Rational functions are ratios of two polynomials, for example $(x+5)/(x^2+3)$.

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Cancelling variable factors in a rational function

Consider the function $\displaystyle\frac{2x−1}{x+5}$. The domain of this function is all real numbers except $x = -5$. Now consider that I do this: $\displaystyle\frac{2x−1}{x+5}⋅\frac{x}{x}$. This ...
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Why do rational functions like $\frac{1}{x}$ have weird shapes? [on hold]

As many know, rational functions have weird shapes like hyperbolas placed diagonally. They are in couple parts, for example $\frac{1}{x}$. Why does that happen?
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How to show that $X^p-t\in\mathbb{F}_p(t)[x]$ is irreducible? [duplicate]

This question is previously asked here, but there is no complete solution of it. I understand that the root $\alpha$ exist in the algebraic closure of $\mathbb{F}_p(t)[x]$, and it is the only root ...
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Does every non-Archimedean ordered field contain $\mathbb Q (x)$ as an ordered subfield?

Every ordered field $\mathbb F$ contains $\mathbb Q$ in a canonical way. If the field is not Archimedean there exists an $x>n$ for all $n \in \mathbb N$. Since we are dealing with a field any ...
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32 views

Simplify $\frac{5x}{x^2 - x - 6} + \frac{4}{x^2 + 4x + 4}$

Simplify $\frac{5x}{x^2 - x - 6} + \frac{4}{x^2 + 4x + 4}$. How come the answer is left as $\frac{5x}{(x+2)(x-3)} + \frac{4}{(x+2)^2}$. Why don't we go any further?
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Continuity of solutions for systems of rational ODEs

If you have a system of ODEs of the from $\frac{d\mathbf{x}}{dt}=\mathbf{f(x,p)}$ where $\mathbf{x}$ is a vector valued variable, and $\mathbf{p}$ is a vector of parameters, and $\mathbf{f(x,p)}$ is a ...
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Showing that the field of rational functions is not dense

I am going through Counter Examples of Analysis but I am having trouble understanding a claim it makes. The book establishes that the set of rational functions defines an ordered field where the ...
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How would I divide $\frac{4x^2-2x}{x^2+5x+4}\div \frac{2x}{x^2+2x+1}$? [closed]

How would I divide $\frac{4x^2-2x}{x^2+5x+4}\div \frac{2x}{x^2+2x+1}$? Work would be appreciated. What would I do first?
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70 views

How do I multiply $\frac{8y-4}{10y-5}\times \frac{5y-15}{3y-9}$?

I am not sure how to multiply these fractions. Do I cross multiply or multiply the numerators together and multiply the denominators together? How do I multiply $\frac{8y-4}{10y-5}\times ...
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1answer
229 views

The best way of integrating irrational functions

Ok, so, here is the example integral: $$I=\int\frac{x-2-\sqrt{-x^2-4x+4}}{x^2-\sqrt{-x^2-4x+4}}dx$$ I always solve these types of integrals using Euler's substitutions, but, recently, I came across ...
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28 views

Why is this a correct way to multiply 2 terms together?

In a problem, I noticed the author did this: $$\frac{1}{(a+2)+(z-2)} = \frac{1}{(a+2)}\cdot \frac{1}{1+\frac{z-2}{a+2}}$$ What he is saying is to take the entire $(a+2)$ term and multiply it by $1$ ...
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68 views

What is known about this group reminiscent of the anharmonic group?

The anharmonic group is this nonabelian group of six rational functions with the operation of composition of functions: \begin{align} t & \mapsto t & & \text{order 1} \\[8pt] t & ...
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143 views

Has this chaotic map been studied?

I have recently been playing around with the discrete map $$z_{n+1} = z_n - \frac{1}{z_n}$$ That is, repeatedly mapping each number to the difference between itself and its reciprocal. It shows some ...
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1answer
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$\{a x^{z}: a\in \Bbb{Q}, z \in \Bbb{Z}\} \approx \Bbb{Q}^{\times} \otimes_{\Bbb{Z}} \Bbb{Z}^+ \implies$? what about $\Bbb{Q}$-linear sums?

Consider all functions $f: \Bbb{Q} \to \Bbb{Q}$ of the form $f(x) = a x^z$ where $a \in \Bbb{Q}, z \in \Bbb{Z}$, call it $G$. It forms an abelian group under usual multiplication. I think it's ...
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Rational function not in lowest term with same zeros in top and bottom always has hole in the graph or not?

I'm studying about rational function and I have watch this video: https://www.youtube.com/watch?v=sRgmtojZGFg&list=PLDE28CF08BD313B2A&index=17 in the minute (1:09:50) the instructor talks ...
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Quickest way to factorize $\frac{w^2 + 5kw + 4k^2}{w^2+kw}$

I would say 90% of polynomials in my textbook are factorable e.g. $$\frac{w^2 + 5kw + 4k^2}{w^2+kw}$$ This gives $$\frac{(w+k)(w+4k)}{(w+k)w}$$ $$\frac{w+4k}{w}$$ This took me far too long to ...
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Does $f(x) = 1 = \frac{x-1}{x-1}$ have a hole at $x=1$?

