Polynomial Definition and 1000 Threads

Homework Statement I have a limit problem, however I do know how to work limits, I guess what I need is more of a refresher on how to work third degree polynomials. The polynomial(s) I am trying to work with are the following: x3-2x2+2x-15 -and- x3-5x2+10x-12 The limit is a limit where x...

So I was just wondering why when you approximate using the Taylor Polynomials for something like e^x/x at x = 0 you can just find the approximation for e^x and make it all over x, could you do the same for like e^x/x^2 or e^x/x^3? I hope my question makes sense... thanks

Homework Statement Let Tn(x)=cos(narccosx) where x is real and belongs to [-1,1] and n E Z+ Find T1(x). Show that T2(x)=2x^2 - 1. Show that Tn+1 (x) + Tn-1 (x) = 2xTn(x) Hence, prove by induction that Tn(x) is a polynomial of degree n. The Attempt at a Solution Since cosθ=x and arccosx=θ we...

Homework Statement Homework Equations The Attempt at a Solution I keep getting the answer 1- 96/4!*x^4. Can't find where I'm going wrong.

I'm working on the following: "Prove that x^2 - 1=0" has infinitely many solutions in the division ring Q of quaternions." The Quaternions are presented in my book in the representation as two-by-two square matrices over ℂ. The book gives that for a quaternion (sorry for the terrible...

Homework Statement Let h: \mathbb{P_2} \rightarrow \mathbb{P_2} represent the transformation h(p(x)) = xp'(x) + p(1-x) for every polynomial p(x) \in \mathbb{P_2}. Find the matrix of h with respect to the standard basis \{1, x, x^2\} Homework Equations Matrix A of transformation: {\bf A}...

Hi all, I've learned that in unconstrained polynomial regression, the optimal order can be determined using two F tests : one to test for the significance of the overall regression, the other to test for the significance of the higher coefficients (assuming the first test passed of course)...

Hi. Can someone explain to me how to sort a Series so that the terms are in increasing powers of the exponent? For example the code: myseries = Normal[ Series[Sqrt[1 - w], {w, 0, 5}]] /. w -> 1/z produces 1-\frac{7}{256 z^5}-\frac{5}{128 z^4}-\frac{1}{16 z^3}-\frac{1}{8 z^2}-\frac{1}{2...

Homework Statement Suppose that A is a 6x6 matrix with real values and has a min. poly of p(s) = s^3. a) Find the Characteristic polynomial of A b) What are the possibilities for the Jordan form of A? c) What are the possibilities of the rank of A? Homework Equations See below...

Homework Statement Factor x^3+3x+6 over Z5, Z10, Z, Q and ℝ. 2. The attempt at a solution For Z5, I have the roots of x^3+3x+1 in Z5, x=2=-3 and x=3=-4, so x+3 and x+4 are factors. By long division of x^3+3x+1, it is found that x+3 is a repeated factor, so x^3+3x+6=(x+4)(x+3)^2. For Z and...

xy'^2 + yy' = 0 where y' = dy/dx The answer is C1 = y and C2 = xy but I get this: y'(xy' + y) = 0 where y' = 0 and thus y = C1 For the other solution: xy' + y = 0 y' = -y/x y = -y ln x + C2 C2 = y + y ln x Full question is here: http://www.cramster.com/solution/solution/640396

Homework Statement Given a random variable X with a known distribution (e.g. a beta distribution), find the distribution of f(X) = X^2 + X The Attempt at a Solution I've tried the normal approaches: the standard transformation theorem; conditioning on X; Laplace transformation, etc...

Homework Statement Prove that the polynomial $(x-1)(x-2)...(x-n) + 1$ is irreducible over Z for n\geq 1 and n \neq 4Homework Equations N/AThe Attempt at a Solution Let $f(x) = (x-1)(x-2) \cdots (x-n) + 1$ and suppose $f(x) = h(x)g(x)$ for some $h,g \in \mathbb{Z}[x]$ where $\deg(h), \deg(g) <...

I'm trying to find a general formula for an algebraic equation, I'm studying the behavior of ∏_{i=2}^n(1-\frac{1}{i^m}) for m=3 and so far I've seen that I can find a general formula if n^2 + n + 1 produces only prime numbers. if not, it would get way harder to find a general formula for it by...

Hi guys I'm getting into a little trouble when differentiating polynomial functions. How do you differentiate f(x)=ax+b/cx+d ? Is there other ways of calculating this apart from the chain rule ? Thanks for any help.

My quantum text, leading up to the connection formulas for WKB and the Bohr-Sommerfeld quantization condition states that for \begin{align}u'' + c x^n u = 0 \end{align} one finds that one solution is \begin{align}u &= A \sqrt{\eta k} J_{\pm m}(\eta) \\ m &= \frac{1}{{n + 2}} \\ k^2 &=...

