Polynomial Definition and 1000 Threads

Homework Statement Let V be the linear space of all real polynomials p(x) of degree < n. If p \in V, define q = T(p) to mean that q(t) = p(t + 1) for all real t. Prove that T has only the eigenvalue 1. What are the eigenfunctions belonging to this eigenvalue? Homework Equations Not sure...

Homework Statement Polynomials p(x) and q(x) are given by relationship q(x)=p(x)(x5-2x+2). a) If x-2 is a factor of p(x)-5 , find the remainder when q(x) is divided by x-2. b) If p(x) is of the form x2+ax+b and x-1 is a factor of p(x)-5, find the values of a and b. Homework...

Homework Statement [H_{n}(x)=-xH_{n-1}(x)-(n-1)H_{n-2}(x) ,for,n>=2 H_{0}(x)=1\ and H_{1}(x)=-x a)Show that H_{n}(x) is an even function when n is even and an odd function when n is odd. Also show by induction that: b)H_{2k}(x)=(-1)^k(2k-1)(2k-3)...1 hat is the value o H_{n}(0) when n is odd...

Homework Statement Let R be a commutative ring and let fa: R[x] -> R be evaluation at a \in R. If S: R[x] -> R is any ring homomorphism such that S(r) = r for all r\in R, show that S = fa for some a \in R. Homework Equations The Attempt at a Solution I don't get this at all...

Homework Statement Find all p, prime for which x+2 is a factor of f(x) = 5x4 - 2x3 + 3x2 + 4x - 1 in Zp Homework Equations The Attempt at a Solution So in Zp, x = p-2 I tried the first 4 primes and got the following results: Z3, x=1, f(x) = 9 = 0 Z5, x=3, f(x) = 390 - 1 =/ 0...

Homework Statement I must show that t H_n satisfies a diferential equation. By diferentiating H_n(x) = -xH_(n-1)(x) - (n - 1)H_(n-2)(x) (1) and using induction on n, show that, for n >= 1, H'_n(x) = -nH_(n-1)(x) (2) I have to use (2) to express H_(n-1) and H_(n-2) in terms of derivatives...

Homework Statement Let V be the vector space of all real-coefficient polynomials with degree strictly less than five. Find the eigenvalues and their geometric multiplicities for the following maps from V to V: a) G(f) = xD(f), where f is an element of V and D is the differentiation map...

Homework Statement Let J be the nxn matrix all of whose entries are equal to 1. Find the minimal polynomial and characteristic polynomial of J and the eigenvalues. Well, I figure the way I'm trying to do it is more involved then other methods but this is the easiest method for me to...

Homework Statement Find the Taylor polynomial of degree 3 of \frac{1}{2+x-2y} near (2,1). Homework Equations The Attempt at a Solution I have already solved this problem by evaluating the R^2 Taylor series; I'm mostly curious about another aspect of the problem. By substituting u = x-2y, it...

Homework Statement The polynomials are defined as follows H_n(x) for n = 0; 1; : : : as follows: first, set H_0(x) =1 and H_1(x) = -x; then, for n >= 2, H_n is defined by the recurrence H_n(x) = -xH_(n-1)(x) - (n - 1)H_(n-2)(x): (1) I have to use (1) to verify that H_2(x) = x^2 -1 and...

Homework Statement Show that if the polynomial p(z)=anzn+an-1zn-1+...+a0 is written in factored form as p(z)=an(z-z1)d1(z-z2)d2...(z-zr)dr, then (a) n=d1+d2+...+dr (b) an-1=-an(d1z1+d2z2+...+drzr) (c) a0=an(-1)rz1d1z2d2...zrdr Homework Equations Taylor form of a polynomial? p(z) = \sum...

1. I am having trouble finding a inverse of a polynomial. http://img218.imageshack.us/i/problemqk.png/ http://img218.imageshack.us/i/problemqk.png/

I was trying to calculate when two ellipses of the form ((x-x0)/a)^2+((y-y0)/b)^2=1 have intersections. Most of the time I get 4th order equations. Actually I only want to know a condition IF they just touch. Any ideas on this? After playing around I had various equations. For example the...

Homework Statement Slope of: y=.00002715x^2-.04934171x+44.18240907 Homework Equations d/dx The Attempt at a Solution d/dx[.00002715x^2-.04934171x+44.18240907] = .0000543x-.04934171 This is the derivative (slope) of the function though it's looking for a numerical value. It is...

Homework Statement Hello. I don't understand this: Let f(x) be a polynomial function of degree k+1, then f(x) has the form ak+1xk+1 + ... + a1x + a0 Now the polynomial function has degree h(x) = f(x) - ak+1(x-a) has degree <= k How? Homework Equations The Attempt at a...

