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

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Rational Funtion Integration

This looks to be a simple problem, but it has me stumped. I already have the answer, but a step-by-step solution would be appreciated. $$\int\frac{x+4}{x^2+2x+5}$$
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What are the asymptotic considerations in the following?

The following is from this paper that discusses polynomials and classic number theory functions. The proof of theorem 1.3 has a final statement saying that $R$ must be null because we arrive at ...
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Range of a Rational Function

How to find the Range of function $$f(x)= \frac{x^2-3x-4}{x^2 - 3x +4}$$ I tried to equate the expression to $y$, then cross multiplied $$ y= \frac{x^2-3x-4}{x^2 - 3x +4}$$ $$ y(x^2 - 3x +4)= ...
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Solving a rational equation with multiple and nested fractions

This is the equation to solve: $\dfrac{\dfrac{x+\dfrac{1}{2}} {\dfrac{1}{2}+\dfrac{x}{3}}}{\dfrac{1}{4}+\dfrac{x}{5}}=3$ What I did: $x+\dfrac{1}{2}=\dfrac{2x+1}{2}$ ...
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Rational Expression Subtracting

Did I do this correctly? Because the length of the line $\overline{AB}$ is negative. My work:
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finding the behavior of the asymptotes in a rational function

I'm having trouble understanding how to graph this function: $f(x) = \frac{x-2}{(x-4)(x+4)}$. The part I undertand: The x-intercept is (2,0) since x=2 makes the numerator zero. The y-intercept is ...
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Extracting variable from a couple of rational functions

Let $P$ and $Q$ be two rational functions of $z$ (coefficients over $\mathbb{C}$). How can one decide whether $z = R(P(z), Q(z))$ for some rational function $R$ of two variables, and if it is the case ...
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Trouble Solving a system of 3 equations

I'm having trouble solving a system of 3 equations. The set of equations in question is shown below $C_a=\frac{R_a}{\frac{R_a}{r_a}+\frac{R_b}{r_b}+\frac{R_c}{r_c}}, \quad ...
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Polynomial inequalities vs rational inequalities

A question from one of the comprehension questions I have is: How would the intervals of the solution set differ between a polynomial inequality and a rational inequality? I have tried to research ...
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Factoring Trick - Adding Up Coefficients

My professor told me this for factoring polynomials: Add up the coefficients and if they equal 0 then the polynomial has root of 1. Add up, but switch the signs of the coefficients with odd ...
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A question about the domain of definition of rational functions

Is the function $f(x) = \frac{x^2}{x}$ defined for every $x \in \mathbb R$, or only defined on $\mathbb R \setminus \{0 \}$? Background: Say we are given $P(x) = x^2 - 4x + 3$ and $Q(x) = (x - ...
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How can one calculate the limit of $\frac{1}{x^2-9}$ as x approaches -3 and 3 by hand? [closed]

Reviewing math for college after a gap year and so I know this is probably a pretty elementary question, but let me know if it has any interesting implications or alternative solutions or if it ...
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Solve for $y$: $\frac{y+1}{y-1} = 10^{x^2}$ [closed]

Can someone please show me the steps (all of them… yeah, even the obvious ones) to go from $$\begin{align}\frac{y+1}{y-1} = 10^{x^2}\end{align}$$ to ...
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How to integrate $\int\frac{3x+2}{x^2-x-2}dx$

This is the indefinite integral I have to evaluate: $$\int\frac{x^3}{x^2-x-2}dx$$ so by using the long division on polynomials technique, I got to: $$\frac{x^2}{2}+x+\int\frac{3x+2}{x^2-x-2}dx$$ How ...
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Complex analysis: Prove a meromorphic function to be rational.

I come across a problem about complex analysis: Show that a meromorphic function on the complex plane, which achieves any complex number no more than fixed given times, must be rational. The only ...
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How do you determine the end behavior of a rational function?

Example $$\frac{6x + 2}{x^2 - 9} = \frac{6x + 2}{(x + 3)(x - 3)}$$ I know how to find the vertical and horizontal asypmtotes and everything, I just don't know how to find end behavior for a ...
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How to convert the parametric equation into implicit form?

