Question

Form the equation of lowest dimensions with rational coefficients, one of whose roots is
$\sqrt{5+2\sqrt{6}}$.

Answer 1

Given root of the equation is $\sqrt{5+2\sqrt{6}}$

For equation with rational coefficient, irrational roots occurs in conjugate pairs.

$\therefore$ other roots of the equation to complete the conjugate pairs are: $\sqrt{5-2\sqrt{6}},-\sqrt{5+2\sqrt{6}},-\sqrt{5-2\sqrt{6}}$

Since the least number of roots is $4$, the required equation is of degree $4$

We can now find the coefficients of the required equation by calculating the various sums and products of the roots.

$S_1=0,S_2=-10,S_3=0,S_4=1$

$\therefore$ the required equation is $x^4-10x^2+1=0$