`(3x^2 - 7x - 2)/(x^3 - x)` Write the partial fraction decomposition of the rational expression.

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`(3x^2-7x-2)/(x^3-x)=(3x^2-7x-2)/[x(x+1)(x-1)]`

`(3x^2-7x-2)/(x^3-x)=A/x+B/(x+1)+C/(x-1)`

 

Multiply through by the LDC `x^3-x.`

`3x^2-7x-2=A(x^2-1)+Bx(x-1)+Cx(x+1)`

`3x^2-7x-2=Ax^2-A+Bx^2-Bx+Cx^2+Cx`

`3x^2-7x-2=(A+B+C)x^2+(-B+C)x-A`

 

Equate the coefficients of like terms. Then solve for A, B, and C.

`3=A+B+C`

`-7=-B+C`

`-2=-A`

 

`A=2`

 

`3=A+B+C`

`3=2+B+C`

`1=B+C`

 

Use the elimination method to solve for B and C.

`1=B+C`

`-7=-B+C`

_______________

`-6=2C`

`-3=C`

 

`1=B+C`

`1=B-3`

`4=B`

 

`A=2, B=4, C=-3`

 

`(3x^2-7x-2)/(x^3-x)=2/x+4/(x+1)-3/(x-1)`

 

 

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