Let$\;\,r=a+\sqrt{b}+\sqrt{c},\;\,$where
- $a,b,c\;$are rational numbers, with $b,c > 0$.
- Each of$\;\,b\,,c,\,bc\;\;$is not the square of a rational number.
- $r > 0$.
Let $f(x)$ be the minimal monic polynomial with rational coefficients for which $r$ is a root.
Based on data from numerical examples, the following claim appears to be true:
- If $f$ has more than one positive root, then $r$ is the largest of them.
Is it true?
Square both sides twice.
$(x-a)^2=b+c+2\sqrt{bc} $
$\big((x-a)^2-b-c\big)^2=4bc $
Then $x=a+\sqrt b-\sqrt c$, $x=a-\sqrt b+\sqrt c$, $x=a-\sqrt b-\sqrt c$ are solutions, $r$ is maximum.