Question

if 2 + root 5 and 2 minus root 5 are the zeros of the polynomial then find the polynomial

Answer 1

Answer:

x^2-4x-3 is the polynomial

Answer 2

Heya!!

So, the given zeroes (or roots) of the polynomial are (2+√5) and (2-√5).

Now, if the two roots are known, then the polynomial can be easily derived by using-

${x}^{2} -( \alpha + \beta ) \times x+ \alpha \beta = 0 \\ where \: \alpha \: and \: \beta \: are \: the \: two \: roots \: of \: the \: given \: equations$
now, in this case, let
$\alpha = 2 + \sqrt{5}$
and
$\beta = 2 - \sqrt{5}$
so, sum of roots will be
$( \alpha + \beta ) = 2 + \sqrt{5} + 2 - \sqrt{5} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = 4$
and the product of the roots will be
(2+√5)×(2-√5) =4-5=-1

Thus, the polynomial becomes
x^2-4x-1=0