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Abstract Algebra

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Binary Operations from Logic, Integer Coefficients

ABSALG-I3NOCX

Let $\mathbb{Z}[x]$ be the set of all polynomials in $x$ with integer coefficients.

Let $p,q\in \mathbb{Z}[x]$.

Identify which of the following binary operations are not closed on $\mathbb{Z}[x]$.

Let $deg(p)$ denote the degree $p$ and use the convention that degree of the zero polynomial is $-1$.

A

$(p,q)\mapsto 2p-5q$

B

$(p,q)\mapsto \cfrac{p}{q+2}$

C

$(p,q)\mapsto p\circ(q(0))$

D

$(p,q)\mapsto \textrm{deg(p)}$

E

All of the above binary operations are closed on $\mathbb{Z}[x]$