<s>[INST] Modulo polynomial in ring theory Let $x^4-16$ be an element of the polynomial ring $E= \mathbb{Z}[x]$ and use the bar notation to denote passage to the quotient ring $\mathbb{Z}[x]/(x^4-16)$. Find a polynomial of degree $\leq 3$ that is congruent to $7x^{13} -11x^9 + 5x^5-2x^3+3$ modulo $x^4-16$. Can anyone help understand how to mod out polynomials. 
 [/INST] ${\rm mod}\,\ x^4-2^4\!:\,\ x^4\equiv 2^4\,\Rightarrow\, \color{#c00}{x^{4n}}\equiv 2^{4n}$
$\! \begin{align}{\rm therefore}\quad &\ \ 7\,\color{#c00}{x^{12}}x -11\,\color{#c00}{x^8}\,x + 5\,\color{#c00}{x^4}\,x-2x^3+3\\
 \equiv &\  \ 7\, 2^{12}\,x - 11\, 2^8\, x + 5\, 2^4\, x -2\,x^3+3\quad {\bf QED}\end{align}$
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