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

Explanation

Given expansion is `(1 + x + 2 x^(3)) ((3)/(2) x^(2) - (1)/(3x))^(9)`
Now, consider `((3)/(2) x^(2) - (1)/(3x))^(9)`
`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) x^(-r) = .^(9)C_(r) ((3)/(2))^(9 - r) (-(1)/(3))^(r) x^(18 - 3r)`
For term independent of x, putting `18 - 3r = 0, 19 - 3r = 0 " and " 21 - 3r = 0`, we get
`r = 6, r = 19//3, r = 7`
Since, the possible value of r are 6 and 7
Hence, second term is not independent of x.
`:.` The term independent of x is `.^(9)C_(6) (3^(9 - 6))/(2) (-(1)/(3))^(6) + 2. .^(9)C_(7) (3^(9 - 7))/(2) (-(1)/(3))^(7)`
`= (9 xx 8 xx 7 xx 6!)/(6! xx 3 xx 2 x) .(3^(3))/(2^(2)).(1)/(3^(6)) - 2. (9 xx 8 xx 7! )/(7! xx 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)`