<s>[INST] A polynomial is divisible with another one For what $m$ and $n$ is the polynomial $2X^{19}+X^{13}+mX^{11}+X^8+2X^6+nX^2+2$ divisible by $X^4+X^3+X^2+X+1$.
I tried to find the real solutions for g but couldn't
 [/INST] Note that $x^5-1=(x-1)(x^4+x^3+x^2+x+1)$
\begin{align}
&\;2x^{19}+x^{13}+mx^{11}+x^8+2x^6+nx^2+2\\
=&\;2x^4(x^5-1+1)^3+(x^3+mx)(x^5-1+1)^2+(x^3+2x)(x^5-1+1)+nx^2+2\\
=&\;2x^4[(x^5-1)^3+3(x^5-1)^2+3(x^5-1)+1]\\
&\quad+(x^3+mx)[(x^5-1)^2+(x^5-1)+1]+(x^3+2x)[(x^5-1)+1]+nx^2+2\\
=&\;P(x)(x^5-1)+2x^4+x^3+mx+x^3+2x+nx^2+2
\end{align}
for some polynomial $P(x)$.
$2x^{19}+x^{13}+mx^{11}+x^8+2x^6+nx^2+2$ is divisible by $x^4+x^3+x^2+x+1$ if and only if $2x^4+x^3+mx+x^3+2x+nx^2+2$ is divisible by $x^4+x^3+x^2+x+1$.
\begin{align}
&\;2x^4+x^3+mx+x^3+2x+nx^2+2\\
=&\;2x^4+2x^3+nx^2+(m+2)x+2
\end{align}
It is obvious that $m=0$ and $n=2$.
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