Question

Write a quadratic polynomial whose zeroes are $2+\sqrt{3}$ and $2-\sqrt{3}$

Answer 1

Zeroes of a polynomial represents its roots.

Let $\alpha =2+\sqrt{3},\beta =2-\sqrt{3}$

Sum of the roots: $\alpha +\beta =2+\sqrt{3}+2-\sqrt{3}=4$

Product of the roots: $\alpha \beta =(2+\sqrt{3})(2-\sqrt{3})=2^2-(\sqrt{3}{)}^2=4-3=1$

We know a polynomial can be written as $x^2-\left(\text{sum of the roots}\right)x+\text{product of the roots}=x^2-4x+1$

Hence the quadratic polynomial is $x^2-4x+1$