If $ x-\sqrt{3} $ is a factor of the polynomial $ a x^{2}+b x-3 $ and $ a+b=2-\sqrt{3} $. Find the values of $ a $ and $ b $.

Given:

\( x-\sqrt{3} \) is a factor of the polynomial \( a x^{2}+b x-3 \) and \( a+b=2-\sqrt{3} \).

To do:

 We have to find the values of \( a \) and \( b \).

Solution:

$a+b=2-\sqrt{3}$.............(i)

\( x-\sqrt{3} \) is a factor of the polynomial \( p(x) = a x^{2}+b x-3 \)

This implies,

$p(\sqrt3)=a(\sqrt3)^2+b(\sqrt3)-3=0$

$3a+\sqrt3b-3=0$

$3a+\sqrt3b=3$

$\sqrt3(\sqrt3a+b)=(\sqrt3)^2$

$\sqrt3a+b=\sqrt3$.........(ii)

Subtracting (ii) from (i), we get,

$a+b-\sqrt3a-b=2-\sqrt3-\sqrt3$

$a-\sqrt3a=2-2\sqrt3$

$a(1-\sqrt3)=2(1-\sqrt3)$

$a=2$

This implies,

$2+b=2-\sqrt3$

$b=-\sqrt3$