Suppose $x,y,z$ are three variables satisfying $yz=zy, zx=xz,xy=yzx$.
Could anyone give me two (non-commutative) polynomials $f$ and $g$ in the above three variables such that the following equality holds: $$ f(x,y,z)\cdot(3z^2+zyx+x^2)=g(x,y,z)\cdot(3+yx+x^2), $$ i.e., the product of $f$ and $3z^2+zyx+x^2$ is equal to the product of $g$ and $3+yx+x^2$?
Are there any computer software to solve problems of this type? Especially, 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 $B_i(y,z), n$. Especially, what about the case $B_0=3, B_1=y^2$?
Thanks in advance!
$(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) $
I got this by computation, but I suspect there should be some explanation for this equality, any idea on this?