How do I find the congruence of a polynomial modulo another polynomial?

Question

Self studying abstract algebra and could use some help. I want to find the polynomial of degree $$\le 3$$ that is congruent to $$7x^{13}-11x^9+5x^5-2x^3+3$$ modulo $$(x^4-16)$$ and to obtain as much info as possible about the quotient ring $$Z[x]/(x^4-16).$$ How do I proceed? Dividing polynomials was my first, "brute force" idea? Is there a more efficient way to do it?

Answer 1

$$x^4 \equiv 16 \pmod{x^4-16}$$

\begin{align} 7x^{13}-11x^9+5x^5-2x^3+3 &\equiv 7(x^4)^3x - 11(x^4)^2x+5(x^4)x-2x^3+3 \pmod{x^4-16}\\ &\equiv 7(16)^3 x-11(16)^2x+5(16)x-2x^3+3 \pmod{x^4-16} \end{align}

Answer 2

Even the "brute force" Euclidean division is not all that bad in this case because the powers $4$ and greater are conveniently spaced $4$ apart:

$$ 7x^{13}\color{red}{-16 \cdot 7 x^9} \\ +(\color{red}{16 \cdot 7}-11)x^9\color{blue}{- 16 \cdot (16 \cdot 7-11)}x^5 \\ +\big(\color{blue}{16 \cdot (16 \cdot 7-11)}+5\big)x^5 - \color{green}{16 \cdot \big(16 \cdot (16 \cdot 7-11)+5\big)}x \\ \boxed{\,-2x^3 + \color{green}{16 \cdot \big(16 \cdot (16 \cdot 7-11)+5\big)}x+3\;} $$