Question

Obtain all the zeroes of the polynomial ${2x}^3-4x-x^2+2,$ if two of its zeros are $\sqrt{2}$ and $-\sqrt{2}$

Answer 1

Given

$2x^3-x^2-4x+2$

Let $\alpha ,\beta$ and $\gamma$ are the zeros of the polynomial and $\sqrt{2},-\sqrt{2}$ are the zeros of the polynomial

$\Rightarrow \alpha +\beta +\gamma =\frac{-b}{a}=\frac{-(-1)}{2}=\frac{1}{2}$

$\Rightarrow \sqrt{2}+\left(\sqrt{-2}\right)+\gamma =\frac{1}{2}$

$\Rightarrow \gamma =\frac{1}{2}$

$\therefore$ the other zeros of the given polynomial is $\frac{1}{2}$