Question

In the following expansion, find the indicated coefficient.

x³ in `(x^2 + (3sqrt(2))/x)^9`

Answer 1

Here, a = x², b = `(3sqrt(2))/x`, n = 9

We have, t_r+1 = ⁿC_r . a^n–r . b^r 

= `""^9"C"_"r" (x^2)^(9-"r") ((3sqrt(2))/x)^"r"`

= `""^9"C"_"r"  x^(18 - 2"r").(3sqrt(2))^"r".x^(-"r")`

= `""^9"C"_"r"(3sqrt(2))^"r".x^(18 - 3"r")`

To get the coefficient of x³, we must have

x^18–3r = x³

∴ 18 – 3r = 3

∴ 15 = 3r

∴  r = 5

∴ Coefficient of x³ = `""^9"C"_5(3sqrt(2))^5`

= `(9!)/(5!4!) (3sqrt(2))^5`

= `(9 xx 8 xx 7 xx 6)/(4 xx 3 xx 2 xx 1) xx 243 xx 4sqrt(2)`

= `122472 sqrt(2)`

∴ Coefficient of x³ is `122472 sqrt(2)`