Polynomials Definition and 784 Threads

Homework Statement When trying to generate solutions to the harmonic oscillator, I'm trying to use hermite polynomials. I understand that there's a recursive relationship between the hermite polynomials but I'm confused in how each hermite polynomial is generated. Homework Equations...

Homework Statement Let W equal the set of all polynomials in F[x] with degree less than or equal to n-1 such that the sum of the coefficients of the terms is 0. Find a basis of W over F. The Attempt at a Solution I don't know where to begin to find the basis. Do I use the fact that the sum...

"Let f is a polynomial from k[x,y], where k is a field. Suppose that x appears in f with positive degree. We view f as an element of k(y)[x], that is polynomial in x whose coefficients are rational functions of y." I think I am missing something...why do we need rational functions here? can't...

I don't understand how to factor a polynomial over Z3 [x], Z7 [x], and Z11 [x] I need to factor the polynomail x3 - 23x2 - 97x + 291 PLEASE HELP!

Homework Statement lim x-> -1 x3-6x2+11x - 6 x3-4x2+19x + 14 Homework Equations The Attempt at a Solution

Lagrange Polynomals are defined by: lj(t)= (t-a0) ...(t-aj-1)(t-aj+1)...(t-an) / (aj-a0)...(aj-aj-1)(aj-aj+1)...(aj-an) A) compute the lagrange polynomials associated with a0=1, a1=2, a2=3. Evaluate lj(ai). B) prove that (l0, l1, ... ln) form a basis for R[t] less than or equal to n...

Hello! Is there any way of calculating the integral of H_n(x) * H_m(x) * exp(-c^2 x^2) with x going from -infinity to +infinity and c differs from unity. I'm aware that c=1 is trivial case of orthogonality but I'm really having a problem with the general case. (I should say that this isn't a...

Hi everyone, I have to demonsrate that for every real polynomial, P Q and R, I have : P²=X(Q²+R²) ==imply==> P=Q=R=0 Using degrees, we can easily demonsrate the above. However, I'm looking for another way, without using that.

It is basic knowledge that if a polynomial P(x) of nth degree has a root or zero at P(a), then (x-a) is a factor of the polynomial. However, can this be proved? or is this more of a definition of roots of polynomials?

Hello, I'm having trouble understanding the computation involved in the resultants from algebraic geometry. I've checked out the textbook Using Algebraic Geometry by Cox et al, so I'll be using terminologies from that text. For systems of 2 polynomials I can compute the resultant the "long...

Would the GCD of x^2+x+c and (x-a)^2+(x-a)+c always be 1?

Whats a good book to learn about these topics?

I'm not reading a text in English, so I should clarify some notation first: P[X], is the set of all polynomials. P[[X]], is the set of formal power series, which may infinitely many nonzero coefficients. I'm asked to prove that X is the only prime(?correct term?) element in P[[X]]. By saying...

I'm looking right now at what purports to be the normalisation condition for the associated Laguerre polynomials: \int_0^\infty e^{-x}x^k L_n^k(x)L_m^k(x)dx=\frac{(n+k)!}{n!}\delta_{mn} However, in the context of Schroedinger's equation in spherical coordinates, I find that my...

Homework Statement Find all the monic irreducible polynomials of degree \leq 3 in \mathbb{F}_2[x], and the same in \mathbb{F}_3[x].Homework Equations n/aThe Attempt at a Solution n/a

Hello again my favourite helpers (or should I say, saviours!) :-p Long time no speak, but I am in more need of an explanation than an answer. In this mathematics textbook I have, it gives an explanation under the heading Polynomials in general. It goes as follows: "If f(x)=...

Homework Statement 3/x^2+2x - 2/x^2+x-2 + 4/x^2(x-1) Find the lowest common denominator and solve. Homework Equations The Attempt at a Solution I factored x(x+2) - (x-1)(x+2) + x^2(x-1) It looks like (x+2), (x-1) are common but what to do with the x & x^2 left over? Thank...

Homework Statement Let V=Pn(C) (polynomials of nth degree with complex coefficients), where n >=1. Find a basis for W=V s.t. every f(x) belonging to the basis satisfies f(0)f(1)= -1. (Demonstrate that the set you find is linearly independent and spans W.) Homework Equations For...

Hi! Brief question: I wonder which conditions should a polynomial function of odd degree fulfill in order to be symmetric to some point in the plane. Are there such conditions?

