Module 2 - Introduction
Implicit in SMR 1.2a, is the need to articulate (1) a proof of why the graph of a linear inequality is a half plane and (2) a proof of the fact that the graph of a linear function is a straight line. The latter proof requires the use of basic properties of similar triangles.
The proof of the result that the roots of real polynomials come in complex conjugate pairs (SMR 1.2b) allows one to see how to make use of the Fundamental Theorem of Algebra in a nontrivial way. In the process, one gains a better understanding of both the Fundamental Theorem of Algebra and the Quadratic Formula.
The rational root theorem for polynomials with integer coefficients (SMR 1.2b) is one that students and textbooks often mistake as a recipe for locating all the roots of such a polynomial. By reviewing the proof carefully, a prospective teacher is likely to understand the full meaning of this theorem.
The Binomial Theorem (SMR 1.2b) occupies a place of honor in algebra and has important connections in other areas of mathematics. Prospective teachers should be able to understand one of its most accessible proofs, and thereby learn a substantive application of mathematical induction. (CTC)
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