Question

Find the coefficient of the x^9 in the expansion of (3x^4 − 1/x)^6

Answer 1

By the binomial theorem:

`\left(3x^4-\frac1x\right)^6=\displaystyle\sum_(k=0)^6 \binom6k \left(3x^4\right)^(6-k) \left(-\frac1x\right)^k`

where

`\dbinom nk=(n!)/(k!(n-k)!)`

is the binomial coefficient. Now,

`\dbinom6k \left(3x^4\right)^(6-k) \left(-\frac1x\right)^k=\dbinom6k 3^(6-k) (-1)^k x^(24-4k) x^(-k)=\dbinom 6k 3^6 \left(-\frac13\right)^k x^(24-5k)`

and the x⁹ term occurs when 24 - 5k = 9, or k = 3, so the coefficient is

`\dbinom63 3^(6-3) (-1)^3=-(6!)/((3!)^2) 3^3=\boxed{-540}`