`(5 - x)/(2x^2 + x - 1)` Write the partial fraction decomposition of the rational expression.

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

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

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

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

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

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

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

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

`:.(5-x)=x(A+2B)+A-B`

`A+2B=-1`

`A-B=5`

solving the above two linear equations to get A and B ,

subtracting the second equation from the first equation,

`3B=-1-5`

`3B=-6`

`B=-6/3`

B=-2

Plug the value of A in second equation,

`A-(-2)=5`

`A+2=5`

`A=5-2`

A=3

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

 

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