Question

Does the expansion of $\left(2x^2-\frac{1}{x}\right)$ contain any term involving x⁹?

Answer 1

Suppose x⁹ occurs in the given expression at the (r + 1)th term.
Then, we have:
$$\begin{gathered} T_{r+1}={}^{20}C_r(2x^2{)}^{20-r}{\left(\frac{-1}{x}\right)}^r \\ =(-1{)}^r{}^{20}C_r{\left(2\right)}^{20-r}{\left(x\right)}^{40-2r-r} \\ \mathrm{For}\mathrm{this}\mathrm{term}\mathrm{to}\mathrm{contain}{x}^9,\mathrm{we}\mathrm{must}\mathrm{have} \\ 40-3r=9 \\ \Rightarrow 3r=31 \\ \Rightarrow r=\frac{31}{3} \\ \mathrm{It}\mathrm{is}\mathrm{not}\mathrm{possible},\mathrm{as}\mathit{}r\mathrm{is}\mathrm{not}\mathrm{an}\mathrm{integer}. \end{gathered}$$

Hence, there is no term with x⁹ in the given expression.