Rational Definition and 632 Threads

5x^2 + 8/x3 + x2 I got a 5x^2 + 8 = A/x^2 + B/x + 1 A(x+1) + B(x^2) (Ax+A) + (Bx^2) (Ax + Bx^2) + A 5x^2 + 8 = (A + B) x + (A)x^2 5x^2+x+8=(A+B) + (A) This is about as far as I can get but I think I made a mistake somewhere but I don't know where? Can someone help me?

given a rational function R(x,y,z) that has no poles on the R^3 plane my question is if for big (x,y,z) \rightarrow \infty it has the following asymptotic expansion R(x,y,z) \sym \sum_{m,n,l= -\infty}^{N}a_{m,n,l}x^{m}y^{n}z^{l}

Hi, I'm just new here, I don't know if I'm on the right thread.:D Homework Statement Let F be a subfield of K. A, B be elements of Mn(F). Show that if A and B are similar over K, then A,B are similar over F. (Hint: what can be said about the rank of f(C(f(x)^m))^n? about the rank of...

Trying to help out a friend. I appolagise if yet again this is in the wrong part of the forum, i haven't an idea what it is categorised as, I am an apprentice engineer and simply that lol Can someone explain how the following is done: x^2 – x + 1 / (x - 2)(x^2 + 1) ≅ A / (x - 2) + Bx +...

I have the following function to differentiate: f(x) = x^\frac{4}{5} * (x-4)^2 My derivative is: f'(x) = x^\frac{4}{5} * 2(x-4) + (x-4)^2 * (\frac{4}{5}) * x^\frac{-1}{5} My calculator gives me: f'(x) = \frac{2(x-4) * (7x-8)}{5x^\frac{1}{5}} How does my expression simplify into...

Hi everyone, this isn't really a homework question- this integral has come up during project work- but this seems like a sensible place to ask it nontheless. Homework Statement Compute I(t)=\int^{\infty}_{0} dx \frac{x^5}{(t+x^2)(1+(t+x^2)^4)} where t is real and positive. Homework...

Homework Statement The actual problem is: "Does x^2+y^2=3 have any rational points? If so, find a way to describe all of them. If not, prove it."Homework Equations NoneThe Attempt at a Solution I found a book on Google Books (can't find it again) that said that this circle has no rational...

Homework Statement write out the form of the partial fraction decomposition of the function, do not determine the numerical values of the coefficients x^2/(x^2 + x + 2) Homework Equations The Attempt at a Solution since the numerator is not less of a degree than the...

How can I find the values for ψ that the following sqaure root equals to a rational number?: √(87600-1.44ψ^2) Also, I don't want ψ to be greater than 10. What's the procedure to find the answer?

Im preparing for a CLEP test in precalculus. As part of my prep, I need to review identifying domains of functions. I have a question about writing domains in standard notation. I was hoping someone could explain a bit the style. For an example: x-2 / x^2 -2x -35 As a rational...

Integration of Rational Functions by Partial Fractions? Ok I'm working on some homework problems and I don't even know how to do the first one, here is my problems and the steps that I did thus far ( I don't know if I did them right) 5x-13/(x-3)(x-2)= A/x-3 + B/x-2\rightarrow ...5x-13=...

Homework Statement the title says it all x \rightarrow 4 for f\left(x\right) = \frac{\sqrt{1+2x}-3}{\sqrt{x}-2} I have multiplied both top and bottom by conjugate, \sqrt{x}+2: f\left(x\right) = \frac{\sqrt{x(1+2x)}+2\sqrt{1+2x} -3\sqrt(x)-6}{x-4}but don't know how to take this further...

graph/analyze a function of a rational /w complex root?!? Homework Statement The function is y=x^2/(x^2+3) Homework Equations First and Second Derivatives Chart to find intervals of increase/decrease and concavity. The Attempt at a Solution 1) Domain {XeR} 2) Intercepts If...

b]1. Homework Statement [/b] lim f(x) = 2 x -> infinty lim f(x) = -2 x-> - infinty lim f(x) = - infinty x-> -4 lim f(x) = - infinty x-> 2- lim f(x) = infinty x-> 2+ relative min of 0 at x=2 relative max of -0.900466 at x=0.442818 concave down (-infinty, -4) (-4,-2)...

How can i prove that addition and mulitiplication of rational numbers follow commutative and associative law?

Homework Statement Let x_{1}, x_{2}, ... be a sequence of rational numbers in which each rational number in (0,1) occurs exactly once. Define the function, H(x) = 0 if x \leq 0, and 1 if x > 0. Next, define the function F(x)= \sum^{\infty}_{k=1} 2^{-k} H(x - x_{k}). Prove that F is...

