Question

find the quadratic polynomial whose zeros are (5+2root3) and (5-2root3)

Answer 1

(x+5+2√3)(x+5-2√3)
x²-7
x=√7.....product of zeroes
(x+5+2√3)+(x+5-2√3)
2x+10-8√3
x²+2x+10-8√3-√7

Answer 2

$\alpha = 5 + 2 \sqrt{3} \\ \beta = 5 - 2 \sqrt{3}$
now use sum of zeros and product of zeros with coefficient of polynomial
$\alpha + \beta = \frac{ - b}{a} = 5 + 2 \sqrt{3} + 5 - 2 \sqrt{3} \\ \alpha + \beta = - \frac{b}{a} = 10 \\ \alpha \beta = \frac{c}{a} = (5 + 2 \sqrt{3} )(5 - 2 \sqrt{3} ) \\ \alpha \beta = 25 - 12 = 13 \\ {x}^{2} - ( \frac{ - b}{a} )x + \frac{c}{a} \\ {x}^{2} - 10x + 13$
is the answer