Question

Find all the zeros of the polynomial 2x³ + x² − 6x − 3, if two of its zeros are `-sqrt3` and `sqrt3`

Answer 1

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`

= x² - 3

x² - 3 is a factor of f(x). Now , we divide f(x) = 2x³ + x² − 6x − 3 by g(x) = x² - 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)

2x³ + x² − 6x − 3 = (x² - 3)(2x + 1) + 0

2x³ + x² − 6x − 3 `= (x+sqrt3)(x-sqrt3)(2x+1)`

Hence, the zeros of the given polynomial are `-sqrt3`, `+sqrt3`, `(-1)/2`.