Homework Statement Prove Proposition 1.15. Proposition 1.15. If a and b are real numbers satisfying a<b, then there are rational numbers and irrational numbers between a and b. Homework Equations Professor said to use the Archimedean property The Attempt at a Solution a < b...

Homework Statement f(x) = \frac{x}{x-1} + \frac{x+1}{3x} Homework Statement need to take the first derivative of this expression... I can do it but I am curious as too why i cannot take the derivative of \frac{x}{x-1} and then just add it to the derivative of \frac{x+1}{3x}...

Homework Statement Suppose that r is a solution of the equation: anxn + a(n−1)x(n−1) + . . . + a1x + a0 = 0 where the coefficients ak belongs to Z for k = 0, 1, . . . n, and n is greater or equal to 1. If r is a rational solution r = p/q, where p, q belong to Z and p and q are...

A long time ago I took a number theory course and really enjoyed it. At one point we were shown the proof for the theorem that a number is rational if and only if it has a periodic decimal expansion. The (<=) direction is really easy if you know some Calculus, but I remember the (=>) direction...

Prove that Cl(Q) = R in the standard topology I'm really stuck on this problem, seeing as we haven't covered limit points yet in the text and are not able to use them for this proof. Can anybody provide me with help needed for this proof? Many thanks.

Homework Statement Prove via Mathematical Induction that, The sum of n rational numbers is rational. Homework Equations The Attempt at a Solution Let N = 1 The sum of one rational number is the number it's self, which is a rational number. Assume when n = n, the sum of n...

Homework Statement 1. x-2|x| < 3 Homework Equations The Attempt at a Solution Okay, I attempted this equation below but I don't know if I'm right. x-2|x|- 3 < 0 x^2 - 2x - 3 < 0 (x - 3) (x + 1) < 0 x = 3 x = -1

Homework Statement Show that R(x) cannot be made into a complete ordered field, where R(x) is the field of rational functions. Homework Equations Definition of a complete ordered field: An ordered field O is called complete if supS exists for every non empty subset S of O that is...

I've been recently reading a book on abstract algebra and number theory, and I stumbled upon a problem that at first glance looked obvious, but I can't seem to figure out how to formally write the proof. 1.)So, let's say we have 4 integers, r,s,t,u, all greater than or equal to 1. Suppose...

I've been skimming the paper of A.N Schellekens (http://arxiv.org/abs/0807.3249) that discusses the notion of string theory landscape. He argues against uniqueness, and seems to think that the existence of a landscape of possibilities adds explanationary value and is a good thing. Surprised...

how could we calculate the follwing integral ?? \int_{0}^{\infty} \frac{ K(x)}{Q(x)}dx here K(x) and Q(x) are POLYNOMIALS , of course if we had an integral over all R instead of (0 , \infty ) we could apply Cauchy's residue theorem i think there is a 'closed circuit' to perform the...

Homework Statement Show that the set of rational numbers in the interval (0, 1) cannot be expressed as the intersection of a countable collection of open sets. Homework Equations The Attempt at a Solution This sounds like something requiring proof by contradiction. There must be...

Homework Statement Show that the function f(x) = { x/2 if x is rational { x if x irrational is not differentiable at 0 Homework Equations If f is differentiable at 0 then for every e > 0 there exists some d > 0 such that when |x| < d, |(f(x)-f(0))/x - L | < e...

Are whole numbers rational numbers? If so, will any whole number divided by any other whole number result in a rational number?

Hi guys: I've got a problem I've been working on for some weeks and this might be the key to unlocking it. The question is: Given a vector in R^k, what is the measure of the set of vectors whose components are rationally dependent? Rationally dependent means for a given vector, you may...

Homework Statement Find the rational number representation of the repeating decimal. 1.0.\overline{36}Homework Equations The Attempt at a Solution I know it has something to do with infinite geometric sequences but I'm not sure what. what would your ratio be for a repeating decimal, I've...

Homework Statement The definite integral of (t^3 + t -1)/(sin(t)) from 2 to x^2 Homework Equations The Attempt at a Solution First off, I don't have the solution anywhere, my teacher just gave this to us to work on for the final exam review. I can think of a few things. I...

Homework Statement \int \frac{4x^5-1}{(x^5+x+1)^2} dx = ? Homework Equations The solution is - \frac{x}{x^5 + x + 1} The Attempt at a Solution Other than getting lucky and noticing immediately that this could be the derivative of a fraction, I do not see an easy way to solve this. The...

