parmenides51 wrote: Prove that the equation $x^{19} + x^{17} = x^{16 }+ x^7 + a$ for any $a \in R$ has at least two imaginary roots Let $f(x)=LHS-RHS$. $17^{th}$ derivative has no real root and so $f(x)$ has at most $17$ real roots and so at least two non-real roots.