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 ...


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

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

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 ...


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 ...


4

Theorem: For every integer $m$, the polynomial $x^3 - mx^2 - (m+1)x - 1$ is irreducible among polynomials with rational coefficients. Proof: This polynomial has degree $3$, so if it is a product of lower-degree polynomials, the decomposition is (linear)(quadratic) or (linear)(linear)(linear). Either way, there is a linear factor and thus a rational root ($...


3

I really think this questions gets to a bigger issue. This is part of a general problem with our approach to math education. The point of high school math class, from any practical standpoint, is not to teach students things that they will constantly be using all the time (or even ever). Empirically this is verified in myriad examples, not just the rational ...


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