1. Find `x^4` term of `(x-2)^6`

Solution:
`(x-2)^6`

Suppose the `x^4` term exists and it is `(r+1)^(th)` term

`"Here "a=x,b=-2,n=6`

`T_(r+1)=((n),(r))a^(n-r)b^r`

`=((6),(r))(x)^(6-r)(-2)^r`

`=((6),(r))(1)^(6-r)*(x)^(6-r)*(-2)^r`

`=((6),(r))(1)^(6-r)*(-2)^r*(x)^(6-r)`

`=((6),(r))(1)^(6-r)*(-2)^r*(x)^((6-r))`

`=((6),(r))(1)^(6-r)*(-2)^r*(x)^(6-r)`

Determine the value of `r` for the `x^4` term

For the `x^4` term, the power of `x` must be 4

`6-r=4`

`r=2`

To find `T_3`, we let `r=2`

`:. T_3=T_(2+1)`

`=((6),(2))(x)^4(-2)^2`

`=(6!)/(2!(6-2)!)(x)^4(-2)^2`

`=(6*5)/(2*1)(x^4)(4)`

`=15(x^4)(4)`

`=60x^4`

The `x^4` term in the expansion of `(x-2)^6` is `60x^4`

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2. Find `x^1` term of `(2x^2-1/x)^11`

Solution:
`(2x^2-1/x)^11`

Suppose the `x^1` term exists and it is `(r+1)^(th)` term

`"Here "a=2x^2,b=-1/x,n=11`

`T_(r+1)=((n),(r))a^(n-r)b^r`

`=((11),(r))(2x^2)^(11-r)(-1/x)^r`

`=((11),(r))(2)^(11-r)*(x^2)^(11-r)*(-1)^r*(1/x)^r`

`=((11),(r))(2)^(11-r)*(-1)^r*(x^2)^(11-r)*(1/x)^r`

`=((11),(r))(2)^(11-r)*(-1)^r*(x)^(2(11-r))*(x)^(-r)`

`=((11),(r))(2)^(11-r)*(-1)^r*(x)^(22-2r-r)`

`=((11),(r))(2)^(11-r)*(-1)^r*(x)^(22-3r)`

Determine the value of `r` for the `x^1` term

For the `x^1` term, the power of `x` must be 1

`22-3r=1`

`3r=21`

`r=7`

To find `T_8`, we let `r=7`

`:. T_8=T_(7+1)`

`=((11),(7))(2x^2)^4(-1/x)^7`

`=(11!)/(7!(11-7)!)(2x^2)^4(-1/x)^7`

`=(11*10*9*8)/(4*3*2*1)(16x^8)(-1/(x^7))`

`=330(16x^8)(-1/(x^7))`

`=-5280x`

The `x^1` term in the expansion of `(2x^2-1/x)^11` is `-5280x`

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3. Find `x^-2` term of `(x-1/x^2)^16`

Solution:
`(x-1/x^2)^16`

Suppose the `x^-2` term exists and it is `(r+1)^(th)` term

`"Here "a=x,b=-1/(x^2),n=16`

`T_(r+1)=((n),(r))a^(n-r)b^r`

`=((16),(r))(x)^(16-r)(-1/(x^2))^r`

`=((16),(r))(1)^(16-r)*(x)^(16-r)*(-1)^r*(1/(x^2))^r`

`=((16),(r))(1)^(16-r)*(-1)^r*(x)^(16-r)*(1/(x^2))^r`

`=((16),(r))(1)^(16-r)*(-1)^r*(x)^((16-r))*(x)^(-2r)`

`=((16),(r))(1)^(16-r)*(-1)^r*(x)^(16-r-2r)`

`=((16),(r))(1)^(16-r)*(-1)^r*(x)^(16-3r)`

Determine the value of `r` for the `x^-2` term

For the `x^-2` term, the power of `x` must be -2

`16-3r=-2`

`3r=18`

`r=6`

To find `T_7`, we let `r=6`

`:. T_7=T_(6+1)`

`=((16),(6))(x)^10(-1/(x^2))^6`

`=(16!)/(6!(16-6)!)(x)^10(-1/(x^2))^6`

`=(16*15*14*13*12*11)/(6*5*4*3*2*1)(x^10)(1/(x^12))`

`=8008(x^10)(1/(x^12))`

`=8008/(x^2)`

The `x^-2` term in the expansion of `(x-1/x^2)^16` is `8008/(x^2)`

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