It is well known that complex harmonic polynomials of degree n may have more than n zeros. In this paper, we examine a one-parameter family of harmonic trinomials and determine how the number of zeros depends on the parameter. Our proof heavily utilizes the Argument Principle for Harmonic Functions and involves finding the winding numbers about the origin for a family of hypocycloids.

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Making meaningful connections between mathematics and other scientific disciplines has become a priority in education. In recent years, numerous national reports have called for more cross-disciplinary material in entry-level mathematics courses. (Cohen 1995 is one such example.) In this article, we consider an application of quadratic equations to a standard problem in chemistry. We show how quadratic equations arise naturally in solving the problem and then go on to consider a standard approximation using a simpler quadratic. This focus brings to light several important mathematical ideas. They include the sensitivity of the solutions on the equation coefficients and the use of inequalities to obtain upper bounds on the error