Find the term independent of x in the expansion of : (3x-3/x^2)^15

Question

 Find the term independent of x in the expansion of : \(\Big(3x-\frac{2}{x^2}\Big)^{15}\)

Answer 1

To Find : term independent of x, i.e. x⁰

For \(\Big(3x-\frac{2}{x^2}\Big)^{15}\)

a=3x, b = \(\frac{-2}{x^2}\)  and n=15 

We have a formula

Now, to get coefficient of term independent of x that is coefficient of x⁰ we must have, 

(x)^15-3r = x⁰ 

• 15 - 3r = 0 

• 3r = 15 

• r = 5

Therefore, coefficient of x⁰ = (¹⁵₅)(3)^15-5(-2)⁵

= \(\frac{15\times14\times13\times12\times11}{5\times4\times3\times2\times1}\).(3)¹⁰.(-32)

= -3003.(3)¹⁰.(32)

Conclusion : coefficient of x⁰ = -3003.(3)¹⁰.(32)