Prove that $2r^4 +20r^2 = 15r^3 + 15r - 6$ has no rational solutions without solving for $r$.
My first thought was using remainders upon division, but I'm not sure how to apply this with variables.
2026-03-30 05:25:11.1774848311
Prove that there are no rational solutions
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Hint: There are no solutions mod $5$.
Indeed, if $r=x/y$ then $2x^4 \equiv -6 y^4$ or $x^4 \equiv -3 y^4$. Now use Fermat's theorem.