Polynomial Definition and 1000 Threads

Homework Statement Consider the followign function f(x) = x^-5 a=1 n=2 0.8 \leq x \leq 1.2 a) Approximate f with a tayloy polynomial of nth degree at the number a = 1 b) use taylor's inequality to estimate the accuracy of approximation f(x) ≈ T_{n}(x) when x lies in the interval...

Homework Statement The energy density of electromagnetic radiation at wavelength λ from a black body at temperature T (degrees Kelvin) is given by Planck's law of black body radiation: f(λ) = \frac{8πhc}{λ^{5}(e^{hc/λkT} - 1)} where h is Planck's constant, c is the speed of light, and...

How to "Offset" a polynomial Suppose I have a function for a curve, for example y=x2. I want to find a function to "offsets" it by 2 units. That is, I want a larger parabola that is exactly 2 units away from my original parabola. What I have in mind is the offset command in AutoCAD. Is...

I think I solved it a week ago, but I didn't write down all the things and I want to be sure of doing the things right, plus the excersise of writing it here in latex helps me a loot (I wrote about 3 threads and didn't submited it because writing it here clarified me enough to find the answer...

Hi all, I have been stopped by a sextic (6th degree) polynomial in my research. I need to find the biggest positive root for this polynomial symbolically, and since its impassible in general, I came up with this idea, maybe there is a way to approximate this polynomial by a lower degree...

Homework Statement Find the Taylor polynomial approximation about the point ε = 1/2 for the following function: (x^1/2)(e^-x)The Attempt at a Solution I'm trying to get a taylor polynomial up to the second derivate i.e.: P2(×) = (×^1/2)(e^-x) + (x-ε) * [(e^-x)(1-2×)/2(×^1/2)] +...

Here is the question: Here is a link to the question: Write a function for the polynomial that fits the following description.? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Recently I am studying about theorems regarding to polynomial equations and encounter the lower and upper bounds theorem. Which states that if a<0 and P(a) not equals 0, and dividing P(x) by (x-a) leads to coefficients that alternate signs, then a is a lower bound of all the roots of P(x)=0. The...

Here is the question: Here is a link to the question: Math help: factoring? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

The question is: Using the chain rule to prove that a function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ which is polynomial in the coordinates is differentiable everywhere. (The chain rule is for the use under function composition circumstances, how to apply it here to prove that the function $f$...

This is not actually a homework question, but it seemed appropriate to put it here. In an old exam from 1921 I found the following problem. I never learned how to solve this type of thing and I haven't been able to figure it out, so: how does one solve this? Homework Statement Solve for...

Homework Statement obtain the number r = √15 -3 as an approximation to the nonzero root of the equation x^2 = sinx by using the cubic Taylor polynomial approximation to sinxHomework Equations cubic taylor polynomial of sinx = x- x^3/3!The Attempt at a Solution Sinx = x-x^3/3! + E(x) x^2 =...

the error of a taylor series of order(I think that's the right word) n is given by \frac{f^{n+1} (s)}{n!} (x-a)^n I think this is right. The error in a linear approximation would simply be \frac{f''(s)}{2} (x-a)^2 My question is what is s and how do I find it. Use linear...

So I'm in a college algebra class and I know how to do polynomial long division. I'm curious as to why polynomial long division works. I've looked at some proofs, but they use scary symbols that I don't understand (I am quite dumb). Do I need very high-level math to comprehend why polynomial...

See attached PDF for details: How do I calculate errors on the fit parameters, p?

Homework Statement Give an example of a cubic polynomial, defined on the open interval (-1,4), which reaches both its maximum and minimum values. Homework Equations - The Attempt at a Solution I can see that I would need a function such that there is some f(a) and f(b) in...

Homework Statement Let T:P_m(\mathbb{F}) \mapsto P_{m+2}(\mathbb{F}) such that Tp(z)=z^2 p(z) . Would a suitable basis for range T be (z^2, \dots, z^{m+2}) ?

Let A ∈ Mn×n(F ) Why dim span(In, A, A2, A3, . . .) = deg(mA)?? where mA is the minimal polynomial of A. For span (In,A,A2...) I can prove its dimension <= n by CH Theorem but what's the relation between dim span(In,A,A2...)and deg(mA)

I encounter a problem on the fitting ability of a special class of multi-variable polynomials. To be specific, I need find whether a special class of multi-variable polynomials, denoted by p(m), where m is the number of variables, can universally and exactly fit all member in another special...

