30

I would say the standard implementations of the Rational Root Theorem (make a huge list for the sake of making the huge list) indeed feel like a complete waste of time. However, the theorem can sometimes combine nicely with technology tools. I'll provide an example of this. Consider the following algebra puzzle: Solve for $x$: $7x^3 -39x^2+52x+30 = 0$. If ...


24

$$3x^2 -14x - 5 = 0$$ Multiply through by A or here, 3 $$9x^2 -42x - 15 = 0$$ Now, use substitution, u=$3x$ (3X is the square root of this first term, and by using the u substitution, we now have an 'a' of 1. ) $$u^2 -14u - 15 = 0$$ factor to $$(u-15)(u+1)$$ Substitute back u=3x $$(3x-15)(3x+1) $$ last, divide out that 3 we multiplied by - $$(x-5)(...


22

The place I’ve seen this is usually in a History of Math class. This makes the most sense to me since solving polynomial equations (and the methods of reducing one type to another) plays an important role throughout the ages, but especially in 16th century mathematics. students of the sciences certainly don’t need this info for practical reasons, so there’s ...


13

Quadratic Formula (deterministic---no guess and check about it) The QF yields that $-{\frac13}$ and $5$ are roots. So $$3x^2-14x-5=c\left(x+\frac13\right)(x-5)$$ Comparing leading coefficients, $c$ must be $3$: $$\begin{align}3x^2-14x-5&=3\left(x+\frac13\right)(x-5)\\&=(3x+1)(x-5)\end{align}$$ Use Parabola Vertex Form (deterministic---no guess and ...


13

Factoring non-monic quadratic polynomials can be done by factoring with respect to a particular constraint. More precisely, DL Renfro points to the ac Method of Factoring which can be summarized roughly as follows: Given a quadratic $ax^2 + bx + c$, the polynomial can be factored iff there is a factor pair for $ac$ whose sum is $b$; here, I denote by "...


12

Teach them how to complete the square. This is probably the simplest systematic method for factoring quadratic polynomials, and it's also very geometrically intuitive (you can literally visualize it in terms of a square). Guessing and checking usually works for very simple examples — monic quadratic polynomials with small integer factors — but it's not ...


12

I think an important learning outcome of this part of high school Algebra 2 (theory of polynomials) is developing mathematical capacity to juggle several disparate abstract tools all at once, similar to how in calculus, one uses first derivative test, second derivative test, concavity, asymptotes, intercepts, end behavior, etc., all at once, in some ...


11

One drawback I see is that it is only marginally better than the standard algorithm of factoring $c$ and seeing which factors add up to $b$. To apply Eisenstein's, you must find a factor of b or c and then test it on the other factor. So factoring occurs in either situation. While Eisenstein's is a little bit faster, it doesn't lead to a solution in the ...


11

To the first question, I suspect the primary reason is that the mathematical community learned to solve quadratics proceeded in this order - that is, mathematicians realized there were specific, solvable cases before proceeding to the general solution. But I think there are two related arguments to be made for maintaining that order, one historical, one ...


11

At the moment, I can answer bullet point two: Are there any high school textbooks that explicitly acknowledge that the methods included in the text are not adequate to solve all 3rd and 4th degree polynomial equations, and that in higher degrees that are no general methods at all? Yes, you can find this on p. 267 of CME Project's (2009) Algebra 2 text. ...


11

Students learn linear and quadratic equations in high school algebra. And then, if they have forgotten it, re-learn it in college, in courses called "pre-calculus" or something. Unless they specialize in mathematics at the college level, they do not learn any more. Why not? Because we have computers now, so most people do not need to solve polynomial ...


11

Agreed...it's one of the less useful parts of high school algebra. But not because "you could use a computer"--you could say that about almost everything. And then we get some of the same people who push the "use a computer" who are surprised when their kids flounder because of lack of manipulational ability in calculus. ;) The reason ...


11

Let $x = \sqrt 2 + \sqrt 7$ prove that $x$ is irrational. \begin{align} x - \sqrt 2 &= \sqrt 7 \\ x^2 - 2\sqrt 2x + 2 &= 7 \\ x^2 -5 &= 2\sqrt 2x \\ x^4 -10x^2 +25 &= 8x^2 \\ x^4 -18x^2 + 25 &= 0 \end{align} So $\sqrt 2 + \sqrt 7$ is a root of $x^4 -18x^2 + 25$. According to RRT, if $x$ is rational, then $x = \pm 1$ or $x=...


10

To help students understand how we move from specific cases to the general case, I created a visual depiction of the process of completing the square. I used shapes analogous to Algebra Tiles, which are a good manipulative for building understanding of factoring as well. I then took the process one step further and illustrated the derivation of the Quadratic ...


