Question

find the quadratic polynomial where zeros are √3+√5 and √5-√3 ?

Answer 1

Hii ☺ !!


Sum of zeroes = √3 + √5 + √5 - √3 = 2√5.


And,

Product of zeroes = ( √3 + √5 ) ( √5 - √3 ) = (√5)² - (√3)² = 5 - 3 = 2.


Therefore,


Required quadratic polynomial = x²-(sum of zeroes)x + product of zeroes.


=> x² - ( 2√5 ) + 2


=> x² - 2√5 + 2.

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

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