So no one is quite sure that P != NP, although they tend to favor that relation. But I was curious, has anyone proved a minimum degree order to any algorithm that solves NP complete problems in polynomial time? In other words, they don't know if it can be done in polynomial time, but do they...

Let p be a prime number. Find all roots of x^(p-1) in Z_p I have this definition. Let f(x) be in F[x]. An element c in F is said to be a root of multiplicity m>=1 of f(x) if (x-c)^m|f(x), but (x-c)^(m+1) does not divide f(x). I'm not sure if I use this idea somehow or not.

Started my first year of Electronic Engineering a few months back and I'm already struggling with the mathematics. I have been to this forum a number of times over the last year and finally decided to join just 10 minutes ago!Homework Statement Find the roots of P(x) given that two roots are...

Hi guys i am a bit confused about this problem, a particle of mass, m, moves in potential a potential u(x)=k(x4 - 7 x2 -4x) I need to find the frequency of small oscillations about the equilibrium point. I have worked out that x=2 corresponds to the equilibrium point as - dU/dx = F =...

Homework Statement Assume F is a field of size p^r, with p prime, and assume f \in F[x] is an irreducible polynomial with degree n (with both r and n positive). Show that a splitting field for f over F is F[x]/(f). Homework Equations Not sure. The Attempt at a Solution I know from...

Hey guys, I am new to this forum and I have just found you after some long and unsuccessful research on the following question: Homework Statement The question is a combined matrix and polynomial question. First I am given the following matrix A: 2 1 1 5 4 -3 2 1 3...

Homework Statement Is it possible to find general solution for the following 3rd degree polynomial differential equation: dx/dt=-a1*x+a2*x^2+a3*x^3 Homework Equations The Attempt at a Solution I understand that its is possible to integrate 1/(-a1*x+a2*x^2+a3*x^3), however, end equation...

I have hard time to come with Maclaurin Polynomial of a given order [lets say 3] for a composite function like ln(cosx). Will appreciate help of how to approach such a problem.

Find a (complex) polynomial function f of x and y that is differentiable at the origin, with df/dz = 1 at the point z=0, and differentiable at all points on the unit circle x^2 + y^2=1, but is not differentiable at any other point in the complex plane. (Bruce Palka, Page 101) I think we use...

Question: Find the cubic polynomial, which take the following values: y(0) = 1, y(1)=0, y(2)=1 and y(3)=10 Hence obtain y(4).

Homework Statement Is 3cos22x + cosx2 - 1 a polynomial function? Homework Equations The Attempt at a Solution

Homework Statement the problem that i attached bellow is related to how you can obtain a transfer function from its squared magnituded. my question is not on the problem it self as its just a solved example from my book. what i find difficult to understand as you can see from my...

1. Suppose that f:R-->R is of class C2. Given b>0 and a positive number \epsilon, show that there is a polynomial p such that |p(x)-f(x)|<\epsilon, |p'(x)-f'(x)|<\epsilon, |p"(x)-f"(x)|<\epsilon for all x in [0,b]. The Attempt at a Solution First I choose a polynomial q...

Homework Statement B = |a 1 -5 | |-2 b -8 | |2 3 c | Find the characteristic polynomial of the following matrix. Homework Equations None The Attempt at a Solution So basically I have to find the det(B-λI). No matter what I do to the matrix I can't make the...

Hi: There are 3 invariants. The first one is a trace. The third one is a determinant. So they are invariants. The strange thing is the 2nd one. It is a hybrid term. Why is it also an invariant?

Spivak's "Calculus," chapter 3 - problem 7 - a Homework Statement Prove that for any polynomial function f, and any number a, there is a polynomial function g, anad a number b, such that f(x) = (x - a)*g(x) + b for all x. (The idea is simply to divide (x - a) into f(x) by long division, until...

Question from spivak's calculus - 3rd edition - chapter 3, question 6(a). Homework Statement If x1, ..., xn are distinct numbers, find a polynomial function fi of degree n - 1 which is 1 at xi and 0 at xj for j =/= i. Hint, the product of all (x - xi) for j =/= i, is 0 at xj if j =/= i...

Homework Statement Let F be a field, F[x] the ring of polynomials in one variable over F. For a \in F[x], let (a) be all the multiples of a in F[x] (note (a) is an ideal). If b \in F[x], let c(b) be the coset of b mod (a) (that is, the set of all b + qa, where q \in F[x]). F[x]/(a), then is the...

a is a root of order k of the polynomial p provided that k is a natural number such that p(x)=[(x-a)^k]r(x), r is a polynomial and r(a) not equal to 0. Prove a is a root of order k of the polynomial p iff p(a)=P'(a)=...=[p^(k-1)](a)=0 and [p^(k)](a) not equal to 0. Note: [p^(k-1)](a) :=...

