Polynomial Definition and 1000 Threads

Homework Statement Let A be an nxn matrix with real number entries, in which all entries are 1. Find the characteristic polynomial of A. Homework Equations characteristic polynomial: f(t)=det(A-tI), I is identity matrix The Attempt at a Solution I've tried to do this by various...

How do you convert binary into polynomial form? I understand hexadecial conversion so {4e} = 01001000 now how do I go about changing that into a polynomial?

1.Let F=K(u) where u is transcedental over the field K. If E is a field such that K contained in E contained in F, then Show that u is algebraic over E. Let a be any element of E that is not in K. Then a = f(u)/g(u) for some polynomials f(x), g(x) inK[x] 2.Let K contained in E...

Homework Statement Consider the polynomial p(x) = x^5+45x^3+324x-3x^4 - 135x^2 - 972 . Given that p has some integer roots on the real and imaginary axes, factorise p into linear and quadratic factors with real coefficients. Enter your answer a set of factors in the form { x-1, x+4...

What would be the best way to show that if F is an infinite field and f(x) is a polynomial in F[x] and f(a)=0 for an infinite number of elements a of F, that f(x) must be the zero polynomial? It kind of just makes logical sense to me, so I can't think of a way to actually show this. please help

I started to answer this question, and I have quite a bit an answer, but still not complete, let's say that we write a Lagrangian in QFT, which an unknown function of the scalar field \phi and its derivative \partial \phi. We can always Taylor-expand it and get: L(\phi,\partial\phi) = a + b \phi...

Homework Statement Show the polynomial f(x) = x^p + x^{p-1} + ... + x - 1 is irreducible over Z_p where p is a prime.Homework Equations The Attempt at a Solution I know f(x) has no roots in Zp, but other than that, I'm stuck. Thanks for the help.

I cannot for the life of me remember where I learned to do this nor can I remember why this works, but I know a method for finding the simplest polynomial for a series. I had something half typed up and then I found this site: http://www.johansens.us/sane/technotes/formula.htm which explains a...

Homework Statement At how many points in the xy-plane do the graphs of y=x^{12} and y=2^{x} intersect? Homework Equations none The Attempt at a Solution I have no idea what to do. I thought of trying to narrow it down to some intervals where the graphs may cross, but, since they're...

Homework Statement if p is prime, prove that p divides A, where A satisfis 1+\frac{1}{2}+...+\frac{1}{p-1}=\frac{A}{\left(p-1\right)!} Homework Equations The chinese remainder theorem? Eulers theorem? The Attempt at a Solution So as the question marks imply, I'm at a loss as to...

Homework Statement Prove that for n>=2, n not prime, the following polynomial is reducible f_n(x) = x^{n-1} + x^{n-2} + ... + x + 1 Homework Equations The Attempt at a Solution If n is even, the polynomial has -1 as a root, so it is reducible. But when n is odd (and not prime) I'm not sure...

Hey guys, I really need some help please! I would really appreciate it if anyone can help out, if we have F16 = F2/(x^4+x+1). can anyone explain to me how can I compute the minimal polynomials and the characteristic polynomils over F2 of elements of F16 and to point out the primitive ones...

Hello, I am trying to solve a degree three polynomial, but unfortunately I am stuck t3+2t2+1=1 This is as far as I can get: t(t2+2t)+1=0 Where do I go from here? Thanks!

Homework Statement y = -5x3 + 4/x5 + 1.7pi Homework Equations Differentiate each Function. Derivatives The Attempt at a Solution I have done question like these before but this is the first time using pi. I used all derivative rules none of them give me the right answer. The answer I am...

Homework Statement Factor f(x) = 3x4 + 2 into a product of irreducible polynomials in Z5[x] Homework Equations The Attempt at a Solution I don't get it. I tried dividing it using the division logarithm, but then I can only get it to a point where it's like, 3(x-1)(..) <-...

Let p(x)=a0+a1x+a2x2\in Real Numbers and let z\in Complex Field. Show that p(conjugate of z)=conjugate of p(z)

Homework Statement Find the Taylor polynomial for f(x) = 1/(1-x), n = 5, centered around 0. Give an estimate of its remainder. The Attempt at a Solution I found the polynomial to be 1 + x + x2 + x3 + x4 + x5, and then tried to take the Lagrange form of the remainder, say, for x in [-1/2, 1/2]...

If the polynomial x^4-16x^2-25x+10 is divided by another polynomial x^2-2x+k, the remainder comes out to be x+a find k and a. My approach: I tried dividing the first polynomial by the second and then equating the remainder to (x+a). It didn't work out. Are my calculations wrong or am i on the...

Homework Statement Hi, I have been given the polynomial function P(z)=4z^4 -12z^2 +3z +19 I need to Establish the main technique/s required to solve the polynomial. Then I need to find the solution of the polynomial? Any help would be greatly appreciated because I have no idea where to...

Homework Statement Let R = Z[x] be a polynomial ring where Z is the integers. Let I = (x) be a principal ideal of R generated by x. Prove I is a prime ideal of R but not a maximal ideal of R.Homework Equations The Attempt at a Solution I want to show that R/I is an integral domain which...

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 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...

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...