题目

Problem

In the expansion of $\left(kx+\dfrac{2}{x}\right)^4$ , where $k$ is a positive constant, the term independent of $x$ is equal to $150$ .

Find the value of $k$ and hence determine the coefficient of $x^2$ in the expansion.

[4]

题目中文翻译

在 $\left(kx+\dfrac{2}{x}\right)^4$ 的展开式中， $k$ 是正的常数，与 $x$ 无关的项等于 $150$ 。

求 $k$ 的值，并由此确定展开式中 $x^2$ 的系数。

解答

展开

若选 $j$ 个 $2/x$ ，总幂次就是 $4-2j$ 。先求常数项确定 $k$ ，再找幂次为 $2$ 的项。

展开

The general term is $\binom4j k^{4-j}2^j x^{4-2j}$ . The constant term occurs at $j=2$ , so

\binom42 k^2 2^2=24k^2=150.

This gives $k^2=25/4$ . As $k>0$ , $k=\frac52$ . The term in $x^2$ occurs at $j=1$ , and its coefficient is

\binom41\Big(\frac52\Big)^3\cdot2=125.