# Tag Info

### 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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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 ...
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### 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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### 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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### 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 \...

### 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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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 ...
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### 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 ...
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### 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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### 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 ...
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### 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 $$X\mid P(X) \iff \color{#c00}{P(0)} = 0$$ ...
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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 $$p(x) = c_{n}x^{n}+c_{n-1}x^{n-1}+\cdots+c_0=(x-a)... • 161 6 votes ### Definition of equation vs. expression vs. polynomial Here are informal definitions of the terms that seem confusing to you: A function is a relation between two sets, usually sets of numbers. It maps elements of the first set to elements of the second ... • 29.9k 5 votes ### Why do we teach the Rational Root Theorem? (high school algebra 2) Theorem: For every integer m, the polynomial x^3 - mx^2 - (m+1)x - 1 is irreducible among polynomials with rational coefficients. Proof: This polynomial has degree 3, so if it is a product of ... • 3,536 4 votes ### How to show (x - a) is a factor of a polynomial p(x) if and only if p(a) = 0 (without division) Let \,p(x) = c_n x^n + c_{n-1} x^{n-1} + \ldots +c_1 x + c_0\,, and expand it in powers of \,(x-a)\,:$$ \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\...
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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 ...
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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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $n$ ...
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### 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 ...
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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 ...
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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 ...

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