Question

find the polynomial whose zeros are 2+square root of 3 & 2-square root of 3​

Answer 1

Answer:

$let \: x = 2 + \sqrt{3} \\ y = 2 - \sqrt{3} \\ x + y = 2 + \sqrt{3} + 2 - \sqrt{3} = 4 \\ xy = {2}^{2} - { \sqrt{3} }^{2} = 4 - 3 = 1 \\ we \: know \: that \: the \: equation \: will \: be \\ {a}^{2} - (x + y)a+ xy \\ {a}^{2} - 4 a+ 1$

Hope its help uh❤

Answer 2

★ Question ★

→

find the polynomial whose zeros are 2+square root of 3 & 2-square root of 3.

$\rule{300}2$

Let the quadratic polynomial be = $ax^{2}+bc+c$

$\implies a≠0$

(and it's zeros are $\alpha,\beta$)

Then,

$\implies \alpha=2+\sqrt3 \\ \implies \beta=2-\sqrt3$

sum of the zeroes = $\alpha+\beta$

$\implies 2+\sqrt3+2-\sqrt3 \\ \implies =4$

$\implies \alpha×\beta$(product of zeroes)

$\implies (2+\sqrt3)(2-\sqrt3) \\\implies 2^{2}-\sqrt3^{2}\\ \implies 4-3\\ \implies = 1$

Therefore,

the required quadratic polynomial↓

$\implies ax^{2}+bc+c \\\implies [x^{2}-(\alpha+\beta)+\alpha ×\beta] \\ \implies [x^{2}-4x+1]$

$\rule{300}2$

hence ,

the required quadratic polynomial is

$\implies {\fbox{\fbox{x^{2}-4x+1}}}$