SOLUTION: find the coefficient of x^4 in the expansion of (x+1/x)^(5) x (x-1/x)^(3)

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Question 1160863: find the coefficient of x^4 in the expansion of (x+1/x)^(5) x (x-1/x)^(3)
Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!


%28x%2B1%2Fx%29%5E%285%29%2A%28x-1%2Fx%29%5E%283%29

%28%28x%5E2%2B1%29%2Fx%5E%22%22%29%5E5%28%28x%5E2-1%29%2Fx%5E%22%22%29%5E3

expr%28%28x%5E2%2B1%29%5E5%2Fx%5E5%29%2Aexpr%28%28x%5E2-1%29%5E3%2Fx%5E3%29

%28%28x%5E2%2B1%29%5E5%28x%5E2-1%29%5E3%29%2Fx%5E8

Using Pascal's triangle:



So we only need to multiply the terms which will give us
a term in x12 on top since we have x8 on the bottom, and
the exponents will subtract to give exponent 4.

%28x%5E10%29%283x%5E2%29%2B%285x%5E8%29%28-3x%5E4%29%2B%2810x%5E6%29%28x%5E6%29

3x%5E12-15x%5E12%2B10x%5E12

-2x%5E12

So the coefficient of x4 in the original problem is also -2

Edwin