Discuss the sign scheme of the expesion `((x^(2)-2011x+2010)(x^(2)+2011x-2012))/((x^(2)-4x+4)(x^(2)-8x+15)(x^(2)-5x +6)^(2))`

Explanation

We have,
`((x^(2)-2011x +2010)(X^(2) +2011x -2012))/((x-2)^(2) (x-3)(x-5) (x-2)^(2) (x-3)^(2))`
`= ((x-2010)(x-1)(x+2012)(x-1))/((x-2)^(4)(x-3)^(3) (x-5))`
` = ((x -2010)(x+2012)(x-1)^(2))/((x-2)^(4) (x-3)^(3)(x-5))`
We observe that the critical points for the given rational funcations are 1,2,3,5,2010, -2012

The given expression is position when ` x in ( -oo, -2012) cup ( 3,5) cup( 2010 , oo)` and negative when ` x in (- 2012 , 1) cup ( 1,2) cup ( 1,2) cup ( 2,3) cup ( 5,2010) `