Chapter 7.3, Problem 107E

To determine

To solve: The system of equation and check whether it is correctly solved or not and if not then explained what is wrong.

Answer to Problem 107E

The solution of the system is $A=-1$ and $A=1$ , the student is not correct and the problem is the original fraction’s numerator and denominators have the same degree of 2. So, $x^2+1$ must be divided by $x^2-x$ before performing the decomposition.

Explanation of Solution

Given information:

The system is.

  $\{\begin{array}{l}A+B=0\\ -A=1\end{array}$

Calculation:

From the above system $A=-1$ .

Substitute $-1$ for $A$ in $A+B=0$ to find the value of $B$ .

  $\begin{array}{c}-1+B=0\\ B=1\end{array}$

The partial fraction will be as follows.

  $\begin{array}{c}\frac{x^2+1}{x\left(x-1\right)}=\frac{-1}{x}+\frac{1}{x-1}\\ =\frac{-\left(x-1\right)+x}{x\left(x-1\right)}\\ =\frac{1}{x\left(x-1\right)}\end{array}$

The student has not worked correctly.

The problem is the original fraction’s numerator and denominators have the same degree of 2. So, $x^2+1$ must be divided by $x^2-x$ before performing the decomposition.

Therefore, the solution of the system is $A=-1$ and $A=1$ , the student is not correct and the problem is the original fraction’s numerator and denominators have the same degree of 2. So, $x^2+1$ must be divided by $x^2-x$ before performing the decomposition.