Question

Can you show me where I went wrong in this problem? And what i should correct

Answer 1

The given partial fraction is:

`(x^2+x+1)/(x(x-1)^2)`

There are some constants, A, B, and C such that:

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

Multiplying both sides by x(x - 1)², it follows:

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

Substituting x=1 it follows that:

`\begin{gathered} 1+1+1=C \\ \text{Therefore,} \\ C=3 \end{gathered}`

Substituting C=3 into the equation, it follows that:

`x^2+x+1=A(x-1)^2+Bx(x-1)+3x`

Expand the right side of the equation:

`x^2+x+1=A(x^2-2x+1)^{}+B(x^2-x)+3x`

By comparing coefficients of x², we have that:

`A+B=1-------(1)`

By comparing coefficient of constant, it follows that:

`A=1---------(2)`

Substitute A=1 into equation (1), it follows that:

`\begin{gathered} 1+B=1 \\ \text{Therefore, } \\ B=0 \end{gathered}`

Subtracting equation (2) from equation (1), it follows that:

`\begin{gathered} 2B=2 \\ B=1 \end{gathered}`

Therefore, the partial fraction decomposes to:

`(1)/(x)+(3)/((x-1)^2)`

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