Let $f$, $g$ $\in \mathbb{C}[X]$ be two polynomials, $f=x^m+px^n+q$, $g=x^m+qx^n+p$ with $p\ne q$. When do we have a common root?
Let $a\in\mathbb{C}$ be the common root. So $f(a)=g(a)=0$. $$f(a)-g(a)=0 \implies (a^n-1)(p-q)=0$$ For $a^n=1$ we get $f(a)=g(a)=a^m+p+q$, so the condition we were looking for would be $p+q=0$.
I'd like to know if my reasoning was right in this case, as I'm not 100% convinced about it. Thank you!
We need to solve
$$\begin{cases} a^m +pa^n+q=0\\ a^m +qa^n+p=0\end{cases}$$
This gives $a^n=1$ (because $p\neq q$ by assumption) so $a$ has to be an $n^{th}$ root of unity and therefore
$$p+q=-\zeta_k^m$$
Where $\zeta_k=e^{2i\pi\over k}$