Polynomial Definition and 1000 Threads

I want to find the root(for N) of this equation: \frac{(2N-1)^2}{N(1-N)}=Ce^t The hint says "consider taking a substitution u=N-1/2" ...which is the top bit of the fraction. But what does take a substitution here mean ? This is a part of a loooong modelling problem which involved an ugly...

"Given operators σ,τ on a finite-dimensional space V, show that στ=i, and that σ=p(τ) for some polynomial p in F[x]." The first part was no problem. As for the second, I have a strong suspicion that p is the characteristic polynomial, mostly because I believe I heard of that fact before (that...

Hello, I just want to clear up a confusion. Is f(x,y) = a + bx + cy + dxy a quadratic polynomial? where x and y are variables and a,b,c,d are constants.

This is the end of a triple integration problem. I can get down to what seems like it should be a simple polynomial integral of a single variable. Yet I just can't get the numbers to work out. \int_0^5 \frac{-1}{2}x^3 + \frac{15}{2} x^2 - \frac{75}{2} x + \frac{125}{2} dx The indefinite...

I created a method for both approximating a function and extending a it's domain from a Natural to a Real Domain. Does this have a name already or any interesting application? Basically. Add polynomial of degree 0, 1, 2, 3, etc. Making at the same time the approximation function equal to f(0)...

Homework Statement Find the first 3 non-zero terms of the Taylor polynomial generated by f (x) = x^{3} sin(x) at a = 0. Homework Equations f^{n}(x) * (x-a)^{n} / (n!) The Attempt at a Solution I got the question wrong: my answer was 1/3! + 1/5! + 1/7! Here is the answer below. I...

If $p(x)$ be a polynomial of degree $98$ such that $\displaystyle p(x) = \frac{1}{x}$ for $x=1,2,3,...,98$ Then value of $p(100)=$

How would I go about approaching this problem? Given the polynomial: x^100 - 3x + 2 = 0 Find the sum 1 + x + x^2 + ... + x^99 for each possible value of x.

How would one evaluate $$\Phi = \int_{-\infty}^{+\infty} e^{-(ax+bx^2)} dx$$. I was trying to change it into a product of an error function and a gamma function, but I needed an extra dx. Any other ideas?

Let a,b,c,d be real numbers. Sauppose that all the roots of the equation $z^4 + az^3 + bz^2 + cz + d = 0$ are complex numbers lying on the circle $\mid z\mid = 1$ in the complex plane. The sum of the reciprocals of the roots is necessarily: options a) a b) b c) -c d) d ---------- Post...

In the Turing Machine, the machine accepts a word if the computation terminates in the accepting state. The language accepted by the machine, L(M), has associated an alphabet Δ and is defined by L(M) = {w \in \Delta} This means that the machine understands the word w if w belongs to the...

If $F$ is the field of rational numbers, find the necessary and sufficient conditions on $a$ and $b$ so that the splitting field of $p(x)=x^3+ax+b=0$ has degree exactly $3$ over $F$. ATTEMPT: If $p(x)$ is not irreducible in $F[x]$ then the splitting field of $p(x)$ over $F$ can have degree...

Homework Statement p(x)=((x−1)^2 −2)^2 +3. From here find the full factorization of p(x) into the product of first order terms and identify all the complex roots. Homework Equations I am having trouble doing this by hand. I know there are four complex roots but can't seem to figure out...

Polynomial function f(x)= x^3-12x^2+46x-52 A. List possible rational ZerosB. find all the zeros (real and complex) of the function (test x=2 as a rational zero using the synthetic division?

Can anyone explain ideals and polynomial rings i.e. definitions, examples, the most important theorems, etc.?

If p(x) is a polynomial such that p(0)=5 ,p(1)=4 ,p(2)=9,p(3)=20 , the minimum degree it can have

Hi guys, I'm not entirely good with factoring so I was wondering if any of you can show me how I would go about factoring this polynomial: (With necessary steps if you can Please and thanks!) 36mx + 10y - 24x - 15my

we assume a,b,c,d are unknown variables,whose solution are either 0 or 1.So a power of a variable equals to itself(e.g,a^(n)=a).Would u please help me find a proper way to sovle the following simultaneous equations whose solutions are either 0 or 1...

Is it possible to have a cubic polynomial (ax^3+bx^2+cx+d) which has three REAL roots, with one of them being +/- infinity? If there is, can you give an example? Thanks!

