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.
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 »
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 couple recent questions asked what constitutes “standard form” for a quadratic equation; that will lead us to some older questions about “standard form” for a linear equation. We’ll see that “standard” isn’t quite as standard as you might think.
Let’s look at a nice little challenge: to find a cubic function with maximum and minimum at given locations – without using calculus. We’ll explore how to solve it with graphing software, and using algebra in a couple ways, and finally with calculus. And, surprise! They all give the same answer, though the results look …