Question

The equation of fourth degree with rational coefficients one of whose roots is $\sqrt{3}+\sqrt{2}$, is:

• A. $x^4-10x^2-1=0$
• B. $x^4+10x^2-1=0$
• C. $x^4+10x^2+1=0$
• D. $x^4-10x^2+1=0$

Answer 1

The correct option is D $x^4-10x^2+1=0$

$\because \sqrt{3}+\sqrt{2}$ is a root, its conjugate $-\sqrt{3}-\sqrt{2}$ is also a root
Now we consider the quadratic factor $(x^2-5-2\sqrt{6})$ of the polynomial
The other quadratic factor would be $(x^2-5+2\sqrt{6})$
$\therefore$ The polynomial is $\Rightarrow x^4-10x^2+1$