Question

What is the rational expression as a sum of partial fractions?

Answer 1

`(-x^2+2x-5)/(x^3-x^2)`

Rewrite the expression as:

`(-x^2+2x-5)/(x^2(x-1))`

The partial fraction expansion is of the form:

`(-x^2+2x-5)/(x^2(x-1))=(A)/(x-1)+(B)/(x)+(C)/(x^2)`

Multiply both sides by x²(x - 1):

`\begin{gathered} -x^2+2x-5=Ax^2+(x-1)(Bx+C) \\ -x^2+2x-5=-C+(A+B)x^2+(C-B)x \end{gathered}`

Equate the coefficients on both sides:

`\begin{gathered} -5=-C_{\text{ }}(1)_{} \\ 2=C-B_{\text{ }}(2) \\ -1=A+B_{\text{ }}(3) \end{gathered}`

So, from (1):

`C=5`

Replace C into (2):

`\begin{gathered} 2=5-B \\ B=3 \end{gathered}`

Replace B into (3):

`\begin{gathered} -1=A+3 \\ A=-4 \end{gathered}`

Therefore, the answer is:

`(-x^2+2x-5)/(x^2(x-1))=(-4)/(x-1)+(3)/(x)+(5)/(x^2)`