Algebra

Degenerate Conics I: Mystery of the Missing Case

Degenerate cases are instances of a concept that are just on the edge of fitting its definition. They occur when we stretch a definition to its limits, at which point some of the original properties remain, but others break. We’ll start here with common instances of the phenomenon, in conic sections, pursuing the elusive case …

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A Composite Quartic Equation

A question just after we recently discussed quartic equations, has special features that lead to a unique solution method. We’ll be showing how to use synthetic division, and seeing some interesting graphs.

How Real Are Negative Numbers?

This week we’ll look at some Ask Dr. Math questions like, “How can a number be less than zero?” and “Why do we need negative numbers?” We’ll see a number of examples of their use, and how negative numbers make life easier.

Factoring a Quartic Polynomial

Factoring a quadratic polynomial (degree 2) is a standard topic in algebra; but for higher degrees, things get a lot harder. Here we’ll look at some old questions from the Ask Dr. Math site about factoring quartic (degree 4) polynomials. There is no standard method, but several interesting tricks you might want to know about.

A Hole in a Cube

(A new question of the week) Here is an interesting little question about how drilling a hole affects volume and surface area. We’ll have one answer, and several explanations.

Two Worlds of Relations

Terminology and definitions in mathematics sometimes vary according to context. Here we’ll look at the ideas of relations, functions, and their domains, and discover that they look different from different perspectives.