For #4, I'm mostly confident I did it correctly. In determining the error, we're supposed to find the maximum absolute value on an interval I. I set I = (0,2pi). Is that right? http://i111.photobucket.com/albums/n149/camarolt4z28/4-1.png For #5...

Homework Statement Let M be a subspace of the vector space \mathbb{R}_2[t] generated by p_1(T)=t^2+t+1 and p_2(T)=1-t^2, and N be a subspace generated by q_1(T)=t^2+2t+3 and q_2(T)=t^2-t+1. Show the dimension of the following subspaces: M+N, M \cap N, and give a basis for each...

The first hypothesis is that f is continuous on [a,b]... Is there a more concise mathematical way of saying... "because the function f is a polynomial it is continuous in its domain."? Because I rather not write that on my test it looks sloppy and non professional...

As part of solving a DE I need to make this friendlier to integrate: \frac{2u}{1+u^{-2}} I figured trying to divide it couldn't hurt. I got: u^{-1}+u^{-3} I can't type out all the steps easily, I'm on a mobile device at the moment. That answer looks suspect, did I do it correctly?

Homework Statement Prove, using Rodrigues form, that Hn+1=2xHn -2nHn-1 Homework Equations The Rodrigues form for Hermite polynomials is the following: Hn = (-1)nex2\frac{d^n}{dx^n}(e-x2)The Attempt at a Solution Hn+1 = (-1)n+1ex2\frac{d^(n+1)}{dx^(n+1)}(e-x2) where...

Homework Statement f:ℝ→ℝ x:→sqrt(x^2+1)-lxl Homework Equations calculate f^-1(]0,1]) The Attempt at a Solution well i chose a y from ]0,1] and tried to find an x that solves the problem like the following. 0<sqrt(x^2+1)-lxl≤1 at the end i got the following...

Homework Statement the question asks to find the quadratic maclaurin polynomial for f(x) Given f(x) = x sin(x)The Attempt at a Solution i know that a maclaurin series is when a=0 in a taylor series. i did the 1st-5th derivatives of f(x) and then used the formula for taylor polynomial and set...

I've heard there's a proof out there of this, basically that (I think) you can use the intermediate value theorem to prove that an Nth-degree polynomial has no more than N roots. I'm not in school anymore, just an interested engineer. Does anyone know where I can find this proof or any...

Homework Statement http://img213.imageshack.us/img213/7867/problemeigen.png Uploaded with ImageShack.us Homework Equations http://img339.imageshack.us/img339/771/equationh.jpg Uploaded with ImageShack.usThe Attempt at a Solution so i tried doing the problem i going to use L for lambda i...

Homework Statement f:[1,+∞[→ℝ x→sqrt(x+2)-sqrt(x-1)Homework Equations show that f([1,+∞[)=]0,sqrt(3)]The Attempt at a Solution Any tips on how to start it.

Homework Statement f:]1,+∞[→]2,+∞[ x→ 2x/(x-1) Homework Equations Prove that f is injective and serjective. The Attempt at a Solution I already proved that it's injective by stating the injectivity law: for every (a,b)ε]1,+∞[: f(a)=f(b) implies a=b so: 2a/(a-1)=2b/(b-1)...

Is there a good way to do this? I have an equation, say x^3 - 4*x^2 + 2, so a=1, b=-4, c=0, d=2. Is there an easy way to express the largest root of such an equation? In this case, the roots are: 3.8661982625090223 -0.65544238154983 0.7892441190408067 But I am trying to find an...

Homework Statement What is the minimal degree Taylor polynomial about x=0 that you need to calculate sin(1) to 3 decimal places? 6 decimal places? Homework Equations R_nx = f^(n+1)(c)(x-a)^(n+1)/(n+1)(factorial) The Attempt at a Solution I have attached my attempt. I am stuck on the...

I am currently working on a gamma ray spectroscopy lab in which i have just fit a polynomial to my calibration points. The calibration points are in a relatively straight line, from x=40 to x=450, and y=34 to y=1300 for the first and last end points respectively. Where X is channel number, and Y...

I am trying to get MATLAB to be able to produce a symbolic polynomial and am having some issues. The polynomial I would like it to produce is given a d and k, X(k+1)-X(k-d)-X(k-d-1)-...-1 What I have so far is k = 5; d = 3; syms X; i = 0; while (k-(d+i) >= 0) terms(i+1) =...

If you have a polynomial like (1+x)^6 = x^6+6 x^5+15 x^4+20 x^3+15 x^2+6 x+1, What function would you use to pick only the last N terms? For instance, for N=3 pick x^6+6 x^5+15 x^4 I've being using sum of a binomial times something, but was wondering if there is anything more simple. Thanks!

