Factors of $x^3+ 5x$ in $\mathbb{Z}_6[x]$ As $f(0)=f(1)=f(2)=f(3)=f(4)=f(5)$, where $0,1,2,3,4,5$ are under mod $6$, can we write $x^3+5=(x-1)(x-2)(x-3)(x-4)(x-5)$?
I am extremely sorry for the typo.

Your decomposition is certainly wrong (look at the degree!), but you can decompose the polynomial as follows:
$$x^3+5x = x(x^2+5)=x(x^2-1)=x(x+1)(x-1)$$
in $\mathbb{Z}_6[x]$.
