If `x=1/(2+sqrt3)`,find the value of `x^(3)-x^(2)-11x+4`

Explanation

as `x=1/(2+sqrt3xx(2-sqrt3)/(2-sqrt3)=(2-sqrt3)/((2)^(2)-(sqrt3)^(2))`
`x=(2-sqrt3)/(4-3)=2-sqrt3`
`x-2=-sqrt3` squaring both sides , we get
`(x-2)^(2)=(-sqrt3)^(2)rArrx^(2)+4-4x=3`
`rArr x^(2)-4x+1=0`
Now `x^(3)-x^(2)-11x+41`
`=x^(3)=4x^(2)+x+3x^(2)-12x+4`
`=x(x^(2)-4x+1)+3(x^(2)-4x+1)+1=xxx0+3(0)+1=0+0=0=1`