I) Find the first 4 terms in the expansion of (2 + x ^(2))^(6) in ascending powers of x. ii) Find the term independent of x in the expansion of (2 + x ^(2))^(6)(1 - \frac{3}{x {…

Question

I) Find the first 4 terms in the expansion of


`(2 + x ^(2))^(6)`
in ascending powers of x.
ii) Find the term independent of x in the expansion of

`(2 + x ^(2))^(6)(1 - \frac{3}{x {}^(2) } ) {}^(2)`
​

Answer 1

(i)
`64+192 x^2+240 x^4+160x^6`

(ii) 1072

Explanation:

Part (i)

Using Binomial series formula:

`(2+x^2)^6=2^6+6C1 \cdot 2^(6-1)\cdot x^2+6C2 \cdot 2^(6-2)\cdot(x^2)^2+6C3 \cdot 2^(6-3)\cdot(x^2)^3`

`=64+6 \cdot 32\cdot x^2+15 \cdot 16\cdot x^4+20 \cdot 8 \cdot x^6`

`=64+192 x^2+240 x^4+160x^6`

Part (ii)

`(1-(3)/(x^2))^2=(1-(3)/(x^2))(1-(3)/(x^2))`

`=1-(6)/(x^2)+(9)/(x^4)`

`(2+x^2)^6(1-(3)/(x^2))^2=(64+192 x^2+240 x^4+160x^6)\left(1-(6)/(x^2)+(9)/(x^4) \right)`

`=64-(384)/(x^2)+(576)/(x^4)+192x^2-1152+(1728)/(x^2)+240x^4-1440x^2+2160+...`

There is no need to keep expanding since the remaining will include the variable.

Therefore, the term independent of x = 64 - 1152 + 2160 = 1072