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33 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 ...
Chris Cunningham's user avatar
13 votes
Accepted

The Way We Teach Square Roots

As he says, this is a convention, and you just have to make a choice and stick with it. The usual convention (which makes the square root a function) is to take the positive branch. The teacher ...
Steven Gubkin's user avatar
13 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 ...
guest's user avatar
  • 131
13 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 ...
Eric Towers's user avatar
12 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 \...
Steven Alexis Gregory's user avatar
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 ...
user52817's user avatar
  • 11k
9 votes

What is the standard for "simplifying your answer"?

Rigid criteria for simplification seem to me largely a bad idea if they are not motivated by contextual considerations. The idea that $\sqrt{2}/2$ should be preferred to $1/\sqrt{2}$ struck me as ...
Dan Fox's user avatar
  • 5,869
8 votes

How to help students understand/remember that $x^2 = a$ has two solutions?

There are possibly two different issues here. Issue 1 is that some students are under the mistaken impression that the symbol $\sqrt{5}$ actually designates two different numbers, one positive and ...
mweiss's user avatar
  • 17.4k
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 ...
Matthew Daly's user avatar
  • 5,629
7 votes

How to help students understand/remember that $x^2 = a$ has two solutions?

Telling and explaining is really not enough. Working exclusively in square roots of numbers all but guarantees blocking vs interleaving problems and encourages shallow robotic thinking and ...
WeCanLearnAnything's user avatar
6 votes

How to show $(x - a)$ is a factor of a polynomial $p(x)$ if and only if $p(a) = 0$ (without division)

One way to avoid (explicit) division is to make a change of change variables $\, X = x-a\,$ which reduces it to the following simpler special case $$ X\mid P(X) \iff \color{#c00}{P(0)} = 0$$ ...
Bill Dubuque's user avatar
  • 1,038
6 votes

How to show $(x - a)$ is a factor of a polynomial $p(x)$ if and only if $p(a) = 0$ (without division)

This answer presumes that the students know polynomial multiplication. In particular, $$(x-a)(x^{n-1}+ax^{n-2}+\cdots+a^{n-1})=x^{n}-a^{n}.$$ Since $$p(x) = c_{n}x^{n}+c_{n-1}x^{n-1}+\cdots+c_0=(x-a)...
Math Lover's user avatar
6 votes

Definition of equation vs. expression vs. polynomial

Here are informal definitions of the terms that seem confusing to you: A function is a relation between two sets, usually sets of numbers. It maps elements of the first set to elements of the second ...
Joseph O'Rourke's user avatar
5 votes

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

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 ...
KCd's user avatar
  • 3,536
4 votes

How to show $(x - a)$ is a factor of a polynomial $p(x)$ if and only if $p(a) = 0$ (without division)

Let $\,p(x) = c_n x^n + c_{n-1} x^{n-1} + \ldots +c_1 x + c_0\,$, and expand it in powers of $\,(x-a)\,$: $$ \begin{align} p(x) &= c_n\big((x-a)+a\big)^n+c_{n-1}\big((x-a)+a\big)^{n-1}+\ldots+c_1\...
dxiv's user avatar
  • 271
3 votes

What is the standard for "simplifying your answer"?

I would say the canonical answer for what constitutes 'simplified as much as possible' is whatever the exam board says it is. 'Simplify' isn't a mathematical function. It is a pedagogical instruction ...
Jessica B's user avatar
  • 5,832
2 votes

What is the standard for "simplifying your answer"?

There is a theorem which says that it is impossible to decide the equivalence of two elementary functions syntactically. So there is not, and cannot, be a uniquely defined "simplest form" for a given ...
Steven Gubkin's user avatar
2 votes

How to help students understand/remember that $x^2 = a$ has two solutions?

Use repetition and carrot/stick (e.g. weekly period-long exams, daily one question pop quizzes, in class games). Question shows an unconscious assumption that clear explanation is the key criteria. ...
guest's user avatar
  • 21
2 votes

How to help students understand/remember that $x^2 = a$ has two solutions?

Not a perfect solution, but I would insist on the meaning of $\sqrt{\cdot}$: I have observed that this symbol quickly becomes a meaningless mantra to students. I regularly ask (notably in 1st year of ...
Benoît Kloeckner's user avatar
2 votes

The Way We Teach Square Roots

Original Answer I first define that a square root of a number $a$ is a number $x$ such that $x^2 = a$. I then give some examples like: What are the square roots of 4? $2$ and $-2$ because $2^2=4$ and ...
LucasSilva's user avatar
2 votes

How to help students understand/remember that $x^2 = a$ has two solutions?

We are all dealing with this, of course. In a theoretical/philosophical sense, you can't "make" students learn anything ("You can lead a horse to water..."). In a practical/punitive sense, this is ...
Daniel R. Collins's user avatar
2 votes

Explaining the intuition for why finding roots of polynomials is hard

We can rescale a polynomial so its leading coefficient is $1$ without changing the roots. So let's focus on polynomials with leading coefficient $1$. If its degree is $n$ then it has $n$ ...
KCd's user avatar
  • 3,536
2 votes

Explaining the intuition for why finding roots of polynomials is hard

I advocate for an explanation of why the strategies for quadratics don't immediately work for higher orders. When finding the roots of a quadratic, there are two predominant strategies: factoring, and ...
okzoomer's user avatar
  • 341
1 vote
Accepted

Explaining the intuition for why finding roots of polynomials is hard

There is a closed form expression for the roots of a quintic, it is just that the solutions do not always simplify to expressions involving field operations and radicals. For example, using Wolfram ...
user52817's user avatar
  • 11k
1 vote

The Way We Teach Square Roots

The answer in the video was a bit unsatisfying for you, and I understand why. You expect a firm answer. IRL, when I get this question, I need to quickly determine the level the student is at in her ...
JTP - Apologise to Monica's user avatar

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