Definition of polynomials, polynomial function and commutativity

Question

When multiplying polynomials in a ring using the definition we get $(a_0,a_1,...)(b_0,b_1,...)=(c_0,c_1,..)$ where $\displaystyle c_n=\sum_{m=0}^{n}a_mb_{n-m}$ and $x$ is $x=(0,1,0,...)$. Now consider the ring of quaternions and the polynmial $$ f(x)=x^2-(i+j)x+k $$ This polynomial can be factored a such $$(x-i)(x-j)=(-i,1)(-j,1)=((-i)(-j),-i-j,1)=(k,-(i+j),1)=x^2-(i+j)x+k $$ Using the definition.
Now if we try to expand $(x-i)(x-j)$ considering $x$ a quartenion we get $$(x-i)(x-j)=x^2-xj-ix+ij $$ This way $xj\neq jx $.
Also if we put in the value $x=i$ we get $$ f(i)=i^2-(i+j)i+k=2k $$ But if we use the factorazation for $x=i$ we get $$ (i-i)(i-j)=0 $$ I think this happens because the definition of polynomials "implies" commutativity between $x$ and the coefficients under multiplication and when using the polynomial function this isn't always the case because $x$ becomes an element of the ring. If somebody could clear things out a bit for me i would be thankfull.