I'm using Michael Artin's book "Algebra". I understand what $\mathbb Z[x]$ is. I just don't understand how to think of $\mathbb Z[x]/(x^2+7)$ or things like $\mathbb Z[x]/(x^2-3, 2x-4)$. I understand $(x^2+7)$ represents an ideal.
I also don't understand how to think of $(x^2-3, 2x-4)$ (considered in $\mathbb Z[x]$), another student said that it's the set of all polynomials that can be divded by both of those polynomials, if this is the case: what does $(x^2-3, 2x-4)$ look like written out formally?
We can explain $\Bbb Z[x]/(x^2-3,2x-4) \cong \Bbb Z_2 \times \Bbb Z_2$:
we have $2\in (x^2-3,2x-4); 2=2(x^2-3)-(x+2)(2x-4)$, so $(x^2-3,2x-4,2)=(x^2-3,2)$ and $\Bbb Z[x]/(2,x^2-3)\cong \Bbb Z_2[x]/(x^2-1) \cong \Bbb Z_2[x]/(x+1)(x-1)\cong \Bbb Z_2 \times \Bbb Z_2 $.
where $\Bbb Z_2[x]/(x^2-1)=\{a_0+a_1x+a_2x^2+a_3x^3+...+a_nx^n+(x^2-1); a_0,...,a_n\in \Bbb Z_2 \}=\{b_0+b_1x+I; b_0,b_1\in \Bbb Z_2 \}; x^2=1, x^3=x,...,I=(x^2-1)=\{0+I,1+I,x+I,(1+x)+I \}$