A Key Limitation to Factor Theorem

I don’t really remember my Factor Theorem from John Abbott College well. But one detail which I think I do recall, is that its use was meant for “Univariate Polynomials”, with “Invariant Coefficients”. This means, that the coefficients needed to be integers or ‘other numbers’, known in advance, but not symbolic constants. In computerized cases where the coefficients aren’t preset, there are other, narrow constraints on them. A similar problem exists with the way I was taught to invert certain matrices in Linear Algebra. The elements are well-behaved in certain cases, but just as with polynomials, if the coefficients are suddenly random, floating-point numbers, those methods no longer work. Then, we must use a brute-force approach. And in the case of polynomials, there is no sure brute-force approach that works beyond the 4th degree.