Suppose $x$, $y$, $z$ are three variables satisfying $yz=zy$, $zx=xz$, $xy=yzx$.


1. Could anyone give me two (non-commutative) polynomials $f$ and $g$ in the above three variables such that the following equality holds:
$$f(x,y,z)\cdot(3z^2+zy^2x+x^2)=g(x,y,z)\cdot(3+y^2x+x^2)
$$


**Especially, I expect $f$, $g$ to be some product of modified polynomials of the given ones, see remark 1.**


1. What about a more general one: solve
$$ f(x,y,z)(B\_0(y,z)z^{2n}+B\_1(y,z)z^nx+x^2)=g(x,y,z)(B\_0(y,z)+B\_1(y,z)x+x^2) $$ for any given (non-commutative) polynomial $B\_i(y,z)$ and some $n$.


Thanks in advance!




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(Added)


I want to emphasize that as observed by eithil below, the nonzero solution $f, g$ always exist, for example, for the question 1, you can simply write $$f(x,y,z)=B\_0(y,z)+B\_1(y,z)x+B\_2(y,z)x^2$$ 
$$g(x,y,z)=A\_0(y,z)+A\_1(y,z)x+A\_2(y,z)x^2$$
and use the equality to solve $A\_i, B\_i, i=0,1,2$, but in general, the expression for these $A\_i, B\_i$'s look very messy, so 
**what I really care is whether we can find $f, g$ to be in a "neat" form in the spirit of the example shown in remark 1.**




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Remarks:


1, I have tried to ask this question in MSE, but no further answer appeared besides the following example that I got by direct computation:


$$(3z^4+yz^4x+x^2)(3+yzx+x^2)(3z^2+yzx+x^2)=(3z^4+yz^3x+x^2)(3z^2+yz^3x+x^2)(3+yx+x^2)$$


Motivated by this example, I guess we can always find the solution $f$, $g$ to be products of some modified form of the given two (non-commutative) polynomials, but I have no idea whether there are some explanation for this (combinatorial) phenomenon.


2, If you like, you can think that we are working within some skew polynomial rings, just thinking $y$, $z$ lies in the base ring, and $x$ to be the monomorphism of the base ring. Or think we are working in the integral group ring of the discrete Heisenberg group.



noncommutative polynomials equality
This is not the answer that I expected, but at least it gives an algorithm to find some $ f, g$ such that $ f(x,y,z) a(x,y,z)= g(x,y,z) b(x,y,z)$ for given $a, b$.


See theorem 8 in the paper by Ore " Theory of non- commutative polynomials", Annals of Mathematics, vol.34, 1933, 480-508. 


