Chapter 11.6, Problem 37HP

To determine

Write a power of a binomial for which the second term of the expansion is $6x^{4}y$

Answer to Problem 37HP

The binomial is ${(x+\frac{6}{5}y)}^5$.

Explanation of Solution

Given:

Write a power of a binomial for which the second term of the expansion is $6x^{4}y$

Concept Used:

The general term of the expression ${(x+y)}^n$ is ${\text{T}}_{r+1}=\text{C}{}_r{x}^{n-r}{y}^r$

Calculation:

Example: ${(x+\frac{6}{5}y)}^5$

Let us find the 2^nd term:

  $\begin{array}{l}{\text{T}}_{r+1}=\text{C}{}_r{x}^{n-r}{y}^r\\ \text{Here}:x=x;y=\frac{6}{5}y;n=5;r=1\\ $ {\text{T}}_2={\text{T}}_{1+1}=\text{C}{}_1{x}^{5-1}{(\frac{6}{5}y)}^1=5·x^4·\frac{6}{5}y=6x^{4}y$\end{array}$   ${\text{T}}_2=6x^{4}y$

The second term of the expression ${(x+\frac{6}{5}y)}^5$ is $6x^{4}y$

Thus, the binomial is ${(x+\frac{6}{5}y)}^5$