Find the term independent of x in the expansion of (1 + x + 2x^{3}) `(3/2 x^2 - 1/(3x))^9`

#### Solution

Given expression is (1 + x + 2x^{3}) `(3/2 x^2 - 1/(3x))^9`

Let us consider `(3/2 x^2 - 1/(3x))^9`

General Term `"T"_(r + 1) = ""^n"C"_r x^(n - r) y^r`

`"T"_(r + 1) = ""^9"C"_r (3/2 x^2)^(9 - r) (- 1/(3x))^r`

= `""^9"C"_r (3/2)^(9 - r) (x)^(18 - 2r) * (- 1/3)^r * 1/(x)^r`

= `""^9"C"_r (3/2)^(9 - r) (x)^(18 - 2r - r) * (- 1/3)^r`

= `""^9"C"_r (3/2)^(9 - r) (- 1/3)^r * x^(18 - 3r)`

So, the general term in the expansion of

`(1 + x + 2x^3) (3/2 x^2 - 1/(3x))^9`

= `""^9"C"_r (3/2)^(9 - r) (- 1/3)^r * (x)^(18 - 3r) + ""^9"C"_r (3/2)^(9 - r) (- 1/3)^r * (x)^(19 - 3r) + 2 * ""^9"C"_r (3/2)^(9 - r) (- 1/3)^r * (x)^(21 - 3r)`

For getting the term independent of x,

Put 18 – 3r = 0, 19 – 3r = 0 and 21 – 3r = 0, we get

r = 6

r = `19/3` and r = 7

The possible value of r are 6 and 7 ```.....(because r ≠ 19/3)`

∴ The term independent of x is

= `""^9"C"_6 (3/2)^(9 - 6) (- 1/3)^6 + 2 * ""^9"C"_7 (3/2)^(9 - 7) (- 1/3)^7`

= `(9 xx 8 xx 7 xx 6!)/(3 xx 2 xx 1 xx 6!) * 3^3/2^3 * 1/3^6 - 2 * (9 xx 8 xx 7!)/(7!2 xx 1) * 3^2/2^2 * 1/3^7`

= `84/8 * 1/3^3 - 36/4 * 2/3^5`

= `7/18 - 2/27`

= `(21 - 4)/54`

= `17/54`

Hence, the required term = `17/54`