Factor Theorem (Plus the Fundamental Theorem of Algebra)
These two theorems put together essentially say that every polynomial has at least one zero, but more specifically a polynomial has as many zeros as its largest exponent. That is a 5th order polynomial has 5 zeros. Note not all zeros have to be real and they can be repeated in pairs.
An nth order polynomial can be written in the form of:(1)
Where $z_n$ is the nth zero and a is simply a coefficient. The usefulness of this comes from the fact that you can write the equation for any polynomial if you know the zeros and one other point. The additional point is needed to find the value of the leading coefficient (a). Also knowing the zeros of a polynomial allows you to write the polynomial in factored form.