SOLUTION: Hi, can help me solve this? Thankss~ (a) Find the term that is independent of x in the expansion of (x - 3/x)^6 in ascending powers of x. (b) Hence, find the term that is indepen

Algebra ->  Test -> SOLUTION: Hi, can help me solve this? Thankss~ (a) Find the term that is independent of x in the expansion of (x - 3/x)^6 in ascending powers of x. (b) Hence, find the term that is indepen      Log On


   

Question 1170629: Hi, can help me solve this? Thankss~
(a) Find the term that is independent of x in the expansion of (x - 3/x)^6 in ascending powers of x.
(b) Hence, find the term that is independent of x in the expansion of (2+2/x^2)(x-3/x)^6.
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

a.
Find the term that is independent of x in the expansion of %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
check expanded form:
x%5E6+%2B+729%2Fx%5E6+-+18+x%5E4+-+1458%2Fx%5E4+%2B+135+x%5E2+%2B+1215%2Fx%5E2+-+540



b. Hence, find the term that is independent of x in the expansion of %282%2B2%2Fx%5E2%29%28x-3%2Fx%29%5E6.

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
expanded form: