Question

What is the coefficient of `x^2` in the expansion of `(3x-2)^7`?

Answer 1

The coefficient of `x^2` in the expansion of `(3x-2)^7` is `-6048`.

Explanation:

According to Binomial theorem, exansion of `(a+b)^n` is

`a^n+C_1^na^(n-1)b+C_2^na^(n-2)b^2+.......+C_r^na^(n-r)b^r+......+C_(n-1)^nab^(n-1)+b^n`

observe that `(r+1)^(th)` term is `C_r^na^(n-r)b^r`

Hence in the expansion of `(3x-2)^7`, `(r+1)^(th)` term is `C_r^7(3x)^(7-r)(-2)^r`

and in this power of `x` is `7-r`.

As we are aiming at power of `x` as `2`, we have `7-r=2`

or `r=7-2=5`

and hence corresponding term is `C_5^7(3x)^2(-2)^5`

i.e. `-C_2^7 9x^2xx32=-(7xx6)/(1xx2)xx288x^2=-6048x^2`

Hence, the coefficient of `x^2` in the expansion of `(3x-2)^7` is `-6048`.