Question

If $x=\frac{1}{2+\sqrt{3}}$, find the value of $x^3-x^2-11x+3$

Answer 1

On rationalising,

$x=\frac{1}{2+\sqrt{3}}\times \frac{2-\sqrt{3}}{2-\sqrt{3}}=\frac{2-\sqrt{3}}{2^2-{\sqrt{3}}^2}$
$x=\frac{2-\sqrt{3}}{4-3}=2-\sqrt{3}$
$x-2=-\sqrt{3}$

On squaring both sides we get,
${(x-2)}^2={(-\sqrt{3})}^2\Rightarrow x^2+4-4x=3$
$\Rightarrow x^2-4x+1=0$
Now,

$x^3-x^2-11x+3$
$=x^3-4x^2+x+3x^2-12x+3$
$=x(x^2-4x+1)+3(x^2-4x+1)=x\times 0+3\left(0\right)$
$=0+0$

$=0$