Question

(x+1/x-1 - x-1/x+1 - 4x/x^2 +1) + 4x/x^4-1​

Answer 1

`(12x)/(x^4-1)`

Step-by-step explanation:

Given

`((x+1)/(x-1) - (x-1)/(x+1) - (4x)/(x^2 + 1)) + (4x)/(x^4 - 1)`

Required

Solve:

Take L.C.M of the first two fractions

`(((x+1)(x+1)-(x-1)(x-1))/((x-1)(x+1)) - (4x)/(x^2 + 1)) + (4x)/(x^4 - 1)`

`(((x+1)(x+1)-(x-1)(x-1))/(x^2-1) - (4x)/(x^2 + 1)) + (4x)/(x^4 - 1)`

`((x^2+2x+1-(x^2-2x+1))/(x^2-1) - (4x)/(x^2 + 1)) + (4x)/(x^4 - 1)`

`((x^2+2x+1-x^2+2x-1)/(x^2-1) - (4x)/(x^2 + 1)) + (4x)/(x^4 - 1)`

Collect Like Terms

`((x^2-x^2+2x+2x+1-1)/(x^2-1) - (4x)/(x^2 + 1)) + (4x)/(x^4 - 1)`

`((4x)/(x^2-1) - (4x)/(x^2 + 1)) + (4x)/(x^4 - 1)`

Solve the expression in bracket

`((4x(x^2+1) - 4x(x^2-1))/((x^2-1)(x^2+1)) ) + (4x)/(x^4 - 1)`

`((4x(x^2+1) - 4x(x^2-1))/(x^4-1) ) + (4x)/(x^4 - 1)`

`((4x^3+4x - 4x^3+4x))/(x^4-1) ) + (4x)/(x^4 - 1)`

`((4x^3- 4x^3+4x +4x))/(x^4-1) ) + (4x)/(x^4 - 1)`

`(8x)/(x^4-1) + (4x)/(x^4 - 1)`

Take LCM

`(8x+4x)/(x^4-1)`

`(12x)/(x^4-1)`

The expression can not be further simplified