Chapter 8.4, Problem 125E

To determine

To find: the error in the calculation.

Answer to Problem 125E

In final result it must be ${\left(-2y\right)}^r$ instead of ${\left(2y\right)}^r$ and correct solution is result is $\underset{\_}{{\displaystyle \sum _{r=0}^4\left(\begin{array}{c}4\\ r\end{array}\right)}{\left(3x\right)}^{4-r}{\left(-2y\right)}^r}$ .

Explanation of Solution

Given information:

The calculation in which error needs to be calculated is as follows

  ${\left(3x-2y\right)}^4={\displaystyle \sum _{r=0}^4\left(\begin{array}{c}4\\ r\end{array}\right)}{\left(3x\right)}^{4-r}{\left(2y\right)}^r$

Formula used:

  ${\left(x+y\right)}^n={\displaystyle \sum _{r=0}^n\left(\begin{array}{c}n\\ r\end{array}\right)}{\left(x\right)}^{n-r}{\left(y\right)}^r$

Calculation:

The calculation in which error needs to be calculated is as follows

  ${\left(3x-2y\right)}^4={\displaystyle \sum _{r=0}^4\left(\begin{array}{c}4\\ r\end{array}\right)}{\left(3x\right)}^{4-r}{\left(2y\right)}^r$

Correct calculation

Need to simplify following

  ${\left(3x-2y\right)}^4$

Apply formula ${\left(x+y\right)}^n={\displaystyle \sum _{r=0}^n\left(\begin{array}{c}n\\ r\end{array}\right)}{\left(x\right)}^{n-r}{\left(y\right)}^r$

  ${\left(3x-2y\right)}^4={\displaystyle \sum _{r=0}^4\left(\begin{array}{c}4\\ r\end{array}\right)}{\left(3x\right)}^{4-r}{\left(-2y\right)}^r$

Hence on comparing the calculation, error is in applying formula that is in result it must be ${\left(-2y\right)}^r$ instead of ${\left(2y\right)}^r$ and correct solution is result is $\underset{\_}{{\displaystyle \sum _{r=0}^4\left(\begin{array}{c}4\\ r\end{array}\right)}{\left(3x\right)}^{4-r}{\left(-2y\right)}^r}$ .