Solving $3x^{4}-7x^{3}+2x^{2}=950$ over the rationals

Question

I am asked to find only rational solutions.

Factoring by $x^{2}$, I get: $$x^{2}(3x^{2}-7x+2)=950$$ By applying the quadratic formula, I have: $$x^{2}(3x-1)(x-2)=950$$ I don't know how to proceed from there.

Thank you for your help.

Answer 1

Your factorization is wrong (see my comment)

Now write $x=p/q$ with $(p,q)=1$. Assuming $x$ is a root, we have

$$3p^4-7p^3q-2p^2q^2=950q^4$$

We deduce $p$ and $q$ being co prime that $p|950$ and $q|3$. It leads to a few cases to test and one can see that $5$ ($p=5$ and $q=1$) is the only one that works.

Answer 2

Write it as:

$$ 3x^4 - 7x^3 - 2x^2 - 950 = 0$$

And then apply the rational root theorem, since you're only looking for rational solutions.

Also this is fourth degree polynomial and there are few algorithms to find all four roots, but they are kind of long and messy.