Question

Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively
$0,\sqrt{5}$

Answer 1

If $\alpha$ and $\beta$ are the zeroes of a quadratic polynmial, then the polynomial can be written as follows-

$p\left(x\right)=x^2-(\alpha +\beta )x+\left(\alpha \beta \right)$

Whereas

$\alpha +\beta =$ Sum of roots

$\alpha \beta =$ Product of roots

Now according to the question,

Sum of roots $=\alpha +\beta =0$

Product of roots $=\alpha \beta =\sqrt{5}$

Therefore the quadratic polynomial will be-

$p\left(x\right)=x^2-\left(0\right)x+\left(\sqrt{5}\right)$

$\Rightarrow p\left(x\right)=x^2+\sqrt{5}$

Hence the quadratic polynomial will be $x^2+\sqrt{5}$.