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

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How to prove that $f$ can be expressed as a ratio of polynomials, given that $|f(z)|=1$ when $|z|=1$? [duplicate]

Given: $f$ is analytic in $| z|\leq1$ and $|f(z)|=1$ when $|z|=1$. Prove that $f(z)=P(z)/Q(z)$ where $P$, $Q$ are polynomials.
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Is any rational function $R(x)$ a real analytic function in its domain?

To begin with, the definition of a rational function $R(x)$ can be found in Wiki. Suppose that $R(x)$ is defined in a subset $D \subseteq \mathbb{R}^n$. Then my question is: Is any rational ...
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Field of rational functions

Let $K$ be a field with characteristic $p>0$ and $M=K(X,Y)$ the field of rational functions in 2 variables over $K$. We consider the subfield $L=K(X^p,Y^p)\subset M$. Show that $[M:L]=p^2$. I ...
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what are the solutions of this biquadratic equation

I tried to choose this equation but I don't know how to find the LCM
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Graph shifting, compression, and stretch

Given $f(x)$, sketch $p(x) = (1/2)f(2x-6)-3$. I can't put the graph here. You can just tell me the order of transformation of the graph. What i did by myself is horizontal compressing (using $2x$ in ...
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Algebraic division with two variables? $\frac{a^3 + b^3}{a+b}$

I know there's a formula for this, but I would like to know how to do algebraic division the long way - would appreciate if you can guide me along. How can I use long division for $$\frac{a^3 + ...
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If $f\in k(\mathbb{A}^1)$ and $f^2\in k[\mathbb{A}^1]$, then is $f\in k[\mathbb{A}^1]$?

Suppose $k$ is algebraically closed, and $f\in k(\mathbb{A}^1)$ is in the field of rational functions over the variety $\mathbb{A}^1$. If we also know that $f^2$ is in the coordinate ring ...
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How to add/subtract complex rational expressions?

I'm studying for my Precalculus final and have noticed I still don't fully grasp performing basic operations on complex rational expressions, or finding if any values must be restricted from the ...
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Partial fraction decomposition of rational functions

How can I start this with Partial fractions? $$ \frac{10}{(s^2-4)(s^2+4)}+\frac{1}{s^2+4}$$ I was thinking of something like: ...
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Determine rational expression

Can somebody help me with the following word problems: Problem #1 Russel's combine can clear a field in 24 tractor hours. Jerome's combine can clear the same field in 30 hours. If they work ...
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Exercise of commutative algebra, rational functions.

This exercise is of my weekly newsletter of the subject of commutative algebra. My knowledge is restricted to the book of William Fulton, Algebraic Curves. I need help to solve it, any hints. ...
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Domain of definition of $(1-y)/x$ on $x^2+y^2=1$?

I'm self-teaching myself some basic algebraic geometry, and I wanted to double check something that seems too easy. An exercise sheet I found asks to compute the domain of definition of the rational ...
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When looking for zeros of a rational function, why is the numerator equated to zero and not the denominator?

If you have a function $F(x)=\dfrac{a(x)}{b(x)}$ and you are asked to find the zero(s) of the function, why do you set the numerator equal to zero, and not the denominator?
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How do I find the zero(s) of a rational function?

I am doing homework and have been given this task: You have the function $$g(x)=\frac{2x^2-8}{x^2+4}$$ and I am asked to find the zeros of the function. My teacher shows the solution as $f(x)=0$ and ...
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Integration of a polynomial

I am facing a problem in finding the integral $$\int\frac{r^2}{-C r^3 + r^2 -2 M r +Q^2}\,dr$$ Here M, Q, and C are parameteres (to be fixed later). Could anybody Please help me in finding it? I ...
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Non-isomorphicity of $(\mathbb{Z}/p^n\mathbb{Z})(x)$ and $(\mathbb{Z}/p\mathbb{Z})(x_1)(x_2)…(x_n)$

How does one go about proving that $(\mathbb{Z}/p^n\mathbb{Z})(x) \not\cong (\mathbb{Z}/p\mathbb{Z})(x_1)(x_2)...(x_n)$ as rings? Intuitively, I understand, but I am not sure how to make it concrete. ...
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Rational function such that $f(\frac{1}{1-x})=e^{\frac{2}{3}\pi i}f(x)$ [closed]

Does there exists such $f\in\mathbb C(x)$ (i.e. a rational function with complex coefficients) that $$f(\frac{1}{1-x})=e^{\frac{2}{3}\pi i}f(x)$$
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Does such a polynomial always exist, for any pole of a rational function?

