Friday, July 10, 2015

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

`(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)`

No comments:

Post a Comment

How does author Elie Wiesel use symbolism to contribute to the meaning of Night?

In his book Night , Elie Wiesel uses symbolism throughout to enhance the text. First of all, the title itself is symbolic. The word "ni...