Question

Find all zeroes of the polynomial `(2x^4 - 9x^3 + 5x^2 + 3x - 1)` if two of its zeroes are `(2 + sqrt3)`  and `(2 - sqrt3)`

Answer 1

It is given that `2 + sqrt3` and `2 - sqrt3` are two zeroes of the polynomial f(x) = `24^4 - 9x^3 + 5x^2 + 3x - 1`

`:. {x - (2 +  sqrt3)} {x - (2-sqrt3)} = (x - 2 - sqrt3) (x - 2 + sqrt3)`

` = (x - 2)^2 - (sqrt3)^2`

`= x^2 - 4x + 4 - 3`

`= x^2 - 4x + 1`

is a factor of f(x)

Now divide f(x) by `x^2 - 4x + 1`

`:, f(x) = (x^2 - 4x + 1)(2x^2 - x - 1)`

Hence, other two zeroes of f(x) are the zeroes of the polynomial `2x^2 - x - 1`

`2x^2 - x - 1 = 2x^2 - 2x + x - 1 = 2x(x - 1)+ 1(x - 1) = (2x + 1) (x - 1)``

Hence the other two zeroes are `-1/2` and 1