Let $S \subseteq \mathbb{R}$ denote a cofinite subset of $\mathbb{R}$, and suppose $r : S \rightarrow \mathbb{R}$ is a rational function. Suppose $a$ is an element of $S^c$ (i.e. suppose $a$ is a pole ...
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Is there an injection between $\mathbf{R}$ and $[0,1)$

I want to find an injective map $f\colon\mathbb{R}\to[0,1)$ that is not a transcendental function (I prefer a rational function). Is it possible to find such a function or do I need a transcendental ...
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If $R = \frac{P}{Q}$ is a rational function, does $f(R) := \deg (P) - \deg (Q)$ have a traditional name/notation?

Suppose $R : C \subseteq R \rightarrow \mathbb{R}$ is a (univariate) rational function. Write $R=P/Q,$ where $P$ and $Q$ are polynomial functions $\mathbb{R} \rightarrow \mathbb{R}$. Is there a ...
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Create an equation for a description of a rational function

A graph has a y-intercept at -5, no x-intercepts, and discontinuous points at (-1,-5) and (3, -5). I want to form an equation for this graph, but I don't know how the y-intercept relates to the graph ...
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Integration of rational function…

$\int\frac{3x^2-x-6}{(x+3)(x^2+x+2)}dx$ I started with $\frac{A}{x+3}+\frac{Bx+C}{x^2+x+2}$ If my first step is right, then I get, $A(x^2+x+2)+(Bx+C)(x+3)=3x^2-x-6$, If my working is correct, how ...
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Key Features of This Rational Function

$$-2x^2-15x-25\over x^2-x-5$$ I'm not sure how you would find the Intercepts or the Asymptote of this function. I've tried factoring the equation but it leaves me with $$-1(2x+5)(x+5)\over ...
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Bivariate rational (quadratic over linear) model fitting by least squares

I am trying to fit a simple model over 2D data points, in the frame of an image formation model with perspective and optical distortion. My model is the ratio of a second degree polynomial over a ...
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Simplify using either Method 1 or 2. [closed]

I need to simplify $$\dfrac{1 - \dfrac 4{t+5}}{\dfrac{4}{t^2-25} + \dfrac{t}{t-5}}$$ I am posting the methods I can use as images. !
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Rational function interpolation?

We know that $n+1$ points is enough to completely determine a polynomial of degree $n$. Are there any techniques which says that a certain number of points is enough to completely determine a rational ...
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Prescribing zeros and poles of a rational function on $\bar {\mathbb C}$ at once

I have to show that for any points $P_1$, $\ldots$, $P_n$ and $Q_1$, $\ldots$, $Q_n$ ($P_i \neq Q_j$ for all $i$, $j$) on $\bar{\mathbb C}$ there exists a rational function $f$ with poles at $P_j$, $j ...
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Beautiful problem on a progression

$\{x_n\}$ is a sequence defined as follows: $x_1=20,\quad x_2=14,\quad x_{n+2}=x_n - \frac{1}{x_{n+1}}$. Prove that $0$ is among the members of this sequence. Find its number. I tried some stuff ...
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Domain and Range of a rational function

given the rational function $\frac{1}{x^{2} - \frac{x}{2}-3 }$ and asked for the domain and range, I multiplied thru by 2 and got $\frac{2}{2x^{2} - x-6 }$ I understand the domain includes all real ...
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What is the possible domain and range for this rational function?

Given a function: y= $\frac{(2x^2+11x+15)^{1/2}}{(x^2-9)^{1/2}}$ What is the domain and range for this function? My attempt: To find domain, I need to find the value of x such that the function ...
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Estimating the distance to the Julia set of a rational map

Suppose that $f \colon \hat{\mathbb{C}} \to \hat{\mathbb{C}}$ is a rational map of degree $d \ge 2$. Let $z_0$ be a point in the Fatou set $F(f)$. I'm interested in finding an estimate for the ...
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Rational function inequality

I've encountered the following inequality in a proof: $$ \left| \frac{P(Re^{it})}{Q(Re^{it})} \right| \leq \frac{R^k|a_k+..+a_0/R^k|}{R^n|b_n-..-b_0/R^n|}$$ Where $P(z)=a_0+a_1z+..+a_kz^k$ and ...
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How to compute the limit of a rational function at infinity?

I am unable to compute the limit, anyone can help ? I only understand some basic knowledge of limit .
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Limit $\lim_{{x\rightarrow 0}}{\left(\frac{\frac{1}{\sqrt{1+x}}-1}{x}\right)}$

I am a high school student in Calculus, and we are finishing learning basic limits. I am reviewing for a big test tomorrow, and I could do all of the problems correctly except this one. I have no ...
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Rational function in both $k(X)[Y]$ and $k(Y)[X]$

If I have a rational function in $X$ and $Y$ and it can be written as both a polynomial in $Y$ with coefficients being rational functions in $X$ (that is, an element of $k(X)[Y]$) and as a polynomial ...
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Possible for variable to cancel in a product of multivariate rational expressions?

