Question

find a quadratic polynomial each with the given numbers as the zeros of the polynomials. √3,3√3​

Answer 1

Given that zeros are

$\sf\alpha = \sqrt{3} \\ \\ \sf \beta = 3 \sqrt{3}$

$\sf SUM \: OF \: ROOTS = s = \alpha + \beta \\ \\ = \sf \sqrt{3} + 3 \sqrt{3} \\ \\ \sf = 4 \sqrt{3} \\ \\ \sf PRODUCTS \: OF \: ROOTS = p = \alpha \beta \\ \\ = \sf \sqrt{3} \times 3 \sqrt{3} = 9$

We Know that quadratic polynomial whose sum is roots is s and product of roots is p is given by

$\sf {x}^{2} - sx + p$

So

Required Polynomial is

$\sf {x}^{2} - 4 \sqrt{3} \:x + 9$

Answer 2

Given that zeros are

$\sqrt{3} \: and \: 3 \sqrt{3}$

So,

$Sum ; s = \sqrt{3} + 3 \sqrt{3} \\ \\ = 4 \sqrt{3} \\ \\ Product;p = \sqrt{3} \times 3 \sqrt{3} = 9$

So Polynomial is

$\sf {x}^{2} - 4 \sqrt{3} \:x + 9$