Chapter 12, Problem 33RE

To determine

To find: The coefficient of $x^2$ .

Answer to Problem 33RE

The coefficient of $x^2$ is $18432$ .

Explanation of Solution

Given information: The expression ${\left(2x+1\right)}^7$ .

Calculation: ${\left(2x+1\right)}^7$

Since, The general term is $T_{r+1}=9_{C_r}{x}^r{2}^{9-r}$

Let’s find the coefficient of $x^2$ here, $r=2$

Now,

  $\begin{array}{l}$ T_{r+1}=9_{C_{{}_r}}{(2x)}^r{2}^{9-r}$$ T_{2+1}=9_{C_2}{(2x)}^2{2}^{9-2}$T_3=9_{C_2}{x}^2{2}^9\end{array}$

Therefore, The coefficient of $x^2$ is,

  $\begin{array}{l}=9_{C_2}\times 2^9\\ =\frac{9!}{2!7!}\times 512\\ =\frac{9\times 8\times 7!}{2\times 7!}\times 512\\ =\frac{72\times 512}{2}\\ =18432\end{array}$

Hence, The coefficient of $x^2$ is $18432$ .