Question

Write the polynomial whose zeroes are $-2-\sqrt{3}\text{and}2-\sqrt{3}$.

Answer 1

Form a polynomial whose zeroes are given by $\alpha =-2-\sqrt{3}\text{and}\beta =2-\sqrt{3}$:

$\begin{array}{rcl}& \Rightarrow & \left(x-\alpha \right)\left(x-\beta \right)=\left(x+2+\sqrt{3}\right)\left(x-2+\sqrt{3}\right)\\ & =& \left(\left(x+\sqrt{3}\right)+2\right)\left(\left(x+\sqrt{3}\right)-2\right)\\ & =& {\left(x+\sqrt{3}\right)}^2-\left(4\right)\left[\because a^2-b^2=\left(a+b\right)\left(a-b\right)\right]\\ & =& x^2+3+2\sqrt{3x}-4\\ & =& x^2+2\sqrt{3x}-1\end{array}$

Hence, the required polynomial is $x^2+2\sqrt{3}x-1$.