Hi, Homework Statement I am expected to show that the polynomial a1xb1 + a2xb2 + ... + anxbn = 0 has at most n-1 solutions in (0,infinity), where a1,a2,...,an are real numbers different than zero, and b1,b2,...,bn are real numbers so that bj is different than bk for j different than k...

Homework Statement http://puu.sh/1QFsA Homework Equations The Attempt at a Solution I'm actually not sure how to do this question. How do i find Δx^2. I kind of understand the question but I don't know how to prove it. I know that Δy becomes dy when the width becomes...

Hello all, I have a series of polynomials P(n), given by the recursive formula P(n)=xP(n-1)-P(n-2) with initial values P(0)=1 and P(1)=x. P(2)=xx-1=x2-1 P(3)=x(x2-1)-(x)=x3-2x ... I am confident that all of the roots of P(n) lie on the real line. So for P(n), I hope to find these roots. I...

Homework Statement Find all polynomials of the form a + bx + cx^2 that: Goes through the points (1,1) and (3,3) and such that f'(2) = 1Homework Equations a + bx + cx^2 f'(x) = x+2cx f'(2) = 2 + 4c polynomial through (1,1) = a + b1 + c1 = 1 polynomial through (3,3) = a + b3+ c3^2 = 3 The...

Hi, Homework Statement I am asked to prove that given all roots of a polynomial P of order n>=2 are real, then all the roots of its derivative P' are necessarily real too. I am permitted to assume that a polynomial of order n cannot have more than n real roots. Homework Equations...

Homework Statement Last exam in my school this exircise was given: From norweagen: " Decide the Taylor polynomial of second degree of x=0 of the function: f(x) = 3x^3 + 2x^2 + x + 1 I found the Taylor polynomial of second degree to be: 2X^2+X+1, which is correct. If I get an...

Hello everyone, first time poster, long time reader here! I'm an ex-math major and while I'm no longer pursuing a degree anymore in mathematics, I still continue onwards in my spare time trying to learn as much as I can about it because it's always been something I've enjoyed partaking in and...

Hi Let p(x,y)≥0 be a polynomial of degree n such that p(x,y)=0 only for x=y=0.Does there exist a positive constant C such that the inequality p(x,y)≥C (IxI+IyI)^n (strong inequality!) holds for all -1≤x,y≤1? The simbol I I stands for absolute value.

Homework Statement Given some ElGamal private key, and an encrypted message, decrypt it. Homework Equations Public key (F_q, g, b) Private key a such that b=g^a Message m encrypted so that r=g^k, t=mb^k Decrypt: tr^-a = m The Attempt at a Solution My problem is finding r^-a...

Ok I promise this time it is not a homework type question. If someone could direct with a name of a theorem here, then I'll go ahead and google it, otherwise have a look. I have a chunk of notes that I'm confused about. We're not shown any proofs here or any explanations, just what is seen. It...

Here is a very difficult cubic polynomial. x^3 - x - 2 = 0 I am wondering whether it is solvable or not. Please think about it.

Homework Statement Given that the minimal polynomial of a over rationals is x^4+x+8, find the minimal polynomial for 1/a over Q. Homework Equations I know there is a lot of work done out there for finding the min. polynomials of a^k for k>0, however I've never seen anything with a^k for...

Hey, guys. Having problems with this question because I don't exactly know how to begin it. Homework Statement The problem states to: "Find the Taylor polynomial of smallest degree of an appropriate function about a suitable point to approximate the given number to within the indicated...

Homework Statement I'm having a bit of trouble with this Maclaurin Series question. It should be simple enough but I can't get the answer which is given as x2. It's been a while since I've done series and my being rusty is a little annoying. Hopefully someone can help :) Consider...

For the years 1998-2009, the number of applicants to US medical schools can be closely approximated by: A(t)= -6.7615t4+114.87t3-240.1t3-2129t2+40,966 where t is the number of years since 1998. a) graph the number of applicants on 0<= t <= 11 b) based on the graph in part a, during what...

