We have learned in our previously classes that whenever we have a polynomial function then it has values of $x$, which satisfy . Are there other structures like Matrix, Vectors or something else which satisfy a polynomial? If so then kindly tell.
A very famous theorem is the Cayley-Hamilton theorem about the characteristic polynomial. It states that a square matrix $A$ does fulfil its own characteristic polynomial for the eigenvalues.
Sure there is. Whenever you have a structure with a multiplication operation and an addition operation that interacts relatively nicely ("ring" is the archetypal definition of "nicely", but you can go somewhat further with less strict requirements if you want), you can make polynomials.
For instance, it is a known result that each matrix is a root of its own characteristic polynomial.
An important part of modern number theory concerns solutions to polynomial equations in the integers modulo prime numbers. For instance, the modularity theorem, famous for being the last building block in the final proof of Fermat's last theorem, was first conceived because it was noticed that the number of solutions of certain types of polynomial equations modulo any prime $p$ was tightly related to the degree $p$ coefficient in the expansion of certain (previously unrelated) functions.
In my opinion polynomials are defined on symbols and we can relate values to these symbols. But they are still symbols nonetheless. Look at this for an example where the point is not to solve the polynomial for a ceertain number rather just to look at how the coefficients work together, in this situation $x$ behaves only as a placeholder. So sometimes there are no solutions at all, we only use the properties of the polynomial.