Obtain all zeros of the polynomial f(x) = x^{4} − 3x^{3} − x^{2} + 9x − 6, if two of its zeros are `-sqrt3` and `sqrt3`

#### Solution

we know that, if x = a is a zero of a polynomial, then x - a is a factor of f(x).

since `-sqrt3` and `sqrt3` are zeros of f(x).

Therefore

`(x+sqrt3)(x-sqrt3)=x^2+sqrt3x-sqrt3x-3`

= x^{2} - 3

x^{2} - 3 is a factor of f(x). Now , we divide f(x) = x^{4} − 3x^{3} − x^{2} + 9x − 6 by g(x) = x^{2} - 3 to find the other zeros of f(x).

By using that division algorithm we have,

f(x) = g(x) x q(x) + r(x)

x^{4} − 3x^{3} − x^{2} + 9x − 6 = (x^{2} - 3)(x^{2} - 3x + 2) + 0

x^{4} − 3x^{3} − x^{2} + 9x − 6 = (x^{2} - 3)(x^{2} - 2x + 1x + 2)

x^{4} − 3x^{3} − x^{2} + 9x − 6 = (x^{2} - 3)[x(x - 2) - 1(x - 2)]

x^{4} − 3x^{3} − x^{2} + 9x − 6 = (x^{2} - 3)[(x - 1)(x - 2)]

x^{4} − 3x^{3} − x^{2} + 9x − 6 `= (x - sqrt3)(x+sqrt3)(x-1)(x-2)`

Hence, the zeros of the given polynomials are `-sqrt3`, `sqrt3`, +1 and +2.