1-2x-5-2-x-6-a-bx-cx-2-dx-3-find-a-b-c-and-d-

Question Number 95888 by i jagooll last updated on 28/May/20

{(1 - 2 x)}^{5} {(2 + x)}^{6} = a + bx + {cx}^{2} + {dx}^{3} + \dots

find : a, b, c and d

Answered by john santu last updated on 28/May/20

\Rightarrow (2 + x) \underset{k = 0}{\sum^{5}} C (5, k) {(2)}^{5 - k} {(- (3 x + {2 x}^{2}))}^{k}

clearly a = 64

term containing x \Rightarrow (2 + x) C (5, 1) 2^{4} (- 3 x - {2 x}^{2})

= 80 (2 + x) (- 3 x - {2 x}^{2})

= - 480 x - {560 x}^{2} - {160 x}^{3}

so b = - 480

term containing x^{2}

= (2 + x) C (5, 2) 2^{3} {(3 x + {2 x}^{2})}^{2}

= 80 (2 + x) ({9 x}^{2} + {12 x}^{3} + {4 x}^{4})

so term x^{2} = {560 x}^{2} + {1440 x}^{2}

then we get c = 2000

term containing x^{3}

= (2 + x) C (5, 3) 2^{2} {(- 3 x - {2 x}^{2})}^{3}

= 40 (2 + x) (- {27 x}^{3} + \dots)

= - {2160 x}^{3} + \dots

so term x^{3} = - {160 x}^{3} + {1920 x}^{3} + {720 x}^{3} - {2160 x}^{3}

= {320 x}^{3}

then d = 320