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# Polynomial Rings: Failure of Division Algorithm in $Z[x]$

ABSALG-1R3ROU

Let $\mathbb{Z}[x]$ be the ring of polynomials with coefficients in $\mathbb{Z}$

For which of the following pairs of polynomials $P(x), G(x)\in \mathbb{Z}[x]$ do there exist $Q(x), R(x)\in \mathbb{Z}[x]$ such that:

$$P=QG+R,\quad{\rm with\;either}\; R=0\;{\rm or}\;\deg(R)< \deg(G)$$

Select ALL that apply.

A

$P(x)=x$, $G(x)=3$

B

$P(x)=x^2$, $G(x)=x+3$

C

$P(x)=x+3$, $G(x)=x^2$

D

$P(x)=x^2-2x+1$, $G(x)=2x-2$

E

$P(x)=2x^2-4x+2$, $G(x)=2x-2$

F

$P(x)=2x^2-4x+3$, $G(x)=\,-2x+2$