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Let $x \in R$ be so small that the powers of $x$ beyond two are insignificant and negligibly small. For such $x,$ if ${(1 - x)}^{3} {(2 + x)}^{6}$ is approximated by $a + b x + c x^{2},$ then $2 (a + b + c) =$

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Detailed Solution

${(1 - x)}^{3} {(2 + x)}^{6}$

$= (1 - 3 x + 3 x^{2} - x^{3}) 2^{6} {(1 + \frac{x}{2})}^{6}$

$= (1 - 3 x + 3 x^{2} .... x^{3}) 2^{6}$

$(1 + 6_{c_{1}} \frac{x}{2} + 6_{c_{2}} {(\frac{x}{2})}^{2} + ..... + 6_{c_{6}} {(\frac{x}{2})}^{6})$

constant $= 2^{6}$

Coefficient of $x = 0$
Coefficient of $x^{2} = \frac{- 9}{4} {.2}^{6}$
$2^{6} + 0. \frac{- 9}{4} {.2}^{6} = - 80$

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