Hi friends, I have been trying to solve the ode of second degree below with respect to z: d2y/dz2=(i/a*z+b)*y i is the complex i, a and b are constants i ended up with the summation of bessel functions of first end second kind. Then I checked with MATLAB ode solver it gives no...

Homework Statement Let T: P2-P3 be the linear transformation defined by T(p(x)) = xp(x). Which of the following are in R(T)? (a) x2 (b) 0 (c) 1 + x Homework Equations R(T) is the the set of all vectors in P3 which are images under T of vectors in P2. The Attempt at a Solution...

Homework Statement Find all zeros of k(x) = 4x^4-7x^2+3x Homework Equations n/a The Attempt at a Solution I don't know if I should find the factors of 0 and 4.. Can you help me what to start out with.. I can do other ones like p(x) = 3x^3-37x^2+84x-24. But this problem k(x) =...

I need help from someone with a graphing calculator. My little brother lost my graphing calculator and synthetic division I can't make work becasue the terms aren't exactly friendly. The equation is, 24.059025x^4 + 94.98287069x^3 + 44.56696001x^2 - 827.1103204x = -1067.832496 The more...

Homework Statement Hi every body I am triyng to find a polynolial approximation to the function: f(x)= (x+2)ln(x+2) using the chebyshev polynomials, the idea is to use MATLAB to find the coeefficients of the approximation poly. using the comand double(int(...)) but this command...

Homework Statement Find the polynomial p(x) of degree 20 satisfying: p(-10) =p(-9) = p(-8) = ...=p(-1) = 0 p(0) = 1 p(1) = p(2) = p(3) = ...p(10) = 0 Homework Equations L(x) := \sum_{j=0}^{k} y_j \ell_j(x) The Attempt at a Solution i tried using the formula above: a =...

Homework Statement Given 2 + 3i is one root of the equation: x^4 - 6x^3 + 26x^2 -46x +65 = 0 Find the remaining roots. The attempt at a solution I am thinking about this as a possible solution although it is too long to be plausible, unless I'm wrong. let x = a + bi and then replace in the...

For any characteristic polynomial determined from A - eI (where A is a nxn matrix, e is an eigenvalue and I is the identity matrix), is it a rule that the coefficient associated with the char. polynomail term of highest degree must be positive ? My tutor made a theory that if the...

Homework Statement A quartic function has x-intercepts of -5, -1, 2, and 4. The point (3.-8) lies on the curve. Determine an equation of the function. Homework Equations The Attempt at a Solution y = k (x+5)(x+1)(x-2)(x-4) Expanded form: f(x)=kx^4 + 9x^2 - 38x + 40 <---My...

Homework Statement 4x5-16x4+9x3+23x2-15x-9 > 0 Homework Equations Synthetic division PQ Rule? The Attempt at a Solution Don't know how or where to begin

Homework Statement 2 People build a sandcastle of 36cm by 48cm by 60cm. Later they learn that the box will be smaller; a certain amount will be taken off each of the length width and height Homework Equations The Attempt at a Solution a) write a p(x) > (36-x) (48-x) (60-x)...

Hello, I'm new here, nice to meet you guys i was in class today and just didn't understand the taylor polynomial approximation, the professor started out approximating a function by polynomials of degree N, he first showed us how a linear polynomial was a crude approximation of the function...

Homework Statement Kaylas lemonade stand has experienced rises /falls in sales caused by temperature changes over 2 summers. Her sales over the first two weeks of summer were tracked according to the model S(x) = x^3 -12x^2 +36x, where x is the number of days and S(x) is the number of sales...

Ive attached the problem and my work in the pic. Questions: Am I even applying the taylor polynomial the correct way? (I never learned taylor series, but I was supposed to be taught in the pre-requisite class) Am I suppose to plug in c=4? I am not so sure about how the U4(t) works...

Homework Statement Why is this not a polynomial function? h(x) = 3x + 2x Homework Equations The Attempt at a Solution 3x+2x = 5x. 5x is a linear function with a degree of 1, why is this not a polynomial funct?

Homework Statement Determien an euation for each polynomial function described below state whether the function is even, odd or neither. Sketch a graph of each. Homework Equations The Attempt at a Solution My question is.. If youre given information like this :: a) A quintic...

Homework Statement What is the difference between f(x)= x^3 , and g(x) = x^3 - x What does that -x do to the function and how do you represent it graphically? What about f(x) = 3x^3 +x Homework Equations The Attempt at a Solution No idea, wolframalphad the graph but i...

Homework Statement We know that a polynomial function is anything with a positive exponent and a rational number. a) - / x^3 -4 d) 3x^-1 - 11 g) y= [sqrt of term]3x^2 -5x Homework Equations The Attempt at a Solution c) 1 / x^3 - 4 (do we use exponents to verify ?)...