Question

Find all zeroes of the polynomial $2x^4-9x^3+5x^2+3x-1$ if two of its zeroes are $2+\sqrt{3}$ and $2-\sqrt{3}$.

Answer 1

Given polynomial

$p\left(x\right)=2x^4-9x^3+5x^2+3x-1$

Zeroes of the polynomial $p\left(x\right)$ re $2+\sqrt{3}$ and $2-\sqrt{3}$

Let $\alpha ,\beta$ be the other two zeroes.

We know that,

Sum of the roots $=2+\sqrt{3}+2-\sqrt{3}+\alpha +\beta =\frac{9}{2}$

$\Rightarrow 4+\alpha +\beta =\frac{9}{2}$

$\Rightarrow \alpha +\beta =\frac{1}{2}$ .................(1)

Product of the roots $=(2+\sqrt{3})(2-\sqrt{3})\times \alpha \times \beta =\frac{-1}{2}$

$\Rightarrow (4-3)\times \alpha \times \beta =\frac{-1}{2}$

$\Rightarrow \alpha \times \beta =\frac{-1}{2}$ ..................(2)

On solving equation (1) and (2) we get

$\alpha =1$ and $\beta =\frac{-1}{2}$