The characteristic polynomial of real matrices possesses real coefficients. This chapter aims to summarize general results on the location and determination of the zeros of polynomials with mainly real coefficients. The operations here are assumed to be performed over the set ℂ of complex numbers. Restricting operations to other subfields of ℂ, such as the set ℤ of integers or finite fields, is omitted because, in that case, we need to enter an entirely different and more complex area, which requires Galois theory, advanced group theory and number theory. A general outline for the latter is found in Govers et al. (2008) and a nice introduction to Galois theory is written by Stewart (2004).

The study of polynomials belongs to one of the oldest researches in mathematics. The insolubility of the quintic, famously proved by Abel and extended by Galois (see art. 196 and Govers et al. (2008) for more details and for the historical context), shifted the root finding problem in polynomials from pure to numerical analysis. Numerical methods as well as matrix method based on the companion matrix (art. 143) are extensively treated by McNamee (2007), but omitted here. A complex function theoretic approach, covering more recent results such as self-inversive polynomials and extensions of Grace's Theorem (art. 227), is presented by Sheil-Small (2002) and by Milovanović et al. (1994), who also list many polynomial inequalities.