Hi. There is a polynomial f = (x^3) + 2(x^2) + 1, f belongs to Q[x]. It will be shown that the polynomial is irreducible by contradiction. If it is reducible, (degree here is three) it must have a root in Q, of the form r/s where (r,s) = 1. Plugging in r/s for variable x will resolve to r^3 +...

Hi. There is a polynomial f = (x^3) + 2(x^2) + 1, f belongs to Q[x]. It will be shown that the polynomial is irreducible by contradiction. If it is reducible, (degree here is three) it must have a root in Q, of the form r/s where (r,s) = 1. Plugging in r/s for variable x will resolve to r^3 +...

-x^3+12x+16 Every single technique I read about online of how to factor 3rd degree polynomials, it says to group them. I don't think grouping works with this. I tried but it didn't work, since there's only 3 terms. Apparently I'm not supposed to have a cubic variable without a squared...

Let k[x,y,z,t] be the polynomial ring in four variables and let I=<x,y>, J=<z, x-t> be ideals of the ring. I want to show that IJ=I \cap J and one direction is trivial. But proving I \cap J \subset IJ has stumped me so far. Anyone have any ideas?

Homework Statement Show that the descriminant of the characteristic polynomial of K is greater than 0. K=\begin{pmatrix}-k_{01}-k_{21} & k_{12}\\ k_{21} & -k_{12} \end{pmatrix} And k_i > 0 Homework Equations b^2-4ac>0 The Attempt at a Solution I have tried the following...

Homework Statement Given the function "P" defined by: P(x) := x^2n + a2n-1*x^2n-1 + ... + a1x + a0; prove that there exists an x* in |R such that P(x*) = inf{P(x) : x belongs to | R} Also, prove that: |P(x*)| = inf{|P(x)| : x belongs to |R} The Attempt at a Solution As the...

I am not going to post my question because I want to find out how to actually use the taylor polynomial and morse potential and then apply that to my problem. Say I have to approximate the morse potential using a taylor series expanding about some value. This will then find me the force...

I've been given a matrix A and calculated the characteristic polynomial. Which is (1-λ)5. Given this how does one calculate the minimal polynomial? Also just to check, is it correct that the minimal polynomial is the monic polynomial with lowest degree that satisfies M(A)=0 and that all the...

Homework Statement Knowing that the equation: X^n-px^2=q^m has three positive real roots a, b and c. Then what is log_q[abc(a^2+b^2+c^2)^{a+b+c}] equal to? Homework Equations a + b + c = -(coefficient \ of \ second \ highest \ degree \ term) = -k_2 abc = -(constant \ coefficient) =...

$\beta m^5 + m^2 + 1 =0$ How do I find the roots?

Here is the correct answer: 2x^3 - x^2 + 4x - 5 My attempt only gives me one cubed term and the other terms are also marginally off, any help on who can show me how to get the correct answer will be hugely appreciated

I am reading Jackson's electrodynamics book. When I went through the Legendre polynomial, I have a question. In the book, it stated that from the Rodrigues' formula we have Consider only the odd terms...

Homework Statement z^4 - z^2 + 1 = 0 is an equation in ℂ. Which of the following alternatives is the sum of two roots of this equation: (i) 2√3; (ii) -(√3)/2; (iii) (√3)/2; (iv) -i; (v) i/2 Homework EquationsThe Attempt at a Solution All I know is that the sum of all roots should equal 0...

Homework Statement The condition that x^4+ax^3+bx^2+cx+d is a perfect square, is Homework Equations The Attempt at a Solution If the above polynomial will be a perfect square then it can be represented as (x-\alpha)^2(x-\beta)^2 where α and β are the roots of it.This means that two...

Find the sequence (B_nf) of Bernstein's polynomials in a) f(x)=x and b) f(x)=x^2 Answers (from my textbook): a) B_nf(x) = x for all n. b) B_nf(x) = x^2 + \frac{1}{n} x (1-x) I know that the bernstein's polynomial is: B_nf(x) = \sum_{k=0}^n f (\frac{k}{n}) \binom{n}{k} x^k...

Homework Statement Find the third degree Taylor polynomial about the origin of f(x,y) = \frac{\cos(x)}{1+xy} Homework Equations The Attempt at a Solution From my ventures on the Internet, this is my attempt: I see that \cos(x) = 1 - \frac{1}{2}x^2 + \frac{1}{4!}x^4 - \cdots...