It is shown that certain sequences of Hankel matrices of finite rank obtained from a given sequence of complex numbers and powers of companion matrices are closely related. This relation is established by investigating the algebraic properties of combinations of polynomial multiples of powers of complex numbers. Among many applications, these properties are used to construct polynomials with zeros being a function of the zeros of given polynomials. For example, Hankel matrices of finite rank are used to develop a method for computing the least common multiple of a finite number of polynomials without factoring them, or computing a polynomial whose zeros are the product of the zeros of two polynomials. A method for computing a factor of the characteristic polynomial of a given matrix is also presented and is established by forming certain types of Hankel matrices whose entries are generated from linear combinations of powers of zeros of its characteristic polynomial. Applications of these ideas to signal processing and computational linear algebra are also given.