Find the co-efficient of x^4 in the expansion of (1+x^2)(x-2/x)^6

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embizze | High School Teacher | (Level 1) Educator Emeritus

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To get a term with `x^4` we need (a) `x^n*(2/x)^(6-n)` to be `x^4` so n-(6-n)=4==>n=5 or (b) `x^2x^n(2/x)^(6-n)` to be `x^4` so 2+n-(6-n)=4==>n=4

(a) if n=5 we get `([6],[5])(-1)^5x^5(2/x)=-12x^4`

(b) if n=4 we get `([6],[4])(-1)^4x^4(2/x)^2=60x^2` . When multiplied by `-x^2` we get `-60x^4`

Adding the coefficients we get `-72x^4` so the coefficient is -72.

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The full expansion:

`(1-x^2)(x^6-6x^5(2/x)+15x^4(2/x)^2-20x^3(2/x)^3+15x^2(2/x)^4-6x(2/x)^5+(2/x)^6)` `=(1-x^2)(x^6-12x^4+60x^2-160+240/x^2-192/x^4+32/x^6)`

`=x^6-12x^4+60x^2-160+240/x^2-192/x^4+32/x^6`

`-x^8+12x^6-60x^4+160x^2-240+192/x^2-32/x^4`

`= -x^8+13x^6-72x^4-220x^2-400+432/x^2-224/x^4+32/x^6`

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