Describing the ring $\mathbb{Z}[x]/(x-8,2x-6)$

Question

How can one describe the ring $\mathbb{Z}[x]/(x-8,2x-6)$? What is meant by describing it? How does an element of the ideal generated by $x-8$ and $2x-6$ look like?

Answer 1

From a well known isomorphism theorem we have $$\displaystyle\frac{\mathbb{Z}[x]}{(x-8,2x-6)}\simeq\frac{\mathbb{Z}[x]/(x-8)}{(x-8,2x-6)/(x-8)}.$$ But the evaluation morphism $\mathbb{Z}[x]\to\mathbb Z$, $x\mapsto 8$, gives rise to an isomorphism $\mathbb{Z}[x]/(x-8)\simeq\mathbb Z$. This isomorphism sends (the residue class of) $2x-6$ to $2\cdot8-6=10$, so $$\displaystyle\frac{\mathbb{Z}[x]/(x-8)}{(x-8,2x-6)/(x-8)}\simeq\mathbb Z/10\mathbb Z.$$

Answer 2

It is easy to see that $(x−8,2x−6)=(10,x+2)$ Further $\mathbb{Z}[x]=\mathbb{Z}[x+2]$ So

$$\mathbb{Z}[x]/(x−8,2x−6)=\mathbb{Z}[x+2]/(10,x+2)=\mathbb{Z}_{10}$$