We’ve discussed the Rational Root Theorem in the past, but not a theorem that is often taught along with it, namely Descartes’ Rule of Signs, which predicts the numbers of positive and negative zeros (roots) of a polynomial. Both are ascribed to Rene Descartes; both are often taught without proof. Here we’ll introduce the theorem, …
A recent question reminded me that we haven’t yet covered completing the square, a technique important for solving quadratic equations, and also in several other applications. We’ll see the traditional method, and a modified method that avoids fractions, including a nice alternative to the quadratic formula.
In a discussion last April, we saw problems arising from integration by substitution, when the substitution is not one-to-one. Last time, I looked back at that issue from the perspective of a similar recent problem. But a question from soon after the first, with some interesting features of its own, led me to examine whether …
A question came in just this month, closely related to the last question in the post When Is an Improper Integral Not an Improper Integral? last April. There, we considered an integral where substitution led to a new integral with limits from 0 to 0, and after some discussion we decided that, while odd, the …
In searching for questions about polynomial division, I ran across several about problems where you are given the remainders when an unknown polynomial is divided by two or three different small polynomials, and have to find the remainder when it is divided by a different, but related, polynomial (typically the product of the others). We’ll …
Finding a Polynomial Remainder, Given Other Remainders Read More »
Last time, we saw how to divide polynomials using long division, which takes a lot of time and space. For the simplest cases, dividing by a monic linear polynomial \((x-c)\), there is a much more efficient method. We’ll again see both how to do it, and why it works; then we’ll see how it ties …
How (and Why) Synthetic Division of Polynomials Works Read More »
Having just looked at the Rational Zero Theorem, I realized we’ve never covered how to divide polynomials, which is used closely with that theorem. Here we’ll look at long division, and then, next time, at synthetic division, its efficient version.
A recent question reminded me that we have not yet covered the Rational Root Theorem (or Rational Zero Theorem), though it has been occasionally mentioned (e.g. here and here and here and here). It allows us to find any roots of a polynomial equation (that is, zeros of the polynomial) that might happen to be …
Last time we solved some of the equations connected with a segment of a circle using Newton’s Method. Let’s take a closer look at the method – how it works, why it works, and a few caveats.
Here is an interesting question we got recently, that turns a common maximization problem (the open-top box) inside-out. What do you do when you’re given the answer and have to find the problem? We’ll hit a couple snags along the way that provide useful lessons in problem-solving.