How Piecewise-Defined Functions Work
I want to first introduce the concept of piecewise-defined functions, and then explore some often-omitted details raised in some recent questions.
Algebra
How to Factor a Quadratic: Power Tools
Last time, we looked at the standard algebraic methods to factor a quadratic trinomial. Those methods all involve some amount of guessing and checking, especially when the numbers involved are large; and they only work when there are binomial factors with integer coefficients. These are the methods a teacher usually expects you to use. But …
Three Standard Ways to Factor a Quadratic
Last time, we looked at some non-recommended ways to factor a quadratic trinomial. Here we’ll explore several recommended methods (the basic case, trial-and-error, and ac-grouping), looking into how to do them, and why they work.
“Magical” Ways to Factor a Quadratic
Recently in my college tutoring center, a student factored a quadratic polynomial using the X method, which I had heard of, but never watched being done for a hard example. Here on the site, a teacher recently asked about a related method I know as “slip-and-slide”. Both are “magical”, with no obvious reason they should …
Hyperbolic and Circular Functions and Complex Numbers
Last time we looked at what hyperbolic functions are, as a parallel world to trigonometric (circular) functions. But there’s another parallel: Each is related to the other through complex numbers. We’ll look at several questions that open the door to this relationship, finding how to calculate trig functions of complex numbers, and then delving into …
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Solving Polynomial and Rational Inequalities
A recent question asked about quadratic inequalities; we’ve touched on that in Domain, Range, and Quadratic Inequalities, but here we’ll introduce three basic approaches. We’ll follow that with old questions about polynomial inequalities (mostly quadratic) and rational inequalities (fractions), expanding on the basic methods.
Exploring Descartes’ Rule of Signs
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, …
Completing the Square for Quadratic Equations
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.
Polynomial Remainder Problems: Changing the Conditions
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 …
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Finding a Polynomial Remainder, Given Other Remainders
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 …
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