Question

Find a quadratic polynomial with zeros 3+5 and 3-5

Answer 1

alpha =3+5= 8

beta=3-5=-2

alpha+beta =8+(-2)=6

alpha×beta=8×-2=-16

equation become,

x²+(alpha+beta)x+alpha×beta

x²+6x-16

Answer 2

The quadratic polynomial whose zeroes are,

$5 \sqrt{3} ,5 - \sqrt{3}$

$\alpha , \beta \: is \: f(x) = k[ {x}^{2} - ( \alpha + \beta )x + \alpha \times \beta ]$

where k is any non-zero real no.

THE QUADRATIC POLY POLYNOMIAL WHOSE ZEROES ARE

$5 \sqrt{3} ,5 - \sqrt{3}$

$f(x) = k[ {x}^{2} - ( \alpha + \beta )x + \alpha \times \beta ]$

$f(x) = k[ {x}^{2} - ( 5 \cancel{ + \sqrt{3}} + 5 \cancel{ - \sqrt{3}} )x + (5 + \sqrt{3} ) (5 - \sqrt{3} ) ]$

$f(x) = k[ {x}^{2} -10x + ( {5)}^{2} - ({ \sqrt{3} )}^{2} ]$

$f(x) = k[ {x}^{2} -10x + (25 - 3)]$

$f(x) = k[ {x}^{2} -10x + 22]$

so, the QUADRATIC polynomial is

$f(x) = k[ {x}^{2} -10x + 22]$