Polynomial Definition and 1000 Threads

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...

[FONT=CMR10][FONT=CMR10]Explain why the polynomial x^4[FONT=CMR7][FONT=CMR7] [FONT=CMSY10][FONT=CMSY10]−[FONT=CMR10][FONT=CMR10]7 is irreducible over [FONT=MSBM10][FONT=MSBM10]Q[FONT=CMR10][FONT=CMR10], 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...

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 [FONT=MathJax_Math]A, [FONT=MathJax_Math]B, [FONT=MathJax_Math]C be random number between [FONT=MathJax_Main]([FONT=MathJax_Main]0[FONT=MathJax_Main],[FONT=MathJax_Main]1[FONT=MathJax_Main]). What is the probability that the polynomial Ax^2+Bx+C=0 has no real roots? I know that this...

Hi all, I would just like to get some clarity on units and zero-divisors in rings of polynomials. If I take a ring of Integers, Z4, (integers modulo 4) then I believe the units are 1 & 3. And the zero-divisor is 2. Units 1*1 = 1 3*3 = 9 = 1 Zero divisor 2*2 = 4 = 0 Now, If I...

Homework Statement Digital filter analysis - this is just one part of a multi-part question I can't move forward with. It's supposed to be an auxilliary question and isn't the "meat" of the problem. Find b, such that maximum of the magnitude of the frequency response function...

I apologize for the rather vague title. It's space-limited and I'm not sure how to concisely state what I want to know. Basically, I understand that the solutions to quadratic equations (and if I remember correctly cubic equations) often had surveying problems land surveying. However, quartic...

Homework Statement Factor x^16-x over the fields F4 and F8 Homework Equations factored over Z (or Q), x^16-x = (x*(x - 1)*(x^2 + x + 1)*(x^4 + x^3 + x^2 + x + 1)*(x^8 - x^7 + x^5 - x^4 + x^3 - x + 1) The Attempt at a Solution I know the that quadratic and higher terms I have left...

Homework Statement 2n - 6m + 5n^2 - 15mn Homework Equations No particular equation since this is factoring The Attempt at a Solution Keep in mind that I struggle when it comes to grouping as I'm not sure where I'm supposed to start but... 2n - 6m + 5n^2 - 15mn Group first 2...

Homework Statement Let f be entire. Then if lim_{z\rightarrow \infty}|f(z)|=\infty then f must be a non-constant polynomial.Homework Equations The Attempt at a Solution So we know f is entire. Thus I suppose it makes sense to go ahead and expand it as a power series centered at zero. Thus...

Homework Statement Let P(n) be the set of all polynomial of degree n with integer coefficients. Prove that P(n) is countable, then show that all polynomials with integer coefficients is a countable set. 2. The attempt at a solution For this problem the book gives me a hint that using...

f(t) = t4 - 23 + 3t2 - 2t + 1 in Q[t] Am i right in thinking I just show by the rational root theorem that the only possible roots are +-1 f(+-1) =/= 0 so there are no repeated factors? Seems too easy..

Homework Statement I uploaded a picture of the question because I didn't want to confuse people from the way it looks because it is pretty long. http://tinypic.com/r/2itpm39/5 Homework Equations -None- The Attempt at a Solution I went from the original equation and took an 'x'...

I have some math people who say you can, and some who say you can't beyond quintic because of the Abel-Ruffini theorem. Which is it? Can I generalize all polynomials? Or at least can I manually make a formula for each individual degree?

Homework Statement \frac{x^3+x^2-5x+3}{x^3-3x+2} Homework Equations The Attempt at a Solution well I'm drawing that long division house with x^3-3x+2 on the outside and x^3+x^2-5x+3 on the inside. I'm seeing that x^3 goes into x^3 one time, so i put a 1 on top of the...

Homework Statement First find all rational zeros of f, then use the depressed equation to find all roots of the equation f(x) = 0. f(x) = x^3 + 5x^2 - 8x + 2Homework Equations The Attempt at a Solution Possible rational zeros: 2, -2, 1, -1 Synthetic division: 1 | 1 5 -8 2 _____1 6 -2...

I may have posted this back in the Old Country, but: let the polynomial: \[P(x)=x^n+a_1X^{n-1}+ ... + a_{n-1}x+1 \] have non-negative coeficients and \(n\) real roots. Prove that \(P(2)\ge 3^n \) CB

Polynomial functions... find "a" Homework Statement When ax3 - 4x2 + 5x - 3 is divided by (x+2) and (x-1), the remainders are equal. Find a. Don't know where to start. A little help? Hints?

Homework Statement Find eight elements r \in \mathbb{Q}[x]/(x^4-16) such that r^2=r. Homework Equations N/A The Attempt at a Solution The elements 0+(x^4-16) and 1+(x^4-16) clearly satisfy the desired properties, but I still need six more elements. Can anyone help me figure out a...

Homework Statement Find (f^{-1})'(a) of: f(x)=\sqrt{x^{3}+x^{2}+x+22} ; a=5. Homework Equations (f^{-1})'(a)=\frac{1}{f'((f^{-1})(a))} The Attempt at a Solution Well, I know to find an inverse: I need to set the equation equal to y, solve for x, then swap x and y. But I don't...

Homework Statement I have a polynimal equation as this - 0.00000000000049125*T^4 + 0.00000000021358333333333333333333333333333*T^3 + 0.00000290233125*T^2 - 0.032444109375*T + 19.891472013020833333333333333333 Homework Equations The Attempt at a Solution I insert those...

Is it possible to compute the Galois Group of a polynomial manually (without a computer)? If so, can someone please explain how? I can't seem to find any information (aside from computer algorithms) on how to find a Galois Group or how to factor a polynomial modulo a prime. If it helps to...

Homework Statement The function y=x^{2}+4x-6 has two inverses. What are they and which domains lead to these inverses? Homework Equations The Attempt at a Solution y=x^{2}+4x-6 x=y^{2}+4y-6 y(y+4)=x+6 Not really sure where to go from here.

Factoring a 4th order polynomial Homework Statement Example: (jw)^{3}+6(jw)^{2}+5jw+30=0 can be re-written into 6(5-w^{2})+jw(5-w^{2}). The fact that there are two identical (5-w^{2}) is a desirable outcome. Imaginary number j=\sqrt{-1} becomes -1 when raised to the power of 2. Homework...

Ok, so obviously, given some polynomial P(x) of degree r, it has r roots in the complex numbers by the FTOA, and if these roots are u_1, u_2,... it can be written as \begin{array}{l} P(x) = (x - {u_1})(x - {u_2})(x - {u_3}) \cdots \\ P(x) = {x^r} - ({u_1} + {u_2} + {u_3} + \ldots ){x^{r - 1}}...

Homework Statement If k is a natural number, prove that 2^{n} > n^{k} for all n \geq k^{2} + 1.Homework Equations We need to use a proof by induction.The Attempt at a Solution Let's do the case when k = 4. We check the base case directly: 2^{17} = 131072 > 83521 = 17^4 Suppose 2^{n} > n^{4} for...

Homework Statement The graph of the quadratic polynomial , y=ax2+bx+c is as shown below in the figure : http://postimage.org/image/nvkxv74yd/ Then : (A) b2-4ac<0 (B) c<0 (C) a<0 (D) b<0Homework Equations y=ax2+bx+c If y=0 , then ax2+bx+c=0 Then , x = {-b+-sqrt(b^2-4ac)}/2a The Attempt at...