30 votes

Why do we teach the Rational Root Theorem? (high school algebra 2)

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 ...
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  • 19.1k
26 votes

Factoring quadratics where the coefficient on the $x^2$ term does not equal 1

$$3x^2 -14x - 5 = 0$$ 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 '...
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22 votes

In which course should we teach solving general cubic and quartic equations?

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 ...
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  • 6,348
14 votes

Factoring quadratics where the coefficient on the $x^2$ term does not equal 1

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, $c$ ...
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13 votes

Factoring quadratics where the coefficient on the $x^2$ term does not equal 1

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 ...
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12 votes

Algebra 2 textbooks that incorrectly claim that all solutions of polynomial equations can be found

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 ...
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12 votes

Factoring quadratics where the coefficient on the $x^2$ term does not equal 1

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 ...
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  • 4,793
12 votes

Why do we teach the Rational Root Theorem? (high school algebra 2)

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 ...
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  • 7,703
11 votes

Where do students learn to solve polynomial equations these days?

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 ...
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  • 6,367
11 votes

Why do we teach the Rational Root Theorem? (high school algebra 2)

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 ...
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  • 111
11 votes

Why do we teach the Rational Root Theorem? (high school algebra 2)

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 \...
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10 votes

Why do we teach the Rational Root Theorem? (high school algebra 2)

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 ...
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9 votes
Accepted

Why is polynomial factorization over the integers part of secondary school curricula?

"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 ...
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9 votes
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Where do students learn to solve polynomial equations these days?

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 ...
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8 votes

In which course should we teach solving general cubic and quartic equations?

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 ...
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  • 13.6k
8 votes
Accepted

How to come up with a Leslie matrix with convenient eigenvalues?

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$...
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  • 7,713
7 votes

Beyond cubic polynomials: Applications?

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. ...
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7 votes

Factoring quadratics where the coefficient on the $x^2$ term does not equal 1

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 ...
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  • 19.1k
7 votes

Factoring quadratics where the coefficient on the $x^2$ term does not equal 1

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 ...
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7 votes

Where do students learn to solve polynomial equations these days?

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 ...
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7 votes

Why do we teach the Rational Root Theorem? (high school algebra 2)

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 ...
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  • 5,539
5 votes

How to work with polynomials in difficult classes?

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 ...
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  • 1,736
5 votes

Why is polynomial factorization over the integers part of secondary school curricula?

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 ...
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  • 583
5 votes
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How to work with polynomials in difficult classes?

There is a great game about polynomials presented by Rachel Kaplove on the eHow YouTube channel. The rules are very simple: Each student gets a card with either expanded polynomial, for example $x^2-...
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5 votes

Motivation for polynomial long division

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 ...
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  • 1,153
5 votes

In which course should we teach solving general cubic and quartic equations?

I'm used to seeing them "starred" in an algebra 2 or "college algebra" course. Starred means the extra topics that very advanced classes could cover. My experience is even for those classes nobody ...
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  • 234
5 votes

Is coefficient same as constant?

I'd say that the video is not using the best word. I would call that constant the coefficient. Constant means that it is a number and not a variable. That's true. But the word coefficient conveys more ...
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  • 17.5k
4 votes

Beyond cubic polynomials: Applications?

Quartics and higher degree polynomials frequently arise when intersecting lower degree curves, e,g, the intersection of two conics, or a line and a torus. Such curve intersections often occur in ...
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  • 1,019
4 votes

How to work with polynomials in difficult classes?

Well, you could start by explaining what $x^2$ means geometrically. If I had a square with side length $x$, then the area would be $x^2$. Explain to them how this is completely different from $2x$. ...
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