 7 }

Exercise 4

4. { (x - 3) }^{ 6 }

{ (x - 3) }^{ 6 }={ _{ 6 }{ C }_{ 0 } }{ (x) }^{ 6 }{ (3) }^{ 0 }-{ _{ 6 }{ C }_{ 1 } }{ (x) }^{ 6-1 }{ (3) }^{ 1 }+{ _{ 6 }{ C }_{ 2 } }{ (x) }^{ 6-2 }{ (3) }^{ 2 }-{ _{ 6 }{ C }_{ 3 } }{ (x) }^{ 6-3 }{ (3) }^{ 3 }+{ _{ 6 }{ C }_{ 4 } }{ (x) }^{ 6-4 }{ (3) }^{ 4 }-{ _{ 6 }{ C }_{ 5 } }{ (x) }^{ 6-5 }{ (3) }^{ 5 }+{ _{ 6 }{ C }_{ 6 } }{ (x) }^{ 6-6 }{ (3) }^{ 6 }

{ (x - 3) }^{ 6 } = 1 . ( { x }^{ 6 } ) (1) - 6 . ( { x }^{ 5 } )( 3 ) + 15 . ( { x }^{ 4 } )( 9 ) - 20 . ( { x }^{ 3 } )( 27 ) + 15 . ( { x }^{ 2 } ) ( 81 ) - 6 . ( x ) ( 243 )  + 1 ( 1 ) ( 729 )

{ (x - 3) }^{ 6 } = { x }^{ 6 } - 18{ x }^{ 5 } + 135{ x }^{ 4 } - 540{ x }^{ 3 } + 1,215{ x }^{ 2 } - 1,458x + 729

Exercise 5

4. { (x - 2y) }^{ 4 }

{ (x - 2y) }^{ 4 }={ _{ 4 }{ C }_{ 0 } }{ (x) }^{ 4 }{ (2y) }^{ 0 }-{ _{ 4 }{ C }_{ 1 } }{ (x) }^{ 4-1 }{ (2y) }^{ 1 }+{ _{ 4 }{ C }_{ 2 } }{ (x) }^{ 4-2 }{ (2y) }^{ 2 }-{ _{ 4 }{ C }_{ 3 } }{ (x) }^{ 4-3 }{ (2y) }^{ 3 }+{ _{ 4 }{ C }_{ 4 } }{ (x) }^{ 4-4 }{ (2y) }^{ 4 }

{ (x - 2y) }^{ 4 } = 1 . ( { x }^{ 4 } ) (1) - 4 . ( { x }^{ 3 } )( 2y ) + 6 . ( { x }^{ 2 } )( 4{ y }^{ 2 } ) - 4 . ( x )( 8 { y }^{ 3 } ) + 1 . ( 1 ) ( 16 { y }^{ 4 } )

{ (x - 2y) }^{ 4 } = { x }^{ 4 } - 8 { x }^{ 3 } y + 24 { x }^{ 2 } { y }^{ 2 } - 32 x { y }^{ 3 } + 16 { y }^{ 4 }

Exercise 6

Find the fifth term for the development of { (x + 2y) }^{ 5 }.

{ T }_{ 5 } = { _{ 5 }{ C }_{ 4 } }{ (x) }^{ 5-4 }{ (2y) }^{ 4 }

{ T }_{ 5 } = 5 . ( x ) ( 16{ y }^{ 3 } )

{ T }_{ 5 } = 80x{ y }^{ 3 }

Exercise 7

Calculate the fourth term for the development of { (2 - 3y) }^{ 4 }.

{ T }_{ 4 } = { (-1)}^{ 3 } { _{ 4 }{ C }_{ 3 } } { (2) }^{ 4-3 } { (3y) }^{ 3 }

{ T }_{ 4 } = - 4 ( 2 ) ( 27 { y }^{ 3 }) { (3y) }^{ 3 }

{ T }_{ 4 } = -216{ y }^{ 3 }

Exercise 8

Calculate the fifth term for the development of { (x - \frac { 1 }{ x }) }^{ 7 }.

{ T }_{ 5 } = { (-1)}^{ 4 } { _{ 7 }{ C }_{ 4 } } { (x) }^{ 7-4 } { (\frac { 1 }{ x }) }^{ 4 }

{ T }_{ 5 } = 35 ( { x }^{ 3 } ) (\frac { 1 }{ { x }^{ 4 } })

{ T }_{ 5 } = \frac { 35 }{ x }

Exercise 9

Find the eighth term for the development of { ({ x }^{ 2 } - 3{ y }^{ 3 }) }^{ 10 }

{ T }_{ 8 } = { (-1)}^{ 7 } { _{ 10 }{ C }_{ 7 } } { ({ x }^{ 2 }) }^{ 10-7 } { (3{ y }^{ 3 }) }^{ 7 }

{ T }_{ 8 } = -1 . 120 . ({ x }^{ 6 }) (2,187 { y }^{ 21 })

{ T }_{ 8 } = -262,440 { x }^{ 6 }{ y }^{ 21 }

Exercise 10

Find the independent term for the development of { ({ a }^{ 3 } - \frac { 2 }{ a }) }^{ 20 }.

{ T }_{ k } = { (-1)}^{ k - 1 } . { _{ 20 }{ C }_{ k - 1 } } . { ({ a }^{ 3 }) }^{ 21 - k } . { (\frac { 2 }{ a }) }^{ k - 1 }

The exponent of a with the independent term is 0, therefore take only the literal part and equal it to { a }^{ 0 }.

{ ({ a }^{ 3 }) }^{ 21-k } . { ({ a }^{ -1 }) }^{ k - 1 } = { a }^{ 0 }

{ a }^{ 63 - 3k - k + 1} = { a }^{ 0 }

64 - 4k = 0

4k = 64

k = \frac { 64 }{ 4 }

k = 16

 

{ T }_{ 16 } = { (-1)}^{ 15 } . { _{ 20 }{ C }_{ 15 } } . { ({ a }^{ 3 }) }^{ 5 } . { (\frac { 2 }{ a }) }^{ 15 }

{ T }_{ 16 } = -508, 035, 072

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.