Question

Find a quadratic polynomial whose zeroes are

$(\sqrt{3}+\sqrt{5})$, $(\sqrt{5}-\sqrt{3})$

Answer 1

The quadratic polynomial with $\alpha$ and $\beta$ as zeroes is

$x^2-(\alpha +\beta )x+\alpha \beta$

Given zeroes of the polynomial are $(\sqrt{3}+\sqrt{5})$ and $(\sqrt{5}-\sqrt{3})$

$\Rightarrow \alpha +\beta =\sqrt{3}+\sqrt{5}+\sqrt{5}-\sqrt{3}=2\sqrt{5}$ ---(1)

$\Rightarrow \alpha \beta =(\sqrt{3}+\sqrt{5})(\sqrt{5}-\sqrt{3})$

$\Rightarrow \alpha \beta =(\sqrt{5}{)}^2-(\sqrt{3}{)}^2$

$\Rightarrow \alpha \beta =5-3$

$\Rightarrow \alpha \beta =2$ ---(2)
Now, the quadratic polynomial is:
$x^2-(\alpha +\beta )x+\alpha \beta$

$=x^2-2\sqrt{5}x+2$