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 …
A recent question about the precise meaning of “difference” led me to some past discussions of the word.
A recent question led me to find a series of old questions, each answering it at a different level. All of them are worth presenting.
Last time, we looked at the Pigeonhole Principle, applying it to geometrical problems, largely about distances, gradually working from almost literal “balls and boxes” (“pigeons and pigeonholes”) to more abstract applications that are harder to see. Here, we will go beyond that, proving facts about sets.
A recent question involved the Pigeonhole Principle; we’ll start with an older question to introduce the idea, then the new question and a few old ones.
Last time, we saw a summary of the difference between population and sample standard deviation, along with other differences in formulas. Here we’ll look at two short answers about the reason for that difference, and then an extensive look at why it’s true. (I’ve taken extra time and space to work through details in order …
Sample Standard Deviation as an Unbiased Estimator Read More »
Last time we introduced standard deviation. Here we’ll look into why two formulas (namely, population and sample standard deviation) are different, and why several different formulas for either are equivalent. We’ll also discover how to update the standard deviation when a new value is added. In doing so, we’ll see some different perspectives than we …
Formulas for Standard Deviation: More Than Just One! Read More »