B Explicit expression for ideal membership

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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?
 
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What is the definition of ##\left<q\right>##?
 
Office_Shredder said:
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.
 
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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.
 

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