This problem is generated from another Green's theorem related question of mine. The original equation of the plane curve is not in rational parametric form. In order to calculate the symbolic ...
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Generating function of Language is rational

Let W be the set of all words over an alphabet $\Sigma$. Let $$L=\{w\in\Sigma^* | w\neq uvu',\text{ with }u,u'\in\Sigma^*,v\in W\}$$ I have to show that the generating function of L is rational. My ...
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Properties of rational polynomials

I have experimental data points that can be modeled by two different rational polynomials. I am wondering if there is a way (e.g. by a transform or integral), to discriminate the following two ...
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Degree theorem for Runge's approximating rational functions

Suppose that $f$ is analytic on an open set $D\subset\mathbb{C}$, and one uses Runge's theorem to obtain a sequence of rational functions $\{r_n\}$ which approach $f$ uniformly on compact subsets of ...
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Finding a (nonidentity) rational map of the plane with period $7$

Does there exist a nonidentity (which also is not a rotation) rational map $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ with period $7$, i.e., for which the seventh iteration $f^7$ is the identity ...
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Why is this horizontal asymptote present and how do I immediately see that from the equation?

This may seem like a stupid question, and I do feel like I should know this. I have been given a simple curve with the following equation and was asked to state the equation of the asymptote of the ...
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Integral of derivative of rational map on unit disk

Let $f:D \rightarrow D$ be a surjective rational map of the unit disk of degree $n$. Prove that $$\iint_D |f'(x+iy)|\:\mathrm{d}x\:\mathrm{d}y\leq \pi \sqrt{n}.$$ Attempt: We know that rational maps ...
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Using the triangle inequality to bound $\frac{x^3 + 3x + 1}{10-x^3}$ for $|x+1|<2$

How do I use the triangle inequality to bound the function $$f(x) = \frac{x^3 + 3x + 1}{10 - x^3}$$ on the interval $|x+1|<2$? I understand how the triangle inequality works, but using fractions ...
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Derivative of Function with Rational Exponents $f(x)= \sqrt[3]{2x^3-5x^2+x}$

I have a question following: $$f(x)=\sqrt[3]{2x^3-5x^2+x}$$ Here's what I did, $$f(x)=\sqrt[3]{2x^3-5x^2+x} \\ = (2x^3-5x^2+x)^{3\over2} \\\\f'(x) = {3\over 2}(2x^3-5x^2+x)^{3\over2}(6x^2-10x+1)$$ ...
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Finding the partial fraction decomposition of $\frac{4s^2 - 5s + 2}{s^2(s^2 +9)}$

I am trying to find the partial fraction decomposition of $\dfrac{4s^2 - 5s + 2}{s^2(s^2 +9)}$ into something of the form $A\dfrac{1}{s} + B\dfrac{1}{s^2} + C\dfrac{1}{s^2+9} + D\dfrac{s}{s^2 + 9}$. ...
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How do I rewrite this rational expression?

How do I rewrite the rational expression: $$\frac{x^3+5x^2+3x-10}{x+4}$$ But in the form of: $$q(x) + \frac{r(x)}{b(x)}$$
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Solve the non-linear system of equations

For real $x,y,z>0$ solve the system of equation \begin{cases} \dfrac{1}{x}-3 y+4 z=5,\\ \dfrac{1}{y}-4 z+5 x=3,\\ \dfrac{1}{z}-5 x+3 y=4, \end{cases} It is easy to check out that $$ x ...
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Rational function regression without poles in a interval, or polynomial regression with positivity constraint

I have some sets of experimental data for some functions $f_i$ from $I=[0,1]$ onto itself, which should satisfy the following physical constraints: $f_i(0)=1$ $f_i(x) \in I= [0,1] \; \forall x \in I ...
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Evaluate the integral $\int \frac{dx}{x^3 + 2x^2 + 2x}$ of a rational function

Evaluate $$\int \frac{dx}{x^3 + 2x^2 + 2x}.$$ I have no idea how to approach this. I know how to solve rational functions with numerator as highest degree polynomial using division and remainder. ...
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$\mathcal{Z}$-transform of differential equations $y(n+2)-3y(n+1)-10y(n)=(-2)^n$

Is defined function: $$y(n+2)-3y(n+1)-10y(n)=(-2)^n$$ with conditions: $$y(0)=0, y(1)=0 $$ And my solution is (Z-transform): $$\mathcal{Z}\{y(n+2)\}=z^2Y(z)-0z^2-2z=z^2Y(z)-2z$$ ...
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What is that function? Polynomial?

