# How to find the Cosets of a Polynomial group

I'm a bit confused if I'm going about this problem correctly, trying to find the cosets of G in $$Z_{3}[x]$$ where G := {$$(x^{3}+ x^{2} + 2x + 1)(g(x))$$: with $$g(x)$$ in $$Z_{3}[x]$$}

I've been given that $$1 + G$$ is a coset since the resulting element is still within G. What I don't understand is how to find the rest of the cosets.

For example would $$x^{3}+ x^{2} + G$$ be another coset. If i'm understanding this correctly then I believe I would have to write out 14 cosets which seems like a lot.

$$G$$ is the principal ideal generated by $$\phi(x)=x^3+x^2+2x+1$$ over $$\Bbb Z_3$$. Each coset $$u(x)+G$$ has the form $$ax^2+bx+c+G$$ where $$a$$, $$b$$, $$c\in\Bbb Z_3$$. So there are $$3^3=27$$ cosets; I don't feel the need to write out all of them, but perhaps you need to.
To see this note that by the division algorithm, if $$f$$ is a polynomial, then $$f(x)=q(x)\phi(x)+r(x)$$ where $$r(x)$$ has degree $$<3$$. Then $$f(x)+G=r(x)+G$$.