Question

If $3$ and $1+\sqrt{2}$ are two roots of a cubic equation with rational coefficients, then the equation is

• A. $x^2-5x^2+9x-9=0$
• B. $x^3-3x^2-4x+12=0$
• C. $x^3-5x^2+7x-3=0$
• D. none of these

Answer 1

The correct option is D none of these

Irrational roots occurs in conjugate pair, then $3,1+\sqrt{2},1-\sqrt{2}$ are the roots.
$s_1=3+1+\sqrt{2}+1-\sqrt{2}=5s_2=3(1+\sqrt{2})+3(1-\sqrt{2})+(1+\sqrt{2})(1-\sqrt{2})=6+1-2=5s_3=3(1+\sqrt{2})(1-\sqrt{2})=-3$
Hence, required equation is
$x^3-s_1{x}^2+s_{2}x-s_3=0\Rightarrow x^3-5x^2+5x+3=0$