1) As far as i think i understand, stability of a polynomial means; the polynomial correspond to a (its inverse laplace transform) differential equation, and the differential equations solution is dependant on the coefficiants of this polynomial? if the polynomial is unstable the solution of...

how would i write the depressed polynomial for the obvious root of f(x) = x5 - .5x4 - 5.5x3 - 3.5x2 - 6.5x - 3

Hi, In Wikipedia it's stated that "... Legendre polynomials are useful in expanding functions like \frac{1}{\sqrt{1 + \eta^{2} - 2\eta x}} = \sum_{k=0}^{\infty} \eta^{k} P_{k}(x) ..." Unfortunately, I am failing to see how this can be true. Is there a way of showing this...

Okay, I'm doing my yr 11 Mathematics assignemnt and I need help. How do you find a Logistic or Polynomial function/equation using data? (in steps please!) I could just give some example data as... an example, but I think I'll just throw 1 small section of my assignment, use the data or...

Can anyone PROOVE how to find out the normalisation of hermite polynomial?

Homework Statement How to use eigenfunction expansion in Legendre polynomials to find the bounded solution of (1-x^2)f'' - 2xf' + f = 6 - x - 15x^2 on -1<= x <= 1 Homework Equations eigenfunction expansion The Attempt at a Solution [r(x)y']' + [ q(x) + λ p(x) ]...

Can anyone tell me how to find the exact number of primitive polynomials of degree n over a finite field F_q? I believe the answer is φ(q^n-1)/n, but I cannot find a proof of this. Thanx in advance.

Since the Hermite Polynomials are orthogonal, could one state that they span all polynomials f where f : R \rightarrow R? This would be EXTREMELY useful for the harmonic oscillator potential in quantum mechanics...

I'm trying to figure out how to prove that every polynomial in \mathbb{Z}_9 can be written as the product of two polynomials of positive degree (except for the constant polynomials [3] and [6]). This basically is just showing that the only possible irreducible polynomials in \mathbb{Z}_9 are the...

Are there any ploynomials p(x) such that p(x)^2 -1 = p(x^2+1) for all x? To cut it short: With CAS software I have verified that there are no solutions except p(x) = 1.618... and p(x) = -0.618... (constants) up to order 53 or so, but I have to prove this (or find the other solutions)...

Homework Statement Recall that two polynomials f(x) and g(x) from F[x] are said to be relatively prime if there is no polynomial of positive degree in F[x] that divides both f(x) and g(x). Show that if f(x) and g(x) are relatively prime in F[x], they are relatively prime in K[x], where K is an...

Hey Everyone, I'm reading a paper by Claude LeBrun about exotic smoothness on manifolds and he is talking about a connection between polynomials and groups that I am not familiar with (or at least I think that's what he's talking about). He's creating a line bundle (which happens to be...

I defined x as a syms, then work with polynomials involving x. But then I can't find an evaluation command for these polynomials at some x value. Anyone knows how I can get it to work? Thank you in advance. There's another system where you can define polynomials just like matrices, e.g. x^2+1...

I need to show that: \sum_{n=0}^{\infty}\frac{H_n(x)}{n!}y^n=e^{-y^2+2xy} where H_n(x) is hermite polynomial. Now I tried the next expansion: e^{-y^2}e^{2xy}=\sum_{n=0}^{\infty}\frac{(-y)^{2n}}{n!}\cdot \sum_{k=0}^{\infty}\frac{(2xy)^k}{k!} after some simple algebraic rearrangemnets i...

I thought this was rather odd, and wanted to just show it to see what you all thought of it. Well, also, if anyone knows what I should read to exactly understand what I did. 1: Let's define each answer of a polynomial such as (ax + b)(cx + d) as x1 = (-b/a), x2 = (-d/c). 2: The...

Homework Statement Prove that f(x)=x^3-7x+11 is irreducible over Q Homework Equations The Attempt at a Solution I've tried using the eisenstein criterion for the polynomial. It doesn't work as it is written so I created a new polynomial...

Homework Statement Suppose there are two polynomials over a field, f and g, and that gcd(f,g)=1. Consider the rational functions a(x)/f(x) and b(x)/g(x), where deg(a)<deg(f) and deg(b)<deg(g). Show that if a(x)/f(x)=b(x)/g(x) is only true if a(x)=b(x)=0. Homework Equations None The Attempt at...