10

The RRT is not taught in isolation. It is taught as a collection of tools for (partially) factoring polynomials. It should be taught with Descartes' rule of signs and some form of polynomial division. It is direct preparation for understanding the proof of Eisenstein's criterion for irreducibility. The "shadow" of that application is via Gauss's ...


9

"any references pointing in the right direction would be greatly appreciated" Reference [1] below is probably where you want to look. A few years ago I tried to obtain a copy of [1], but I was not able to find one. However, I do not have access to interlibrary loan. If you have access to a university’s interlibrary loan, then you might be able to get a copy ...


9

I maintain that what you want to focus on is sense making (MESE 1) (MESE 2). If you can present Eisenstein's criterion as a way of tackling such problems that helps to promote students' critical thinking about algebra, then I would say go for it. (I'd also be interested to look at your curricular materials!) To illustrate my more general feeling, I look at ...


9

The rational root theorem, synthetic division, the remainder theorem, Descartes rule of signs, and similar lower level topics were fairly widely taught in U.S. high school algebra-2 courses before and during the 1980s, but they've slowly been de-emphasized as graphing calculators (allows for numerical equation solving) came onto the scene at the end of the ...


8

I think it is very reasonable in grad abstract algebra to show that these formulas are corollaries of very natural manipulations of Lagrange resolvents. These natural manipulations also illustrate some other points, about "averaging" and Vandermonde determinants, and so on. All these pre-date Galois by decades, and were the context in which Galois, Abel, and ...


8

If I use your simplification that $f_0 = 0$, then I suggest just choosing a real eigenvalue $\lambda$ and writing out the relation for the other parameters: $$-\lambda^3+f_1s_0\lambda + f_2s_0s_1 = 0$$ Now isolate a parameter, say $s_1$: $$s_1 = \frac{\lambda^3-\lambda f_1 s_0}{f_2 s_0}$$ Then just find values that satisfy your requirements. It should be ...


7

Assuming the problem comes from the happy world of textbook problems... a typically successful method is to assume integer factorizations: $$ (3x+a)(x+b) = 0$$ where $ab=-5$. In the world free of those complicated fractions, we have just $a= \pm 1 $ and $b= \mp 5$ to choose. So, our options are: $$ (3x+1)(x-5) \qquad \& \qquad (3x-5)(x+1)$$ $$ (3x-1)(x+5)...


7

Shamir's secret sharing is a cryptographic application of polynomial interpolation where the order of the polynomial depends on the number of shares which you want to be required to access the secret. E.g. if you want 12 stars worth of generals to be required to launch your missiles, you take a random polynomial of degree 11 which passes through $(0, \text{...


7

In the United States, solving linear and quadratic equations is a standard part of Algebra 1, which most students take in 8th or 9th grade. Students will return to polynomials and see long division and synthetic division in Algebra 2. This is also when students will learn the remainder theorem, Descartes’ Rule, and how to identify the only possible rational ...


7

I would reluctantly agree that it's not a particularly powerful tool if you have electronics at your disposal. But I might double down and say that you should be teaching synthetic substitution as well so that students can factor cubic and quartic polynomials without their calculators. About a year ago, I went down a YouTube rabbit hole of watching videos ...


6

The more little tricks and techniques we teach our students, the more they see math as an arcane toolbox of things to remember until the next exam and forget thereafter. Instead of teaching an N-step process for each problem type, whenever possible we should try to find a memorable, generalizable, useful concept that unites all similar problems. In this case,...


5

I see little sense in introducing Eisenstein's criterion (a pretty big cannon) only to have a slight advantage when trying to factorize quadratics. It's really not justified. The strength of Eisenstein's criterion is in proving large polynomials irreducible. This certainly can be taught at the high school level since it's not that complicated and quite ...


5

For multiplying polynomials and combining like terms, this website presents a simple level-appropriate tabular method. For example, to multiply the polynomials $x - 2$ and $2x^2 -3x + 1$ construct the following table: Then fill the table with the products, like the multiplication tables they learned in primary school: Like terms will be located along ...


5

One reason is that it's essential to determining the slant asymptote of a rational expression. Another reason is that it's a useful step in factoring large polynomials. For example, say you're trying to factor $x^3 + 4x^2 -4x-1$. You can determine, using the Rational Root Theorem, that x - 1 is a factor. Since x + 1 isn't, my next step would be to divide ...


5

If we can do something with integers, we can do it with polynomials too Things like adding, subtracting, multiplying, dividing, factoring. At least, that's how I framed these kinds of topics when I taught remedial algebra classes that focused heavily on algebraic manipulations (polynomials, rational expressions, radical expressions). I'm by no means an ...


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