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.
Algebra
Negative vs Minus: Two Words, One Symbol?
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 …
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 Tricks from an Old Textbook
(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 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.
Function Transformations Revisited (II)
(A new question of the week) Last week we examined how a series of transformations affects the equation of a function, in order to write the equation from a graph, or vice versa. We touched on why it works the way it does, but this is something you need to look at from multiple perspectives …
Function Transformations Revisited (I)
(A new question of the week) Transformations of functions, which we covered in January 2019 with a series of posts, is a frequent topic, which can be explained in a number of different ways. A recent discussion brought out some approaches that nicely supplement what we have said before. Here, the focus will be on …
Why Does a² + b² = c² in a Hyperbola?
(A new question of the week) In an ellipse, \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) with focal distance c, parameters a, b, and c all make natural sense, and it is easy enough to see why \(a^2 = b^2 + c^2\). But in the hyperbola, \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\), the equivalent relationship, \(a^2 + b^2 = c^2\), is not nearly as natural, nor …