Question

One root of the equation $x^4-5x^3+a{x}^2+bx+c=0$ where $a,b,c$ are rational number is $(3+\sqrt{2})$ then, if the roots are equal, find the largest value of $C$

Answer 1

Roots are equal

$\Rightarrow$ all roots are $3+\sqrt{2}$

As product of roots $=\frac{C}{I}=(3+\sqrt{2}{)}^4$ $\{4^{th}$ degree equation $\Rightarrow 4$ zeroes$\}$

$\Rightarrow C=(9+2+6\sqrt{2}{)}^2=(11+6\sqrt{2}{)}^2=121+72+132\sqrt{2}$

$\Rightarrow C=193+132\sqrt{2}$

$\Rightarrow$ C is not a rational number.