The following important theorem in mathematics, concerned with the roots of polynomial equations:
Every polynomial equation
where the ai are real or complex numbers and an ≠ 0, has a root in the set of complex numbers.
It follows that, if f(z)=anzn+an−1zn−1+…+a1z+a0, there exist complex numbers α1, α2,⋯,αn (not necessarily distinct) such that
Hence the equation f(z)=0 cannot have more than n distinct roots.