SOLUTION: Find the term that is independent of x in the expansion of (2+2/x^2)(x-3/x)^6.

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Question 1170639: Find the term that is independent of x in the expansion of (2+2/x^2)(x-3/x)^6.
Found 2 solutions by MathLover1, greenestamps:
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

Find the term that is independent of x in the expansion of+%282%2B2%2Fx%5E2%29%28x-3%2Fx%29%5E6
Formula to use
rth term=%28nCm%29%28a%5E%28n-m%29%29%28b%5Em%29

For %28x+-+3%2Fx%29%5E6:
a+=+x
b+=+-3%2Fx
n+=+6

rth term=%286Cm%29%28x%5E%286-m%29%29%2A%28-3%2Fx%29%5Em%0D%0A%7B%7B%7Brth term=

For the rth term involving K%5B1%5Dx%5E0:
6-2m=0
m=3

then
K%5B1%5D=%286C3%29%28-3%29%5E3=-540

For the rth term involving K%5B2%5Dx%5E2:
6-2m=2
m=2
K%5B2%5D=%286C2%29%28-3%29%5E2=135


The constant term in the expansion of+%282+%2B+2%2Fx%5E2%29%28x+-+3%2Fx%29%5E6 is:
K=2K%5B1%5D%2B2K%5B2%5D=2%28-540%29%2B2%28135%29
K=+-810← answer

check with expanded form:





Answer by greenestamps(13198) About Me  (Show Source):
You can put this solution on YOUR website!


The expansion of the second factor will be in the form

Ax%5E6%2BBx%5E4%2BCx%5E2%2BD%2BE%2Fx%5E2%2BF%2Fx%5E4%2BG%2Fx%5E6

When that is multiplied by 2%2B2%2Fx%5E2, we will get constant partial products from two places: %28D%29%282%29 and %28Cx%5E2%29%282%2Fx%5E2%29

So the constant term in the final expansion will be 2C%2B2D

So we need to calculate the coefficients of the constant and x^2 terms, C and D.

D+=+C%286%2C3%29%28%281%5E3%29%28%28-3%29%5E3%29%29+=+%2820%29%281%29%28-27%29+=+-540

C+=+C%286%2C4%29%28%281%5E4%29%28%28-3%29%5E2%29%29+=+%2815%29%281%29%289%29+=+135

Then

2C%2B2D+=+2%28C%2BD%29+=+2%28-540%2B135%29+=+2%28-405%29+=+-810

ANSWER: -810