Let V =Mn(k),n>1 and T:V→V defined by T(M)=Mt (transpose of M). i) Find the minimal polynomial of T. Is T diagonalisable when k = R,C,F2? ii) Suppose k = R. Find the characteristic polynomial chT . I know that T2=T(Mt))=M and that has got to help me find the minimal polynomial

Power Series Representation of a Function when "r" is a polynomial Homework Statement Find a power series representation for the function and determine the radius of convergence. f(x)=\stackrel{(1+x)}{(1-x)^{2}} Homework Equations a series converges when |x|<1...

Let f(x)=\frac{1}{x^2+x+1} Let f(x)=\sum_{n=0}^{\infty}c_nx^n be the Maclaurin series representation for f(x). Find the value of c_{36}-c_{37}+c_{38}. After working out the fraction, I arrived at the following, f(x)=\sum_{n=0}^{\infty}x^{3n}-\sum_{n=0}^{\infty}x^{3n+1} But I dun get how to...

What's the difference between polynomials (as elements of a ring of polynomials) and polynomial functions??

Homework Statement Find the Taylor Polynomial of degree 3 for the function f(x) = ln3x about a = 1/3Homework Equations NoneThe Attempt at a Solution I have found up to the fourth derivative of f(x) along with the values of the derivatives at x = 1/3. At this point i get Σ{(-1)kk!fk(1/3)}, but...

Hi there, I have over the last couple of month worked my way through Algebra Demystified and College Algebra Demystified (well almost), and I just completed the chapter on polynomial functions and I’m stuck at one of the questions in the chapter review where I’m given a complex root and the...

Homework Statement x^4 + 4x^3 - x^2 + 16x - 12 I know that with some higher order polynomials you can substitute say x^4 as a = x^2 thereby making it easier to break the thing apart and find its factors. I know I am looking for 4 roots, but my little substitution method doesn't really work...

f(x) = (x^2)/100 + (x^3)/1000 + (x^4)/10000 + ... till the power infinity

I have to find the complex zeros of the following polynomial: x^6+x^4+x^3+x^2+1 This evidently doesn't have any real solution so I tried to facto it with long division and by guessing I came up with: (x^2-x+1)(x^4+x^3+x^2+x+1) How can I factor the 4th degree polynomial now? Or how can I...

the textbook says that: "a non-constant analytic polynomial cannot be real-valued, for then both the partial derivative with respect to x and y would be real and the cauchyriemann equation cannot be satisfied." why??there's no explanation in the book and this sentence is written as an example...

Homework Statement Show that there is a polynomial function f of degree n such that: 1. f('x) = 0 for precisely n-1 numbers x 2. f'(x) = 0 for no x, if n is odd 3. f'(x) = 0 for exactly one x, if n is even 4. f'(x) = 0 for exactly k numbers, if n-k is odd Homework Equations The...

Homework Statement It seems that I'm a little bit lost about this exercise. It says: Find the taylors polynomial of third degree centered at the origin for z=\cos y \sin x. Estimate the error for: \Delta x=-0.15,\Delta y=0.2. So, I did the first part (the easy one), the taylors polynomial for...

Homework Statement If f(x)=(x+1)p(x) where f(x)=x^{2n}+2nx+2n-1, what is p(x)? Answer given: x^{2n-1}-x^{2n-2}+...-x^2+x+2n-1 Homework Equations The Attempt at a Solution I tried the long division and managed to some terms correct. Is there any other methods of finding this...

Homework Statement Find the inverse of the function. f(x)=2x^{3}+5Homework Equations Possibly the quadratic equation.The Attempt at a Solutionf(x)=2x^{3}+5 y=2x^{3}+5 -2x^{3}=-y+5 x^{3}= \frac{-y+5}{-2} x= \pm\sqrt[3]{\frac{-y+5}{-2}} y= \pm\sqrt[3]{\frac{-x+5}{-2}}So the solution is two...

Factor a 4th order polynomial (Solved) Homework Statement Find the roots of: x^5-1=0 Homework Equations Polynomial long division. The Attempt at a Solution x^5-1 = (x-1)(x^4+x^3+x^2+x+1) = 0 x^4+x^3+x^2+x+1 = (x^2+1)^2+x^3+x-x^2 (x^2+1)^2+x^3+x-x^2 = (x^2+1)^2+x(x^2+1)-x^2...

Here is a potentially neat problem. Let x(t),y(t) (for all t\in \mathbb{R}) be polynomials in t. Prove that for any x(t),y(t) there exists a non-zero polynomial f(x,y) in 2 variables such that f(x(t),y(t))=0 for all t. The strategy is to show that for n sufficiently large, the polynomials...