Homework Statement x^2-x-13/(x^2+7)(x-2) hello i am having trouble solving this problem.. could anyone please show me how to do this step by step? i know polynomial long division is required before it can be converted to partial fractions. I also know the answer is 2x+3/x^2+7 - 1/x-2...

Hi, I've been working on this question which asks to show that {{P}_{n}}(x)=\frac{1}{{{2}^{n}}n!}\frac{{{d}^{n}}}{d{{x}^{n}}}{{\left( {{x}^{2}}-1 \right)}^{n}} So first taking the n derivatives of the binomial expansions of (x2-1)n...

Hi guys, I've been working on a question which is as follows: For which real values of c will the set $\{1+cx, 1+cx^2, x-x^2\}$ be a basis for $P_2$? I'm coming up with the answer as no values of c, but am I really wrong? I've only checked linear independence, because it would imply that it...

Homework Statement Prove that the analytic function e^z is not a polynomial (of finite degree) in the complex variable z. The Attempt at a Solution The gist of what I have so far is suppose it was a finite polynomial then by the fundamental theorem of algebra it must have at least...

Homework Statement Let T:P2→P2 be defined by T(a0+a1x+a2x2)=(2a0-a1+3a2)+(4a0-5a1)x + (a1+2a2)x2 1) Find the eigenvalues of T 2) Find the bases for the eigenspaces of T. I believe the 'a' values are constants. Homework Equations None. The Attempt at a Solution The problem I am...

Homework Statement I read that if f'(x) is zero once in [a b] then f(x) has maximum two real roots. Why maximum? Shouldn't it be exactly 2? Or it has something to do with the case of repeated roots? Homework Equations The Attempt at a Solution was thinking as in figure

Homework Statement If P(x) is a polynomial with real coefficients, then if z is a complex zero of P(x), then the complex conjugate \bar{z} is also a zero of P(x). Homework Equations Book provides a hint: assume that z is a zero for P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0} and...

Homework Statement Suppose f is differentiable in \mathbb{C} and |f(z)| \leq C|z|^m for some m \geq 1, C > 0 and all z \in \mathbb{C} , show that; f(z) = a_1z + a_2 z^2 + a_3 z^3 + ... a_m z^m Homework EquationsThe Attempt at a Solution I can't seem to show this. It does the proof...

Homework Statement If x3 + 5x2 + 4x = 3 = 0 and cos (5 - 3x) = √p, find the value of cot (x5 + 2x4 - 6x3 + 16x2 + 8x + 20) Homework Equations trigonometry polynomial The Attempt at a Solution stuck from the beginning...:-p

Homework Statement Show that x+a is a factor of x^{n}+a^{n}for all odd n. The Attempt at a Solution (1) Assume that x+a is a factor of x^{n}+a^{n}for all odd n. This implies that when x^{n}+a^{n} is divided by x+a the remainder is zero. I don't know - is this a sensible 1st step...

Homework Statement Let A = \begin{pmatrix}1 & 1 & 0 & 0\\-1 & -1 & 0 & 0\\-2 & -2 & 2 & 1\\ 1 & 1 & -1 & 0 \end{pmatrix} The characteristic polynomial is f(x)=x^2(x-1)^2. Show that f(x) is also the minimal polynomial of A. Method 1: Find v having degree 4. Method 2: Find a vector v of...

Use the dimension theorem to show that every polynomial p(x) in Pn can be written in the form p(x)=q(x+1)-q(x) for some polynomial q(x) in Pn+1. I need to see all the steps so that I understand how to do it. PLease and Thank you

"Weirdness" of polynomial long division algorithm Hello. So, i just started to learn about the polynomial long division. As an introductory example, the book presents the long division of natural numbers, claiming that its basically the same thing. The example: 8096:23 Solution...

Consider a polynomial of the following type: A_n w^n + A_{n-1} w^{n-1}k + A_{n-2} w^{n-2} k^2 + ... + A_1 k^n =0 What are the general conditions on {A_i} in order for the roots w(k) to be EITHER real OR functions with even imaginary parts, Im[w[k]]=Im[w[-k]]? I would be interested in...

Explain why the polynomial x^4 −7 is irreducible over Q, quoting any theorems you use.

