`(x^3 + 2x^2 - x + 1)/(x^2 + 3x - 4)` Write the partial fraction decomposition of the improper rational expression.

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Since the rational expression is an improper expression , we have to express the expression as a sum of simpler fractions with the degree of the polynomial in the numerator less than the degree of the polynomial in the denominator.

Dividing using the long division yields,


Polynomials do not completely divide , so we have to continue with the partial fractions of the remainder expression,

Let's factorize the denominator of the remainder fraction,




Let `(6x-3)/(x^2+3x-4)=A/(x-1)+B/(x+4)`





equating the coefficients of the like terms,

`A+B=6`          ----- equation 1

`4A-B=-3`   ------ equation 2

Now we have to solve the above equations to get the solutions of A and B,

Adding the equation 1 and 2 yields,




Plug the value of A in equation 1 ,







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