Let $f,g,p_i,q_i$ be polynomials over some field with $\gcd(p_i,q_i)=1$ and $q_i$ are not constants for $i=1,2$. Assume that one or more of $p_i$ or $q_i$ has a term containing a variable $x$ not ...
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Rational equality modulo $p$

Assume that we have two rational expressions $f,g\in \mathbb{Z}(x,y_1,\ldots,y_n)$ with the property that the variable $x$ can only appear in the numerator of $f$ while only in the denominator of $g$. ...
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Find the domain of the function $f(x) =\frac{x+4}{x^2-9}$

I need to find the domain of the function $\;f(x) =\dfrac{x+4}{x^2-9}.$ My answer was: $(-\infty, -3)\cup(3, \infty)$. The book's answer was: $(-\infty, -3)\cup(-3,-3)\cup(3,+\infty)$ It's question ...
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Doing wrong in this fraction simplification?

$$ \frac{5}{2x-3} - \frac{3}{(2x-3)^2} $$ I have to simplify So I had the minimun common multiple in $$(2x-3)^2$$ which is $$(2x-3)(2x-3)$$ Then I divide the first fraction denominator by my ...
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Simplify rational expression

How do I simplfy this expression? $$\dfrac{\frac{x}{2}+\frac{y}{3}}{6x+4y}$$ I tried to use the following rule $\dfrac{\frac{a}{b}}{\frac{c}{d}}=\frac{a}{b}\cdot \frac{d}{c}$ But I did not get the ...
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How to evaluate an integral of the form $\int \frac{dx}{-ax^2 + b}$?

I need to evaluate $\int \frac{dx}{-ax^2 + b}$ while both $a$ and $b$ are positive. I was blocked while I was trying $ x=\tan\theta $ which turned $ dx=\sec^2\theta\, d\theta $ This method didn't ...
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Limit of a rational function

Calculate the limit $$ \lim_{x \to 0} \frac{3x^{2} - \frac{x^{4}}{6}}{(4x^{2} - 8x^{3} + \frac{64x^{4}}{3} )}$$ I divided by the highest degree of x, which is $x^{4}$, further it gave $$ ...
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Integral of $\arcsin$ of a rational function, using integration by parts

I'm a class 12 student and this a question from my textbook: $$I=\int{\arcsin{2x\over 1+x^2}}\mathrm{d}x$$ I did it using integration by parts like this: $$I=\arcsin{\left(2x\over ...
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A simple-looking rational limit

Please help me compute: $$ \lim_{z\to 0}\frac{\sqrt{2(z-\log(1+z))}}{z} $$ I know the answer is 1 because I plugged it into Mathematica. Attempts with L'Hopital's Rule didn't work. This a step in an ...
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Find all values $c$ such that $(x+1)/(x^2+2cx+4)$ has domain R

Word for word: Find all values of $c$ such that $f(x)=\frac{x+1}{x^2+2cx+4}$ has a domain R I really don't know where exactly to start. I'm not sure what it means by "a domain R"
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Why is this rational expressions indeterminate when evaluated?

I have this rational expression to evaluate, $$ {{3a-3}\over {4a(a-1)}} \text { if } a=1. $$ I understand that if you substitute 1, both the numerator and denominator would turn out 0, thus making ...
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Integration of the rational function $ 4/( 1+4t^2) $

So it's August so my memory of math is a little rough right now. I was wondering if someone could help me with integration with a fraction involved? For example: $$\int_0^{1/2} \frac 4{1+4t^2} \, ...
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Is there any rational map from the nonsingular Segre quadric surface in $\mathbb{CP}^3$ to a nonsingular surface of degree greater or equal to 4?

Is there any rational map from the nonsingular Segre quadric surface in $\mathbb{CP}^3$ to a nonsingular surface in $\mathbb{CP}^3$ of degree greater or equal to 4? Someone told me that the answer is ...
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Why is $f(n) =\frac{n(n+1)(n+2)}{(n+3)}$ in $O(n^2)$?

Let: $$f(n) = n(n+1)(n+2)/(n+3)$$ Therefore : $$f∈O(n^2)$$ However, I don't understand how it could be $n^2$, shouldn't it be $n^3$? If I expand the top we get $$n^3 + 3n^2 + 2n$$ and the biggest ...
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Integral of an exponential of rational function

I have an integral of the form $\int_{a}^{b} \text{exp}\left(\frac{\lambda}{\rho^2 m + \sigma^2_u}\right) \frac{1}{m^2}\text{exp}\left(-\frac{\lambda}{m}\right) dm$. Can this integral be found ...