Finding the coefficient of x in an expansion?

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Finding the coefficient of x in an expansion?

I need an example to show me how to implement the methods of finding the coefficient of x in an expansion. The example question I'm using is the following.
Finding the coefficient of $x^{2}$ $x^2$ in the expansion $(1 + x^{2}) {(\frac{x}{2} - \frac{4}{x})}^{6}$ $(1 + x^2)(x/2-4/x)^6$

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Answer 1 · 2017-07-25T13:31:10

$- 145$ $-145$

Explanation:

$expand {(\frac{x}{2} - \frac{4}{x})}^{6} using the binomial theorem$ $"expand "(x/2-4/x)^6" using the "color(blue)"binomial theorem"$

$\Rightarrow {(\frac{x}{2} - \frac{4}{x})}^{6}$ $rArr(x/2-4/x)^6$

$= {(\frac{x}{2})}^{6} + 6 . {(\frac{x}{2})}^{5} (- \frac{4}{x}) + 15 . {(\frac{x}{2})}^{4} {(- \frac{4}{x})}^{2} + 20 {(\frac{x}{2})}^{3} {(- \frac{4}{x})}^{3} + 15 . {(\frac{x}{2})}^{2} {(- \frac{4}{x})}^{4} + 6 . (\frac{x}{2}) {(- \frac{4}{x})}^{5} + {(- \frac{4}{x})}^{6}$ $=(x/2)^6+6.(x/2)^5(-4/x)+15.(x/2)^4(-4/x)^2+20(x/2)^3(-4/x)^3+15.(x/2)^2(-4/x)^4+6.(x/2)(-4/x)^5+(-4/x)^6$

$= \frac{1}{64} x^{6} - \frac{3}{4} x^{4} + 15 x^{2} - 160 + 960 x^{- 2} - 3072 x^{- 4} + 4096 x^{- 6}$ $=1/64x^6-3/4x^4+15x^2-160+960x^-2-3072x^-4+4096x^-6$

$--------------------------------------------------------------$ $color(blue)"--------------------------------------------------------------"$

$rArr(1+x^2)(1/64x^6-3/4x^4+15x^2-160+.......)$

$"we only require the terms in "x^2$

$"multiplying expansion by 1 "to15x^2$

$"multiplying expansion by "x^2to-160x^2$

$"the term in "x^2=15x^2-160x^2=-145x^2$

$rArr"the coefficient of the term in "x^2=-145$

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Finding the coefficient of x in an expansion?

I need an example to show me how to implement the methods of finding the coefficient of x in an expansion. The example question I'm using is the following.
Finding the coefficient of $x^{2}$ $x^2$ in the expansion $(1 + x^{2}) {(\frac{x}{2} - \frac{4}{x})}^{6}$ $(1 + x^2)(x/2-4/x)^6$

1 Answer

Explanation:

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1 Answer

Explanation:

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Impact of this question

I need an example to show me how to implement the methods of finding the coefficient of x in an expansion. The example question I'm using is the following.
Finding the coefficient of $x^{2}$ $x^2$ in the expansion $(1 + x^{2}) {(\frac{x}{2} - \frac{4}{x})}^{6}$ $(1 + x^2)(x/2-4/x)^6$