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# in the expansion #(1 + x^2)(x/2-4/x)^6#

1 Answer
Jim G.
Jul 25, 2017

#-145#

Explanation:

#"expand "(x/2-4/x)^6" using the "color(blue)"binomial theorem"#

#rArr(x/2-4/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#

#=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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