(A new question of the week) A recent question asked us to find errors in solving recurrence relations by the method of undetermined coefficients. We’ll see several things that can go wrong, and correct some misunderstandings.
(A new question of the week) Here is an interesting collection of problems involving logarithms with different bases, which require some unique thinking. And after we’d worked out a good strategy, another problem arose at a whole new level.
Last time we looked at what a degenerate conic section is, and how it relates on one hand to actual cones, and on the other to the general equation of the conic. Here we’ll look at the parameters of conic sections (focus, directrix, axes, and especially eccentricity) and how they apply to degenerate cases. Does …
Degenerate Conics II: Are Their Parameters Meaningful? Read More »
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 …
Degenerate Conics I: Mystery of the Missing Case Read More »
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.
Last time we looked into terminology related to negative numbers; one subtopic was too big to fit, so I’ve broken it out into a separate post. How are the concepts of “negative” and “minus” (subtraction) related? How much do we need to distinguish them? We’ll look at two questions, the first from a child focused …
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.
(A new question of the week) Last week I discussed several Ask Dr. Math questions about factoring quartic polynomials, which had been on my list of potential topics. That list also included a question on that topic from three years ago, that didn’t make it into the blog at the time. That will lead us …
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 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.