Homework Statement find \int\frac{x^{2}-2x-1}{(x-1)^{2}(x^{2}+1)}dx Homework Equations None The Attempt at a Solution \frac{x^{2}-2x-1}{(x-1)^{2}(x^{2}+1)} = \frac{A}{x-1} + \frac{B}{(x-1)^{2}} + \frac{Cx + D}{x^{2}+1} x^{2}-2x-1 = A(x-1)(x^{2}+1) + B(x^{2}+1) + (Cx + D)(x-1)^{2}...

Homework Statement "Prove that the sum of two rational numbers is a rational number." I just started on proof writing, so I'll just like to verify if I'm not missing anything here, and get some comments about the style. The attempt at a solution Theorem. If a,b \in \mathbb{Q} then a+b \in...

Ok so, This summer I will be taking a Pre-calc/trig course intensive, to get ready to take calculus in the fall, to start up my track for physics. I got a Pre Calculus Workbook For Dummies and I have to say so far I'm not too pleased. I have already found a bunch of typos, and when there...

Is there a way ( a theorem ) to find a rational number for a given irrational number such that it is an approximation to it to the required decimal places of accuracy. For example 22/7 is an approximate for pi for 2 decimal places.

Hi there I have a Rational function y = 1 / x^2-1 . I have a good idea what the graph looks it, it will have vertical asymptotes at -1 and 1 and I can work out the y intercept (-1 concave down). However I'm not sure about the other parts to the question. Homework Statement dy/dx...

Homework Statement Find the critical numbers of the function: F(x) = x^(4/5) (x - 4)^(2) Homework Equations None. The Attempt at a Solution I differentiated and got to (1 / 5th root of x) (x - 4)(2x + 4/5(x-4)) but I don't know how I can simplify the expression to be able...

Homework Statement Solve for x. Homework Equations 3/2 + 2/2x-4 = 1/x-2 The Attempt at a Solution LCD = 2(x-2) 3/2 + 2/2x-4 = 1/x-2 Mult. all terms by 2(x-2) 3x - 6 + 2 = 2 3x = 6 x = 2 What am I missing here?

Homework Statement For all a in the set of real numbers, if a is rational, a + \sqrt{2} is irrational. You may use that \sqrt{2} is irrational and the sum and difference of rational numbers is rational. Homework Equations The Attempt at a Solution My proof seems way too simple, I don't trust...

Homework Statement Hi, I have this function: f(x ) = 0 (x is irrational) or f(x) = 1/q for rational p/q in lowest terms. show that this function is not differentiable anywhere The Attempt at a Solution This is the answer from the solutions book: consider [(f(a+h) - f(a ) ) /...

Hey all, I'm new here so I'm a little noobish at the formatting capabilities of PF. Trying my best though! :P Homework Statement Let a, b, c, d \in Q, where \sqrt{b} and \sqrt{d} exist and are irrational. If a + \sqrt{b} = c + \sqrt{d}, prove that a = c and b = d. Homework...

How to state the equations of a rational functions with the following asymptotes? (1)x=2, y=-3 (2)y=0, x=4 (3)y=0

How do you multiply an equation that has 2 or more numerators. such as : 8x + 8 x - 1 ______ X _______ X2 - 2x + 1 2x + 2 And don't say anything like, "I'm not doing your homework for you" or anything stupid like...

Homework Statement Write the partial fraction decomposition of the rational expression. Check your result algebraically. (x2 – 7x + 16)/[(x + 2)(x2 – 4x + 5)] The Attempt at a Solution [A/(x+2)] + [(Bx+C)/(x2-4x+5)] x2-7x+16= A(X2-4x+5)+(Bx+C)(x+2)...

I have been playing around with numbers in different bases and then I thought, what if they were in fractional bases. I found a way to convert numbers to fractional bases and have been searching on the internet and not found a similar way to do this. Anyway, here is an example of how I would do...

Hello, I am trying to prove the following... lim (x+3) \left|x+5\right|/x+5 x\rightarrow-5 from the left, L=+2 from the right, L=-2 I used delta-epsilon on the right hand limit and got \delta = \epsilon However, I'm not sure how to proceed when I get to this step while trying to prove the...

In the field of formal rational functions, construct a nest of closed, bounded intervals whose intersection is empty. (That is, show that the Nested Intervals property fails in this field) I know it has to involve radical 2 but because that is the only number we know is irrational but other...