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`x/(16x^4 - 1)` Write the partial fraction decomposition of the rational expression. Check your result algebraically.

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Let's factorize the denominator,




Let `x/(16x^4-1)=A/(2x+1)+B/(2x-1)+(Cx+D)/(4x^2+1)`






equating the coefficients of the like terms,

`8A+8B+4C=0`        ----- equation 1

`-4A+4B+4D=0`     ----- equation 2

`2A+2B-C=1`            ----- equation 3

`-A+B-D=0`             ------ equation 4

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

From equation 1,



Subtract equation 3 from the above equation,




From equation 2,



Now subtract equation 4 from the above equation,




Now plug in the values of C in the equation 3,





`A+B=1/4`       ----- equation 5

Plug in the value of D in the equation 4,


`-A+B=0`      ---- equation 6

Now add the equations 5 and 6,



Plug in the value of B in the equation 6,






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