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.
A couple recent questions about integration turn out to be related to our last couple posts, in which we discussed substitutions that are not one-to-one. It’s better disguised here (and isn’t intertwined with symmetry issues!). You’ll also see a couple errors I made in correcting the student’s errors, which I am not hiding because you …
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 …
The start of a new year seems like a good time to take a look at the big picture. A question from September raises a big topic: What is mathematics, in general? What does all math – arithmetic, algebra, geometry, trigonometry, calculus, and beyond – have in common?
Last time, we looked at a problem that had a fairly easy solution, but I had to convince myself that the situation it assumed was actually possible, because the work did not imply its existence, but merely assumed it. Here I want to show several examples of what I feared: problems where a perfectly good …
Last time, we saw several problems where we were given remainders when an unknown polynomial was divided by two or three linear polynomials, and had to find the remainder on division by the product of those divisors. Here we’ll look at a recent question, and an older one, that add a twist: we’ll be given remainders …
Polynomial Remainder Problems: Changing the Conditions Read More »
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 »