What could be more fundamental than the Fundamental Theorem of Algebra? A lot of things, actually. This theorem isn't actually fundamental to algebra at all—why else haven't we heard of it yet? It's very important for polynomials though.
The total number of roots in a polynomial—the number of real roots plus the number of complex roots—will add up to the degree of the polynomial.
Have a degree of 5? Then you have 5 roots. It's as simple as that. The only complication is that complex roots always come in conjugate pairs. That's the ± we always see hanging around complex numbers.