No closed form solution means we can't write out the solution cleanly using standard algebraic operations. You can still write iterative algorithms that converge on the solution, such as by solving for the eigenvalues of a matrix.
When the solution is irrational, as is common for polynomials, it might be difficult to guess what it is based on knowing an approximation of it.
When the problem posed is "factor this polynomial", it's unheard of. This is the conceptual difference between "needing a number to work with" and "studying math".