Homework Statement For polynomials over a field F, prove that every non constant polynomial can be expressed as a product of irreducible polynomial. Homework Equations No relevant equations. The Attempt at a Solution Well a hint the teacher gave me was that the degree of the...

are hermite interpolationg polynomials necessarily cubic even when used to interpolate between two points? this page would have me believe so in calling it a "clamped cubic" : http://math.fullerton.edu/mathews/n2003/HermitePolyMod.html

Hello, I'm having trouble with this question and was wondering if someone could give me hints or suggestions on how to solve it. Any help would be greatly appreciated thankyou! :) Find the Taylor polynomial of degree 3 of f (x) = e^x about x = 0 and hence find an approximate value for...

I need to expand the next function in lengendre polynomial series: f(x)=1 x in (0,1] f(x)=0 x=0 f(x)=-1 x in [-1,0). Now here's what I did: the legendre series is given by the next generating function: g(x,t)=(1-2tx+t^2)^(-1/2)=\sum_{0}^{\infty}P_n(x)t^n where P_n are legendre...

Due to too much wrong information being posted on my behalf, I am resubmitting a cleaned up version of my last post. I have 2 hours to get this problem done :(. Essentially, I don't know at all how to find the Taylor Polynomial for g(x) = \frac{1}{x^5 ( e^{b/x} -1)} [/URL]

Homework Statement f(\lambda) = \frac{8\pi hc\lambda ^{-5}}{e^{hc/\lambda kT}-1} Is Planck's Law where h\ =\ Planck's\ constant\ =\ 6.62606876(52)\ \times\ 10^{-34}\ J\ s; c\ =\ speed\ of\ light\ =\ 2.99792458\ \times\ 10^{8}\ m\ s^{-1}; and\ Boltzmann's\ constant\ =\ k\ =\ 1.3806503(24)\...

[SOLVED] Algebta (factoring polynomials) Homework Statement x^2 * [ (1/2) * (1-x^2)^(-1/2) * (-2x)] + (1 - x^2)^(1/2) * (2x) factor this down. Homework Equations n/a The Attempt at a Solution -x^3*(1- x^2)^(-1/2) + 2x(1 - x^2)^(1/2) I understand how you get to this part...

Homework Statement Okay, basically why does (a_{}n + b_{}n) ignore the Fundamental Theory of Polynomials? Homework Equations ... I could post them here, but basically when n is odd (a_{n} + b_{n}) = a series that looks like this: (a+b) (a_{n-1} b_{0} - a_{n-2}b +a_{n-3}b_{2} + ... +...

(1): Find all irreducible polynomials of the form x^2 + ax +b , where a,b belong to the field \mathbb{F}_3 with 3 elements. Show explicitly that \mathbb{F}_3(x)/(x^2 + x + 2) is a field by computing its multiplicative monoid. Identify [\mathbb{F}_3(x)/(x^2 + x + 2)]* as an abstract group...

Homework Statement Show that the set of polynomials with integer coefficients with at least 1 real root is decidable. The Attempt at a Solution The question did not ask for specific language, just an intuitive finite algorithm will do.

Hello! First post here. My question is, is it possible to use a matrix to solve a system where you have the same variable, but a different degree. i.e. 2x^2 + 2x + y = 2 -3x^2 - 6x + 2y = 4 4x^2 + 6x - 3y = 6 Now, I know this is possible other ways, but seeing as each of those would...

Homework Statement Show that x^{p^n}-x is the product of all the monic irreducible polynomials in \mathbb{Z}_p[x] of a degree d dividing n.Homework Equations The Attempt at a Solution So, I want to prove that the zeros of all such monic polynomials are also zeros of x^{p^n}-x and vice versa. I...

Help, I am infinitly confused :) When solving the limit for this type and factoring the largest power of a variable in the polynomial in order to make its coefficient become a limit multiplied by another limit of Infinity I get lost. I just do not understand how (Infinity)(5 + Infinity +...

Hey! Could someone please help me find out how to express the product of two Legendre polynomials in terms of a sum of Legendre polynomials. I believe I have to use the recursion formula (l+1)P_{l+1}(x)-(2l+1) x P_l(x) + l P_{l-1}(x)=0 but I am not sure how to do this. What is basically...