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

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

Divide by applying long division method,

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

Now continue with the partial fraction of the remainder expression,

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

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

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

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

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

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

`:.(2x+1)=Ax+2A+Bx-B`

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

Equating the coefficients of the like terms,

`A+B=2`  ---- equation 1

`2A-B=1`   --- equation 2

Add the equation 1 and 2,

`A+2A=2+1`

`3A=3`

`A=1`

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

Divide by applying long division method,

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

Now continue with the partial fraction of the remainder expression,

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

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

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

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

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

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

`:.(2x+1)=Ax+2A+Bx-B`

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

Equating the coefficients of the like terms,

`A+B=2`  ---- equation 1

`2A-B=1`   --- equation 2

Add the equation 1 and 2,

`A+2A=2+1`

`3A=3`

`A=1`

Plug the value of A in the equation 1,

`1+B=2`

`B=2-1`

`B=1`

`:.(x^3-x+3)/(x^2+x-2)=x-1+1/(x-1)+1/(x+2)`

 

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