While learning about asymptotes and holes in rational functions in Precalculus, I came across a problem that shouldn't happen, but I don't understand. Something just doesn't add up in my head... Here ...
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Does this equation has a closed form solution?

We have $K$ non-negative coefficients: $a_1,a_2,\dots,a_K,A_1,A_2,\dots,A_K$, where $A_i\geq0,\;a_i\in(0,1),\;\sum A_i<T$. The equation is: $$\sum_{i=1}^K\frac{A_i}{1-a_ix}=T,\quad x\in(0,1)$$
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integrate $\int \frac{x^2+x+1}{3x^2-2x-5}$

$$\int \frac{x^2+x+1}{3x^2-2x-5}$$ I can see where t start, maybe $$\int \frac{x^2+x+1}{3x^2-2x-5}=\int \frac{6x-2}{3x^2-2x-5}+\frac{x^2-5x+3}{3x^2-2x-5}=\ln|3x^2-2x-5|+\int ...
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Indefinite integral of rational functions

This question seemed very trivial at the first glance, but evidently it is a non-trivial problem, at least, for me. Let $p, q$ be polynomials in $x$, and $\deg q > \deg p$, with $q (x)$ being ...
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How to solve integrals where you can't factor a polynomial?

Hi there guys I don't know if the title of the question should be the one for this but the thing is that I'm trying to solve this integral $\int \frac {\frac 12-u^2}{2u^4-2u^2+1}$$du$ and I have this ...
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What does it mean that “$f = g$ in $k(h(t))$?”

Let $k$ be a field and consider the rational function field $k(t)$. I was just reading that if $f(t),g(t),h(t)\in k(t)$ are such that $f(h(t)) = g(h(t))$, then "$f = g$ in $k(h(t))$." What does that ...
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Does this method of finding the range of rational functions always work?

Consider the irreducible rational function in $\mathbb{R}^2$. $$y=\frac{A(x)}{B(x)}$$ where at least one term is quadratic and the other term has degree either 0, 1 or 2. The classic way of ...
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29 views

Degree of Rational Function

This might sound like a very trivial question but I found different answers on the web. Assume one has a rational function $$\frac{f(x)}{g(x)} ,$$ where $f(x)$ and $g(x)$ are polynomials. What is ...
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Is there a rational function $f$ satisfying $f(x) =f\left( \frac{1}{1-x} \right)$ for all $x$?

I would like to find a pair of relatively prime polynomials $p,q \in k[x]$ (where $k$ is a field) such that $$\frac{p(x)}{q(x)} = \frac{p \left( \frac{1}{1-x} \right)} {q\left( \frac{1}{1-x} \right)} ...
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How do I perform u-substitution on this problem?

I am having trouble with this problem: $$\int {\frac{3x + 5}{5x^2 - 4x - 1}} dx$$ I can't seem to find a u where the du exists in the numerator so that it will cancel. If I choose: $$u = 5x^2 - 4x - ...
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199 views

How to integrate $\frac{1}{(1 + x^5)(1 + x^7)}$

My cousin who is in high school asked me if it is possible to integrate $$ \int \frac{1}{(1 + x^5)(1 + x^7)} \, dx $$ I checked the list of integrals of rational functions on Wikipedia link and it ...
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why does the equation $(-x^2 + 2x)/(5x - 4) = 6$ have 2 solutions?

Hmm, I have been wondering about this when I went to solve the following equation: $$\frac{-x^2+2x}{5x-4} = 6$$ How come the above equation has two solutions, $-14 + 2\sqrt{55}$ and $-14 - ...
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Is there a rational surjection $\Bbb N\to\Bbb Q$?

The question is in the title. Is there a one-dimensional rational function $f\in\Bbb R(X)$ which restricts to $\Bbb N\to\Bbb Q$, which is a surjection onto $\Bbb Q$? My guess is no. Expanding the ...
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Piecewise from Rational Absolute Value Function

How would one separate a function like the following into piecewise? $$f(x)={\left|4-x\right|\over{\left|x-4\right|}}$$ I've been taught that with a rational function with an absolute value in the ...
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coefficients of Laurent series of rational function

Let $F(z)$ be a rational function $\frac{P(z)}{Q(z)}$ such that the degree of $P(z)$ is less than the degree of $Q(z)$ and suppose that all the zeros of $Q(z)$ are contained in the open disk $|z| < ...
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Finding the value of a parameter for which a rational function is tangent to x-axis.

I'm stuck on this problem and hope someone can help: The curve $C$ has equation $$ y= \frac{px^2+4x+1}{x+1}$$ where $p$ is a positive constant and $p\neq 3$. Find the values of $p$ for which ...
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A rational orbit that's provably dense in the reals?

Iterating the map $\ \ x\ \mapsto\ x-\frac{1}{x},\ \ $ the orbit of initial point $2$ is "probably" dense in $\mathbb{R}$. Is there an explicit rational mapping together with an initial rational ...
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How can I integrate rational functions with denominator with just quadratics as factors?

