Explicit expression for ideal membership

[aheight]

TL;DR Summary

Construct an explicit expression for all elements in an ideal of Q[x]?

Derive an explicit expression for all $f\in\langle q\rangle\subseteq \mathbb{Q}[x]$. I think it's doable and was wondering if there is a published formula?

 

[Office_Shredder]

What is the definition of $\left<q\right>$?

 

[aheight]

What is the definition of $\left<q\right>$?

$\langle q \rangle$ is the Ideal generated by the polynomial $q(x)\in \mathbb{Q}[x]$.

$$
\langle q\rangle=\{q(x)h(x): h(x)\in \mathbb{Q}[x]\}.
$$

For example $x^6-1\in \langle x-1\rangle$. And in this case, it's easy to derive an explicit expression for all $a_0+a_1x+\cdots+a_n x^n\in\langle x-1\rangle$ right? It's $\{a_0+a_1 x+\cdots+a_nx^2\in \mathbb{Q}[x]:\sum a_i=0\}$. So I was wondering if there is a known formula for the general case:
$$
a_0+a_1x+\cdots+a_n x^n\in\langle b_0+b_1x+\cdots+ b_n x^k\rangle
$$
say for $k<n$ or maybe any $k,n$. Not sure though.

 

[Office_Shredder]

That formula about the sum of coefficients is a slightly neat trick but unnecessary.

The polynomials in $<b_kx^k+...+b_0>$ are the ones of the form $b_k a_n x^{k+n}+ (b_{k-1} a_n + b_k a_{n-1}) x^{k+n-1}+...$

If you're wondering given a polynomial how you can check if it's in the right form, that's also easy. Polynomial division is a straightforward algorithm that you can perform to see if your polynomial is dividing.