The value of `f(x)=x^4+4x^3+2x^2-4x+7,` when `x=cot(11pi)/8` is _________

Answer

6

Explanation

`x = cot""(11pi)/(8) =cot(pi+ ( 3pi)/(8)) = cot ""(3pi)/(8) = sqrt2 -1 `
`rArr (x+1)^(2) = 2 `
`therefore x^(2) + 2x - 1 =0`
Now, `f(x) = x^(4)+ 4x^(3) + 2x^(2) - 4x +7`
`" " = x^(2)(x^(2) + 2x - 1) + 2x^(3)+ 3x^(2) -4x + 7`
`" " =0 + 2x^(3) + 3x^(2) - 4x +7`
`" " = 2x (x^(2) + 2x - 1) - x^(2) - 2x +7`
`" " =-x^(2) - 2x +7`
` " " =-(x^(2)+ 2x - 1) +6 =6`