Is it a polynomial or rational polynomial or else? $y = \dfrac{a}{x^4} + \dfrac {b}{x^2} + c$ I need to fit a curve to a discrete data of that form, so I need to know what fitting to use.
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Solve the following integral: $ \int \frac{x^2}{x^2+x-2} dx $

Solve the integral: $ \int \frac{x^2}{x^2+x-2} dx $ I was hoping that writing it in the form $ \int 1 - \frac{x-2}{x^2+x-2} dx $ would help but I'm still not getting anywhere. In the example it was ...
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Partial fractions expansion problem $\frac{x^3-1}{4x^3-x}$

I want to calculate integral of the fraction, but first how to find the partial fraction expansion of $\frac{x^3-1}{4x^3-x}$. How to expand denominator? I am a bit lost here.
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What is the limit of a rational function as it approaches its vertical asymptote?

For example, take the function $f(x)=\frac{1}{(x-3)^2}$. What is the the limit as x approaches 3? (sorry, I don't know how to format this question) My teacher says that there is no limit at x=3, ...
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Is long division must in integrating improper rational functions

I came across an integration question, which I tried to solved through substitution, but my answer is wrong. I entered the same question in Wolfram Alpha engine and saw the "Step-by-Step" solution ...
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Where to find an algorithm for decomposing rational functions into elementary fractions?

Specifically I need to decompose $\frac1{(1-x)(1-x^n)^2}$ into $\frac{f(x)}{(1-x)^3}+\frac{g(x)}{1-x^n\vphantom{()^2}}+\frac{h(x)}{(1-x^n)^2}$ where $f(x)$, $g(x)$, $h(x)$ are polynomials. Surely ...
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Defining the rational function field in n variables.

Reading over an editing my dissertation "Elementary functions" and i am having trouble with my definition of a rational functions in n variables, this is what i have written but its missing one part: ...
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Rational Exponents

I'm just checking to see if have the correct answer because my teacher didn't give us an answer key and i like to know that I have done one question properly before doing the rest. Evaluate. ...
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Limit as $x$ tend to zero of: $x/[\ln (x^2+2x+4) - \ln(x+4)]$

Without making use of LHôpital's Rule solve: $$\lim_{x\to 0} {x\over \ln (x^2+2x+4) - \ln(x+4)}$$ $ x^2+2x+4=0$ has no real roots which seems to be the gist of the issue. I have attempted several ...
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How do I know if a fractional linear transformation exists?

I have a feeling I'm missing another obvious point about FLTs. How do I know if a specific fractional linear transformation exists? I think I can find specific transformations by using the ...
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Proving a property of Fractional Linear Transformations

I'm having some trouble showing that FLTs send circles and lines to circles or lines. I know that they are compositions of linear maps and inversions. Showing that the linear maps send circles to ...
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Is $\sqrt{-x^2-\frac{1}{x}}$ a rational function?

I have to construct a rational function with the range being $[-1,0)$, which is pretty much just $-1$. I came up with the solution $\sqrt{-x^2-\frac{1}{x}}$. It works for the range, but I'm not sure ...
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Integral of rational function with trigonometric functions

$$ \int \frac{dx}{(\sqrt{\cos x}+ \sqrt{\sin x})^4} $$ I saw this problem online and it looked like an interesting/difficult problem to try and tackle. My attempt so far is to use tangent half-angle ...
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Integral with logarithm - residue

Let $R(x)$ be rational function. It is any general method to calculate $\int_{0}^{\infty}R(x) \log(x)dx$ ? I can do it in special cases, but I am looking for a general method. What should be a minimal ...
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Is algebraic closure of rational functions field Puiseaux series?

Consider a field of rational functions over algebraicly closed field. Is its algebraic closure isomorphic to Puiseaux series over the field?
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Simplify a rational expression

Suppose I want to simplify this expression: $$\frac{bx-bc-dx+ad}{a-c}$$ More specifically, I want to minimize the number of operations. Counting each addition, subtraction, and multiplication, the ...
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Rational Functions

Part 1: You are planning a school field trip that costs $120$ dollars to rent the bus. A. How much will it cost per student if $10$ students go? $12$? $15$? B. Write an equation to represent the ...
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finding the free energy of a van der waals gas (integration)

I have the following integral, $\int{ \frac{-nrtV}{(v-nb)^{2}} dV}$ could anyone tell me how to do this?
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Simplifying $\frac{1/(\frac{1}{z_1}(1-t)+\frac{1}{z_2}t) - z_1}{(z_2 - z_1)}$

This drives me mad! I am not very good in math but thought I could at least do basic things like this one, but can't figure it out and I spent a day on it. I am trying to simplify: ...