You are tutoring a student in algebra. In trying to find a partial fraction

Chapter 8, Problem 107

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Error analysis   You are tutoring a student in algebra. In trying to find a partial fraction decomposition, your student writes the following.

\(\frac{x^{2}+1}{x(x-1)} &=\frac{A}{x}+\frac{B}{x-1}\)

\(x^{2}+1 &=A(x-1)+B x\)          Basic equation

\(x^{2}+1 &=(A+B) x-A\)

Your student then forms the following system of linear equations.

\(\left\{\begin{aligned} A+B &=0 \\ -A &=1 \end{aligned}\right.\)

Solve the system and check the partial fraction decomposition it yields. Has your student worked the problem correctly? If not, what went wrong?

Text Transcription:

x^2+1/x(x-1)=A/x+B/x-1

x^2+1=A(x-1)+Bx

x^2+1=(A+B)x-A

{ A+B=0 _-A=1