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Contoh soal binomial theorem


Tentukan koefisien x13  pada binomial [ ( x3 +1 )2  ( x2 –  )8 ]


 

[ ( x3 +1 )2  ( x2 –  )8 ] = [ ( x3 +1 )2  ( x2 – 2x-1 )8 ]

(x3 +1)2  = x6  + 2x3 + 1

( x2 – 2x-1 )8  = [ x2 + ( – 2x-1 ) ]8

Misalkan : a = x2 , b = (– 2x-1 )

(a+b) 8 = a8 + 8a7b + 28a6b2 + 56a5b3 + 70a4b4 + 56a3b5 + 28a2b6 + 8ab7 + b8

= (x2)8 + 8 (x2)7 (– 2x-1 ) + 28 (x2)6 (– 2x-1 )2 + 56 (x2)5 (– 2x-1 )3 + 70 (x2)4

(– 2x-1 )4 + 56 (x2)3 (– 2x-1 )5 + 28 (x2)2 (– 2x-1 )6 + 8(x2) (– 2x-1 )7 +

(– 2x-1 )8

=  x16 +(-16) x13 + (28*4) x10 + [ 56 *(-8) x7 ] + …………….

[ ( x3 +1 )2  ( x2 – 2x-1 )8 ]

x6                x7         =  1 * 56 * (-8) = – 448

x3                   x10        =  2 * 28 * 4     =   224

x0                x13        = 1 * (-16)        =   – 16  +

koefisien x13 = – 240 .

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