# Tagged Questions

Linear, exponential, logarithmic, polynomial, rational, and trigonometric functions, conic sections, binomial, surds, graphs and transformations of graphs, equation- and system-solving, and other symbolic-manipulation topics.

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### What to keep in mind when attempting proof of basic properties of divisibility/what techniques are useful/what's the intuition for showing them?

So I am currently trying to prove some basic divsiibility relations, as follows. If $a \mid b$ and $a \mid c$, then $a \mid (b + c)$. If $a \mid b$ and $s \in \mathbb{Z}$, then $a \mid sb$. ...
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### How can I construct a solution for this system of many inequalities?

Let there be types $\omega\in\{0,1\}^n$ drawn according to some probability distribution. Suppose that these types are relayed through some imperfect message service. Specifically, any type $\omega$'s ...
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### Is there an elementary introduction to higher order functions?

I am teaching a pre-calculus course (using the textbook by Michael Sullivan if it helps), and I realized that higher order functions seem to show up in with some frequency in pre-calculus and calculus....
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### Intuition behind the proof of the validity of the Euclidean algorithm

As the question title suggests, could anybody explain to me their intuition behind the proof of the validity of the Euclidean algorithm?
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### Use a direct proof to show that if $x\gt 1$ then $x^5 +x+1\gt 2$.

Use a direct proof to show that if $x\gt 1$ then $x^5 +x+1\gt 2$. It's obvious that if $x\gt 1$ then $x+1\gt 2$ , also $x^5\gt0 \;as \;x\gt1\gt0$, thus $x^5+x+1\gt 2$. Is it fine ? I can't ...
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### Writing sum of square roots with symmetric polynomials

I want to write the function $$F_N=\sum_{i=1}^N\sqrt{x_i}$$ in terms of the $N$ elementary symmetric polynomials of the $N$ positive variables $x_1,\dots,x_N$. The $N=1$ case is trivial, as we ...
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### Irreducibility of some polynomial

Let $p(x) = (1+ \cdots +x^k)^2 + (1+ \cdots +x^k) + 1$, for some $k \geq 2$ fixed. I would like to know if $p(x)$ is irreducible in $\mathbb{Q}[x]$.
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### Is there a name for systems of equations with min and max functions included?

In a big project I'm working on, I'm running into systems of equations that look like the following: $$a = \min(b, c)$$ $$b = d^2 + a$$ $$c = \max(a + b, d)$$ Basically, nonlinear systems of ...
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### Existence of Solutions of Two Cubic Equations in a Particular Region

If I have two cubic equations in two variables, $ax^3 + bx^2 y + cxy^2+\dots=0$ and another one with different coefficients, and I know that $(x,y)=(0,0)$ or $(1,1)$ are solutions, are there any nice, ...
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### Polynomial bound

Let $P(x)=a_4 x^4+a_3 x^3+a_2 x^2+a_1 x+a_0$ such that $$\forall i\in \{0, 1, 2, 3, 4\};\phantom{;}a_i\in\mathbb{Z} \wedge |a_i|\leq T\phantom{.}(T\in\mathbb{Z}^+ )$$ Suppose that $P(x)> 0$ for all ...
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### Digits of two irrational numbers, given their power with fixed number of digits

I have $a, b \in \mathbb{R} \setminus \mathbb{Q}$, I want to know the result of $a^b$, but I don't know exact $a, b$ because I write them in numeric form. My question is how many digits of $a, b$ have ...
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### Mandelbrot set's border in parametric form

I've post this question just because I'm curious, Mandelbrot set is defined as: $z_{n+1} = z^2_n + c$, if $n \rightarrow \infty$ and it doesn't diverge we get the border. This border is unlimited ...
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### Selling oranges when people queue up in a line

We have $a$ oranges to give to $b$ people. Each person has a value $f(n)$ for receiving $n$ oranges, where $f$ is a nondecreasing, nonnegative function that is the same for everyone. Let $X$ be the ...
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### In search of a College Leve Pre Algebra Applications Textbook

I am looking for a textbook at the college level that mainly focuses on applying algebra to situations. I want students to know how to set up the equations, not just solve them.
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### Closed formula for the numbers of the form $\sqrt{1+\sqrt{4+\sqrt{9}}}$

how can i find the formula for the nth term of this series? SQ = square root $\sqrt{1} = 1$ $\sqrt{1 +\sqrt{4}} = \sqrt{3}$ $\sqrt{1 +\sqrt{4+\sqrt{9}}} \approx 1.909385061$ \$\sqrt{1 +\sqrt{4+\...