Homework Statement Prove whether the below equations are linear or not. (iii) U = P^2 -> V = P^6; (Tp)(t) = (t^2)p(t^2) + p(1). (iv) U=P^2 -> V =P^6;(Tp)(t)=(t^2)p(t^2)+1. Homework Equations None. The Attempt at a Solution I really don't know. Thanks Tom

Homework Statement I need to generate coefficients of hermite polynomials up to order k. Recursion will be used. Homework Equations a[n+1][k] = 2a[n][k-1] - 2na[n-1][k] The Attempt at a Solution Its not pretty, but here is my code. #include <iostream> #include <iomanip>...

Homework Statement Let P_{n}(x) denote the Legendre polynomial of degree n, n = 0, 1, 2, ... . Using the formula for the generating function for the sequence of Legendre polynomials, show that: P_{n}(-x) = (-1)^{n}P_{n}(x) for any x \in [-1, 1], n = 0, 1, 2, ... . Homework Equations...

Maria designed a rectangular storage unit with dimensions 1m by 2m by 4m. By what should he increase each dimension to produce an actual storage that is 9 times the volume of his scale model? v= (1) (2) (4) v= 8 v has to be 9 times larger v= (x+1) (x+2) (x+4) How do i find the value of x?

The height,h, in meters, of a weather balloon above the ground after t seconds can be modeled by the function h(t)=-2t^3 + 3t^2 +149t + 410 for 0< t < 10. When is the balloon exactly 980m above the ground? 980 = -2t3 + 3t2 +149t + 410 0 = -2t3 + 3t2 +149t - 570

Hey, I'm working on a proof for a research-related assignment. I posted it under homework, but it's a little abstract and I was hoping someone on this forum might have some advice: Homework Statement Suppose T:V \rightarrow V has characteristic polynomial p_{T}(t) = (-1)^{n}t^n. (a) Are...

Homework Statement Suppose T:V \rightarrow V has characteristic polynomial p_{T}(t) = (-1)^{n}t^n. (a) Are all such operators nilpotent? Prove or give a counterexample. (b) Does the nature of the ground field \textbf{F} matter in answering this question? Homework Equations Nilpotent...

How do you prove that there does not exist numbers a,b\in\mathbb{Q} such that 0 = a + b\sqrt[3]{2} + \sqrt[3]{2}^2

There is a theorem in algebra, whose name I don't recall, that states that given a polynomial and its roots I can easily factor it so for instance : p(x)=x^2-36 , assuming that p(x) is a real function, p(0)=0 \Leftrightarrow x=6,-6 then p(x) can be written as : P(x)=(x-6)(x+6) I...

Hi, I've got an equation stating p=a(r-1). If p represents prime number and r is a positive integer, and a is a constant, what can we conclude for the constant a? Like a $\in${-1, 1, -p, p}? I suspect this has something to do with modular arithmetic...:confused: Thanks.

Homework Statement I was given the following problem, but I am having a hard time interpreting what some parts mean. We're given the equation sinθ+b(1+cos^2(θ)+cos(θ))=0 Assume that this equation defines θ as a function, θ(b), of b near (0,0). Computer the Taylor polynomial of...

I can understand most of Galois Theory and Number Theory dealing with factorization and extension fields, but I always run into problems that involve factorization mod p, which I can't seem to figure out how to do. I can't find any notes anywhere either, so I was wondering if someone could give...

I need to find the splitting field in \mathbb {C} of x^3+3x^2+3x-4 (over \mathbb{Q} ). Now, I plugged this into a CAS and found that it is (probably) not solvable by radicals. I know that if I can find a map from this irreducible polynomial to another irreducible polynomial of the same...

Homework Statement Hello there! I'm trying to find the roots of the following cubic polynomial x^3 - 10x + 18 = 0 The Attempt at a Solution I did the following: I rewrite 18 as 18 = - (x^3 - 10x) then I did back substitution and factored out x^3 - 10x - x^3 + 10x = 0 or x(x^2-10) -...

Homework Statement Approximate the function f(x)=sin(\pi x) on the interval [0,1] with the polynomial ax^{2}+bx+c with finding a, b and c. Homework Equations f(x)=a_{0}+\sum^{\infty}_{n=1}(a_{n}cos(nx)+b_{n}sin(nx)) a_0=\frac{1}{2\pi}\int^{\pi}_{-\pi}f(x)dx...

Let A, B, C be random number between (0,1). What is the probability that the polynomial Ax^2+Bx+C=0 has no real roots? I know that this question is a kind of c.r.v problem (uniform distribution). Also, it has something to do with exponential random variables. My problem is, exponential random...