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`x^2/(x^4 - 2x^2 - 8)` Write the partial fraction decomposition of the rational expression. Check your result algebraically.

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Let's factorize the denominator,






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







Equating the coefficients of like terms,

`A+B+C=0`     --- equation 1

`-2A+2B+D=1`   ------ equation 2

`2A+2B-4C=0`     ------ equation 3

`-4A+4B-4D=0`   ----- equation 4

Now we have to solve the above four equations to find the solutions of A,B,C and D.

Rewrite equation 1 ,


Substitute above in equation 3,






Rewrite equation 2 as,


Substitute the above in equation 4,








Plug the values of C in equation 3 ,




`A+B=0`    ------ equation 5

Plug the value of D in equation 4,




`-A+B=1/3`        ---- equation 6

Solve equations 5 and 6 to find the solutions of A and B,

Add the equations 5 and 6.



Plug the value of B in equation 5,





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