Question 1170639

Find the term that is independent of {{{x}}} in the expansion of{{{ (2+2/x^2)(x-3/x)^6}}}

Formula to use 

{{{r}}}th term={{{(nCm)(a^(n-m))(b^m)}}}
 

For {{{(x - 3/x)^6}}}:
    {{{a = x}}}
    {{{b = -3/x}}}
    {{{n = 6}}}


{{{r}}}th term={{{(6Cm)(x^(6-m))*(-3/x)^m
{{{r}}}th term={{{(6Cm)(-3)^m*(x^(6-2m))


Let {{{(6Cm)(-3)^m=K[j]}}}


For the {{{r}}}th term involving {{{K[1]x^0}}}:

{{{6-2m=0}}}

{{{m=3}}}


then

{{{K[1]=(6C3)(-3)^3=-540}}}


For the {{{r}}}th term involving {{{K[2]x^2}}}:

{{{6-2m=2}}}
{{{m=2}}}

{{{K[2]=(6C2)(-3)^2=135}}}
 


The constant term in the expansion of{{{ (2 + 2/x^2)(x - 3/x)^6}}} is:

{{{K=2K[1]+2K[2]=2(-540)+2(135)}}}

{{{K= -810}}}←   answer


check with expanded form:


{{{1458/x^8 + 2x^6 - 1458/x^6 - 34x^4 - 486/x^4 + 234x^2 + 1350/x^2 - 810}}}