<s>[INST] Solve $2000x^6+100x^5+10x^3+x-2=0$ One of the roots of the equation $2000x^6+100x^5+10x^3+x-2=0$ is of the form $\frac{m+\sqrt{n}}r$, where $m$ is a non-zero integer and $n$ and $r$ are relatively prime integers.Then the value of $m+n+r$ is?

Tried to use the fact that another root will be $\frac{m-\sqrt{n}}r$ as coefficients are rational but there are six roots and using sum and product formulas would allow many variables in the equations.

 [/INST] The usual trick is to divide through by $x^3.$ Then notice that taking $w = 10x - \frac{1}{x}$ seems to allow writing the thing, and we get
$$ 2 w^3 + w^2 + 60 w + 30 = 0.  $$
This has a rational root, namely $-1/2,$ and factors as
$$ (2w+1)(w^2 + 30)  $$
Setting
$$ 10 x - \frac{1}{x} = -\frac{1}{2}  $$
leads to
$$ 20 x^2 + x - 2 = 0 $$
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