I am told about integration by partial fraction method.I usually guess to decompose a fraction into partial fractions and then I solve the constants.In this problem: ...
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$T: \mathbb{Q}[X]\mapsto\mathbb{Q}[X]$. Proving $T$ is injective and not surjective.

Given $T: \mathbb{Q}[X]\mapsto\mathbb{Q}[X]$, $T(f)=X\cdot X\cdot f - (X+1) \cdot f'$. Prove $T$ is injective and not surjective. I try to prove this by applying the definition of injectiveness but I ...
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Can $\frac{P(x)s(x)+Q(x)}{R(x)}$ be a polynomial for some polynomial $s(x)$?

Given some polynomials $P(x), Q(x), R(x), \operatorname{gcd}(P(x), R(x)) = 1$ does there exist some polynomial $s(x)$ so $$ \frac{P(x)s(x) + Q(x)}{R(x)} $$ is also a polynomial? If so how do I ...
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Construct a rational function whose graph in the xy-plane has a vertical asymptotes lines x = 3 and x = 5 oblique asymptote the lines y = 2x -3

I'm studying for a test and I came across this example problem and the oblique part is throwing me off. How do i go about solving this?
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Is it possible to represent a root as a simple rational function with an exponent?

Using the following function:$$y=\frac{mx^p+b}{d}$$... where $m$, $p$, and $b$ may be any integer ... where $d$ may be any integer $\gt0$ ... and where $x$ may be any rational number $\ge0$ Is it ...
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Genus of trancendental curve

I understand that an algebraic curve is genus 0 iff it can be parameterized using rational functions. I am curious if there is a simple way to know if a transcendental curve is genus 0? Specifically ...
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1answer
48 views

I can't understand to solve the question (Khan Academy Algebra Basics - Exponent Properties)

I was learning (practicing to solve) simplifying the rational expressions. I know how to simplify the rational expressions... but I can't understand some part of the questions. The question that I ...
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$\mathbb{Z_p}(x^p,y^p,x-y^n) \neq \mathbb{Z_p}(x^p,y^p,x-y^m)$ for $n \neq m$

I am trying to find an easy proof for the following result: Let $p$ be a prime number. Show that for $n \not\equiv m \mod p$ $$\mathbb{Z_p}(x^p,y^p,x-y^n) \neq \mathbb{Z_p}(x^p,y^p,x-y^m).$$ The ...
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Solving a system of 4 equations

why gives me that system of equations: $-\tfrac{1}{x} +2ux =0$ $-\tfrac{1}{y}+2uy=0$ $1+2uz=0$ a)--> $x^2 = y^2= -z $ ? And using the equation: $x^2+y^2+z^2 = 1$ gives b)$z = 1 (+,-) \sqrt{2}$ I ...
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How to calculate the local minimum of a hyperbola without using derivatives?

I've got the following rational function, which is a hyperbola. $f(x) = \frac{2\cdot\pi (x+ 4)^2}{x}$$\quad$on WolframAlpha There is a minimum in the first quadrant and a maximum in the third ...
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Inverses of Rational Functions

Consider the function $g$, where $$g(x)=\frac{3x}{5+x^2}$$ (a) Given that the domain of $g$ is $x\ge a$, find the least value of $a$ such that $g$ has an inverse function. I know that $g(x)$ ...
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87 views

Why does the extra $.1$ make $\int \frac{\left(x^2+2.1x\right)}{\left(x^3+3x+12\right)^6}dx$ much harder?

I'm a first-time Calc I student currently struggling in class. Yesterday we started on Substitution and Integration with integrals. One problem our professor put on the board was: $$\int ...
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51 views

How can I show that this meromorphic function is a rational function of two polynomials?

Here's my updated attempt: Write$$f(z) = \sum_{n=-1}^{\infty} a_n(z-z_1)^n + ...+\sum_{n=-1}^{\infty} m_n(z-z_m)^n+\sum_{n=+1}^{-\infty} \psi_n(z)^n$$ with the last series being an expansion about ...
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1answer
41 views

A simple complex rational function.

Let $$f(z)=\dfrac{z-a}{z-b},\,\,\,\,\,\,z\not=b\not=a$$ be a complex valued rational function. How can I show that, if $|a|,|b|\lt1,$ then there is a complex number $z_0$ satisfying $|z_0|=1$ and ...
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1answer
35 views

Rational Multivariable limit

I am having some issues with the following multivariable limit: $$\lim_{x,y\to0,0} \frac{x^2+y^2}{x+y}$$ I am trying to show whether it exists and is equal to 0, or whether it does not exist. What I ...
2
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2answers
24 views

If a rational function is even then the numerator and the denominator have same parity

Let $K$ be a field and let $F={P\over Q}\in K(X)$ be a rational fraction, for simplicity we denote also by $F$ the rational function associated to the rational fraction $F$. It is clear that if $P$ ...
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1answer
19 views

howto find where a slant aymptote crosses a function

i have the function $f(x) = \frac {(x+1)(x-1)^2}{x^2}$ and i want to sketch it so i found the following: Vertical Asymptote: is x=0 X-intercepts: x=1, x=-1, x=1 Slant Asymptote: $y = x-1 $ then i ...